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# Money Management ## Making a Personal Budget ### Learning Objectives After completing this section, you should be able to: 1. Create a personal budget with the categories of expenses and income. 2. Apply general guidelines for a budget. “That doesn’t fit in the budget.” “We didn’t budget for that.” “We need to figure out our budget and stick to it.” A budget is an outline of how money and resources should be spent. Companies have them, individuals have them, your college has one. But do you have one? Creating a realistic budget is an important step in careful stewardship of your financial health. Designing your budget will help understand the financial priorities you have, and the constraints on your life choices. You want to have enough income to pay not only for the necessities, but also for things that represent your wants, like trips or dinner out. You also may want to save money for large purchases or retirement. You do not want to just get by, and you do not want the problems associated with overdue balances, rising debt, and possibly losing something you have worked hard to obtain. While creating a budget may seem intimidating at first, coming up with your basic budget outline is the hardest part. Over time, you will adjust not only the numbers, but the categories. ### Creating a Budget You should view creating a budget as a financial tool that will help you achieve your long-term goals. A budget is an estimation of income and expenses over some period of time. You will be able to track your progress, which will help you to prepare for the future by making smart investment decisions. There are several budget-creating tools available, such as the apps Good budget and Mint, and Google Sheets. Getting started, though, begins well before you find an app. The following are steps that can be used to create your monthly budget. 1. Track your income and expenses Review your income and expenses for the past 6 months to a year. This will give you an idea of your current habits. 2. Set your income baseline Determine all the sources of income you will have. This income may from paychecks, investments, or freelance work. It even includes child support and gifts. Be sure to use income after taxes. This allows you to determine your maximum expenditures per month. 1. Determine your expenses Review your bills from the past 6 months. You should include mortgage payments or rent, insurance, car payments, utilities, groceries, transportation expenses, personal care, entertainment, and savings. Using your credit card statements and bank statements will help you determine these amounts. Be aware that some of the expenses will not change over time. These are referred to as fixed expenses, like rent, car payments, insurance, internet service, and the like. Other expenses may vary widely from month to month and are appropriately called variable expenses, and include such expenses as gasoline, groceries. 1. Categorize your expenses These categories may be housing, transportation, or food, for broad categories, or may get more specific, where you categorize car payments, car insurance, and gasoline separately. The categories are your choices. Be sure to account for the cost of maintaining a vehicle or home. The more specific you are, the better you’ll understand your spending needs and habits. 2. Total your monthly income and monthly expenses and compare These values should be compared. If your expenses are higher than your income, then adjustments have to be made. Decisions of what to do with any extra income is part of the planning process also. 3. Make plans for unplanned expenses Ask anyone, an unexpected car repair can ruin a carefully crafted budget. Have a plan for how you can be ready for these random expenses. This often means creating a cushion in your budget. 4. Use your budget to make decisions and adjust for any changes Your budget is a changeable document. Add to it when you wish, refer to it when special purchases are to be made. Keeping your budget up to date helps accommodate changes in income and expenses. In this section, we will focus on income and expenses. One of the easiest ways to manage a budget is to create a table, with one column containing income sources, another with income values, a third with expense categories, and a last containing expenses. An example is shown in . Using the budget process, we can make decisions on adding expenses to the budget. To do so, check the cushion of the budget to see if there is room in the budget for the new expense. ### The 50-30-20 Budget Philosophy It isn’t clear, obvious, or easy to decide how much of your income to allocate to various categories of expenses. Many people pay their bills and then consider all the leftover money to be spending money. However, when developing your own budget, you may want to follow the 50-30-20 budget philosophy, which provides a basic guideline for how your income could be allocated. Fifty percent of your budget is allotted to your needs, 30% of your budget is allotted to pay for your wants, and 20% of your budget is allotted for savings and debt service (paying off your debts). Knowing what expenses are necessary and what expenses are wants is important, since wants and needs are often confused. The following are necessary expenses that represent basic living requirements and debt services. This list isn’t complete: mortgage/rent, utilities, car, car insurance, health care, groceries, gasoline, child care (for working parents), and minimum debt payments. The 50-30-20 budget philosophy suggests that 50%, or half, your income go to these necessities. Wants, though, are things you could live without but still wish to have, such as Amazon Prime, restaurant dinners, coffee from Starbucks, vacation trips, and hobby costs. Even a gym membership or that new laptop are wants. Creating the room to afford these wants is important to our mental health. Not budgeting for things we want will negatively impact our quality of life. The remaining 20% should be set aside, either in retirement funds, stocks, other investments, an emergency fund (recommendations are that an emergency fund have 3 months of income), and perhaps extra spent to pay down debt. This 20% is very useful for addressing those unexpected costs, such as repairs or replacement of items that no longer work. Without budgeting this cushion, any expense that is a surprise can cause us to miss necessary payments. The list of necessary expenses was not complete. There are other expenses that could be included. ### Necessary Expenses and Expenses that are Wants For some people, an expense will be necessary while the same expense for someone else will be a want. A good example of this is internet service. Many people consider internet service as a need, especially those who work from home or who are not able to leave their homes. One could also call internet service a need if they have children in school. For others, internet service is a want. If a person’s job doesn’t require them to be online, if they are not in school, if they do not have kids, then internet service can be dropped. There are public options for internet service. One could even use their phone as a hot spot. Cars often fall into the category of need, but could also fall into the want category, depending on where and how you live. Bikes, public transportation, and walking are all options that could replace a car. This would then remove the cost of gasoline and car insurance. Another consideration when deciding if an item on your budget is a need or a want is about your choices and priorities. A car is a need for many. But the need for a car is not the same as the need for a specific car. If you choose to buy a car with payments that exceed your budgeted amount for the car, then that car is a want. The amount you exceed the budget now belongs in the want category. The same can be said for housing. If you want an apartment that costs $1,250 per month, but your budget only allows for an apartment that costs $900, then $350 of the rent is a want. The point of that is to carefully consider if an expense is a need as opposed to a want. When your expenses exceed your income, you may want to change how you budget your income to line up with these guidelines. This may mean cutting back, finding less-expensive living arrangements, finding a less-expensive (and more fuel-efficient) car, or sacrificing some specialty groceries. Using these guidelines keeps your financial life manageable. Better still, they can guide you as you begin your life after graduation. ### Check Your Understanding ### Key Terms 1. Budget 2. Necessary expenses 3. Fixed expenses 4. Variable expenses 5. 50-30-20 budget philosophy ### Key Concepts 1. A budget is a set of guidelines for how to allocate your income. 2. Budgeting helps to plan for many of life’s expenses 3. Budgets are used to compare income to expenses. When expenses exceed income, changes have to be made. 4. Budgets can help evaluate the affordability of life changes. 5. One guideline for setting a budget is the 50-30-20 budget philosophy. The guidelines suggest that 50% of income is allocated to necessary expenses, 30% to expenses that wants, and 20% to savings and other debt reduction. ### Videos 1. Creating a Budget 2. 50-30–20 Budget Philosophy

# Money Management ## Methods of Savings ### Learning Objectives After completing this section, you should be able to: 1. Distinguish various basic forms of savings plans. 2. Compute return on investment for basic forms of savings plans. 3. Compute payment to reach a financial goal. The stock market crash of 1929 led to the Great Depression, a decade-long global downturn in productivity and employment. A state of shock swept through the United States; the damage to people’s lives was immeasurable. Americans no longer trusted established financial institutions. By October 1931, the banking industry’s biggest challenge was restoring confidence to the American public. In the next 10 years, the federal government would impose strict regulations and guidelines on the financial industry. The Emergency Banking Act of 1933 created the Federal Deposit Insurance Corporation (FDIC), which insures bank deposits. The new federal guidelines helped ease suspicions among the general public about the banking industry. Gradually, things returned to normal, and today we have more investment instruments, many insured through the FDIC, than ever before. In this section, we will first look at the different types of savings accounts and proceed to discuss the various types of investments. There is some overlap, but we will try to differentiate among these financial instruments. Saving money should be a goal of every adult, but it can also be a difficult goal to attain. ### Distinguish Various Basic Forms of Savings Plans There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are three main savings account vehicles. ### Savings Account A savings account is probably the most well-known type of investment, and for many people it is their first experience with a bank. A savings account is a deposit account, held at a bank or other financial institution, which bears some interest on the deposited money. Savings accounts are intended as a place to save money for emergencies or to achieve short-term goals. They typically pay a low interest rate, but there is virtually no risk involved, and they are insured by the FDIC for up to $250,000. Savings accounts have some strengths. They are highly flexible. Generally, there are no limitations on the number of withdrawals allowed and no limit on how much you can deposit. It is not unusual, however, that a savings account will have a minimum balance in order for the bank to pay maintenance costs. If your account should dip below the minimum, there are usually fees attached. Having your savings account at the same bank as your checking account does offer a real advantage. For example, if your checking account is approaching its lower limit, you can transfer funds from your savings account and avoid any bank fees. Similarly, if you have an excess of funds in your checking account, you can transfer funds to your savings account and earn some interest. Checking accounts rarely pay interest. There are some weaknesses to savings accounts. Primarily, it is because savings accounts earn very low interest rates. This means they are not the best way to grow your money. Experts, though, recommend keeping a savings account balance to cover 3 to 6 months of living expenses in case you should lose your job, have a sudden medical expense, or other emergency. Around tax time, you will receive a 1099-INT form stating the amount of interest earned on your savings, which is the amount that must be reported when you file your tax return. A 1099 form is a tax form that reports earnings that do not come from your employer, including interest earned on savings accounts. These 1099 forms have the suffix INT to indicate that the income is interest income. Savings accounts earn interest, and those earnings can be found using the interest formulas from previous sections. The final value of these accounts is sometimes called the future value of the account. ### Certificates of Deposit, or CDs We discussed certificates of deposit (CDs) in earlier sections. CDs differ from savings accounts in a few ways. First, the investment lasts for a fixed period of time, agreed to when the money is invested in the CD. These time periods often range from 6 months to 5 years. Money from the CD cannot be withdrawn (without penalty) until the investment period is up. Also, money cannot be added to an existing CD. Certificates of deposit have features similar to savings accounts. They are insured by the FDIC. They are entirely safe. They do, though, offer a better interest rate. The trade-off is that once the money is invested in a CD, that money is unavailable until the investment period ends. ### Money Market Account A money market account is similar to a savings account, except the number of transactions (withdrawals and transfers) is generally limited to six each month. Money market accounts typically have a minimum balance that must be maintained. If the balance in the account drops below the minimum, there is likely to be a penalty. Money market accounts offer the flexibility of checks and ATM cards. Finally, the interest rate on a money market account is typically higher than the interest rate on a savings account. ### Return on Investment If we want to compare the profitability of different investments, like savings accounts versus other investment tools, we need a measure that evens the playing field. Such a measure is return on investment. ### Annuities as Savings In Compound Interest, we talked about the future value of a single deposit. In reality, people often open accounts that allow them to add deposits, or payments, to the account at regular intervals. This agrees with the 50-30-20 budget philosophy, where some income is saved every month. When a deposit is made at the end of each compounding period, such a savings account is called an ordinary annuity. The formula for the future value of an ordinary annuity is , where is the future value of the annuity, is the payment, is the annual interest rate (in decimal form), is the number of compounding periods per year, and is the number of years. ### Compute Payment to Reach a Financial Goal The formula used to get the future value of an ordinary annuity is useful, finding out what the final amount in the account will be. However, that isn’t how planning works. To plan, we need to know how much to put into the ordinary annuity each compounding period in order to reach a goal. Fortunately, that formula exists. With this formula, it is possible to plan the amount to be saved. ### Check Your Understanding ### Key Terms 1. Savings account 2. 1099 form 3. Certificate of deposit 4. Money market account 5. Return on investment 6. Ordinary annuity ### Key Concepts 1. There are three main types of savings accounts, saving accounts, certificates of deposit (CD), and money market accounts. 2. Savings account are very risk free, and so yield low interest rates. 3. The differences in the three types of savings accounts relate to their convenience. 4. Savings account typically have a lower interest rate that money market accounts, which typically have lower interest rates than CDs. 5. Ordinary annuities more accurately reflect how we save, in that money is deposited repeatedly over time. 6. Spreadsheet software, such as Google Sheets, have built in functions that can be used to quickly calculate both the future value of an ordinary annuity account, but also the payment necessary to reach a goal using an ordinary annuity. ### Videos 1. Return on Investment, ROI 2. Future Value Using Google Sheets ### Formulas

# Money Management ## Investments ### Learning Objectives After completing this section, you should be able to: 1. Distinguish between basic forms of investments including stocks, bonds, and mutual funds. 2. Understand what bonds are and how bond investments work. 3. Understand how stocks are purchased and gain or lose value. 4. Read and derive information from a stock table. 5. Define a mutual fund and how to invest. 6. Compute return on investment for basic forms of investments. 7. Compute future value of investments. 8. Compute payment to reach a financial goal. 9. Identify and distinguish between retirement savings accounts. You can save your money in a safe or a vault (or worse, under the mattress!), but that money does not grow. It would be hard to save enough for retirement that way. What can be done to increase the value of the money you already have? The answer is to invest it. Use the money that you have to earn more money back. For instance, as we saw in Methods of Savings, you can save it in a bank. Or, to reach loftier goals, invest in something more likely to grow, such as stocks. A great example of this is Apple stock. Anyone who bought stock in Apple Inc. (formerly Apple Computer, Inc.) in 1997 and held onto the shares earned a lot of money. To be more specific, $100 worth of Apple shares bought in 1980, when it was first sold to the public, was valued at $67,564 in 2019, or 676 times more! Perhaps you have heard a story like that, of an investment opportunity taken that paid off, or the story of an investment opportunity missed. But such stories are the exceptions. In this section, we’ll investigate bonds, stocks, and mutual funds and their comparative strengths and weaknesses. We close the section with a discussion of retirement savings accounts. ### Distinguish Between Basic Forms of Investments Bonds, stocks, and mutual funds tend to offer higher returns, but to varying degrees, come with higher risks. Stocks and mutual funds also vary in how much they earn. Their predicted rates of return on investment are not guaranteed, but educated guesses based on market trends and historical performance. We will use the methods and formulas we learned earlier to evaluate these forms of investment. ### Bonds Bonds are issued from big companies and from governments. Selling bonds is an alternative to an institution taking a loan from a bank. The funds from the selling of bonds are often used for large projects, like funding the building of a new highway or hospital. Bonds are considered a conservative investment. They are bought for what is known as the issue price. The interest is fixed (does not change) at the time of purchase and is based on the issue price of the bond. The interest rate is often referred to as the coupon rate; the interest paid is often called the coupon yield. The interest paid is often higher than savings accounts and the risk is exceptionally low. The bond is for a fixed length of time. The end of this time is the maturity date of the bond. There are several types of bonds: 1. Treasury bonds are issued by the federal government. 2. Municipal bonds are issued by state and local governments. 3. Corporate bonds are issued by major corporations. There are other types of bonds available, but they are beyond the scope of this section. ### Stocks Stocks are part ownership in a company. They come in units called shares. The performance and earnings of stocks is not guaranteed, which makes them riskier than any other investment discussed earlier. However, they can offer higher return on investment than the other investments. Their value grows in two ways. They offer dividends, which is a portion of the profit made by the company. And the price per share can increase based on how others see that value of the company changing. If the value of the company drops, or the company folds, the money invested in the stock also drops. Most stock transactions are executed through a broker. Brokers’ commissions can be a percentage of value of the trades made or a flat fee. There are full-service brokers who charge higher commission rates, but they also offer financial advice and perform the research that you may not have the time or the expertise to do on your own. A discount broker only executes the stock transactions, buying or selling, so they charge lower rates than full-service brokers. There are also brokers that offer commission-free trading. An important thing to remember is that stocks might provide a very large return on investment, but the trade-off is the risk associated with owning stocks. ### Reading Stock Tables Information about particular stocks is contained in stock tables. This information includes how much the stock is selling for, and its high and low values form the past year (52 weeks). In a newspaper, the stock table may look like this: The symbols and abbreviations are defined here: The formulas for yield and price to earnings is a good way to measure how much the stock returns per share. Their values are calculated in the stock table, but deserve attention here. It should be noted that the price of a stock increases and decreases every moment, and so these value change as the share price changes. The stock table information is now, and has been, available online, from websites such as cnn.com/markets, markets.businessinsider.com/stocks, and marketwatch.com. The same information is available from these sites as from the newspaper listings, but are often accessed one stock at a time. shows the stock table for Lowe’s on September 7, 2022. Other key data is further down on the website, and is shown in , below. Notice that the 52-week high and low are now shown as the 52-week range. However, you get additional information, including the stock performance over the past 5 days, past month, past 3 months, the year to date (YTD), and over the past year. You can also read the number of shares outstanding, the expected date for the dividend (EX-DIVIDEND DATE), and importantly for the P/E ratio, the earning per share (EPS). As mentioned, stocks earn money in two ways, through dividends and increase in share price. ### Mutual Funds A mutual fund is a collection of investments that are all bundled together. When you buy shares of a mutual fund, your money is pooled with the assets of other investors. This pooled money is invested in stocks, bonds, money market instruments, and other assets. Mutual funds are typically operated by professional money managers who allocate the fund's assets and attempt to produce capital gains or income for the fund's investors. A key benefit of mutual funds is that they allow small or individual investors to invest in professionally managed portfolios of equities, bonds, and other securities. This means each shareholder participates proportionally in the gains or losses of the fund. The performance of a mutual fund is usually stated as how much the mutual fund’s total value has increased or decreased. Since there are many different investments inside the mutual fund, the risk is reduced significantly, compared to direct ownership of stocks. Even so, mutual funds historically perform well and can earn more than 10% annually. The investments that make up a mutual fund are structured and maintained to match stated investment objectives, which are specified in its prospectus. A prospectus is a pamphlet or brochure that provides information about the mutual fund. Before buying shares of a mutual fund, consult its prospectus, consider its goals and strategies to see if they match your goals and values and also research any associated fees. ### Return on Investment As in Methods of Savings, the formula for return on investment is . As indicated before, this formula does not take into account how long the investment took to reach its current value. It depends only on the initial value, , and the value at the end of the investment, . As mentioned, the ROI does not address the length of time of the investment. A good way to do that is to equate the ROI to an account bearing interest that is compounded annually. The annual return is the average annual rate, or the annual percentage yield (APY) that would result in the same amount were the interest paid once a year. We apply this to the previous example. In and Your Turn, the annual return was lower than the interest rate of the investment. This is because the interest from a bond is simple interest, but annual yield equates to compounded annually. You should see that the annual return is equal to the annual compounded interest that was assumed for the stocks. ### Compute Payment to Reach a Financial Goal As in Methods of Savings, determining the payment necessary to reach a financial goal uses the payment formula for an ordinary annuity, . If dealing with mutual funds or stocks, an assumed annual interest rate, compounded, will be used. This value is often determined through research and informed speculation. ### Retirement Savings Plans We close this section by investigating the three main forms of retirement savings accounts: traditional individual retirement accounts (IRAs), Roth IRAs, and 401(k) accounts. Each has distinct characteristics that are suited to different investors’ needs. ### Individual Retirement Accounts A traditional IRA lets you contribute up to an amount set by the government, which may change from year to year. For example, the maximum contribution for 2022 is $6,000; $7,000 over age 50. Anyone is eligible to contribute to a traditional IRA, regardless of your income level. Your money grows tax-deferred, but withdrawals after age 59½ are taxed at current rates. Traditional IRAs also allow you to use the contribution itself as a deduction on a current year tax return. Roth IRAs allow contributions at the same levels as traditional IRAs, with a maximum $6,000 for 2022; $7,000 over age 50. However, to be eligible to make contributions, your earned income must be below a certain level. A Roth IRA allows after-tax contributions. In other words, the contribution itself is not tax-deductible, as it is with the traditional IRA. However, your money grows tax-free. If you make no withdrawals until you are age 59½, there are no penalties. IRAs pay a modest interest rate. In either case, IRA deposits have to be from earned income, which in effect means if your earned income is over $6,000 ($7,000) then you can deposit the maximum. ### 401(k) Accounts Your employer may offer a retirement account to you. These are often in the form of a 401(k) account. There are traditional and Roth 401(k) accounts, which differ in how they are taxed, much as with other IRAs. In the traditional 401(k) plans, the money is deposited before tax is assessed, which means you do not pay taxes on this money. However, that means when money is withdrawn, it is taxed. These accounts are similar to mutual funds, in that the money is invested in a wide range of assets, spreading the risk. One of the perks some employers offer is to match some amount of your contributions to the 401(k) plan. For instance, they may match your deposits up to 5% of your income. This is an instant 100% return on the money that was matched. 401(k) plans with matching funds provide great value, as their rates of return are high compared to savings accounts, and are less risky that stocks since such funds invest across many investment vehicles. The next example demonstrates the power of constant deposits into a 401(k) plan that has some employer match. ### Check Your Understanding ### Key Terms 1. Bonds 2. Maturity date 3. Stocks 4. Dividend 5. Mutual fund 6. Prospectus 7. Issue price 8. Shares 9. Stock table 10. Individual retirement account 11. Roth IRA 12. 401(k) ### Key Concepts 1. There are many different investments with different returns and risks. 2. Bonds are loans form the purchaser to the entity selling the bond. 3. Bonds have some tax benefits, low to no risk, and a low return. 4. Stocks represent part ownership in a company. As such, stock holders share in the profits, and losses, of the company. 5. Information, including price, P/E, yearly highs and lows, and dividend amount can be found in online stock tables available on many websites. 6. Mutual funds represent collections of professionally administered investment vehicles. Have shares in a mutual fund has lower risk than ownership of stocks. 7. Retirement accounts employ some of the same strategies as mutual funds, in that they spread the risk and are professionally managed. 8. IRAs and Roth IRAs differ on when taxes are paid on the money, and who can use them. Roth IRAs have income limits while traditional IRAs do not. ### Videos 1. Bonds 2. Reading Stock Summary Online 3. Mutual Funds 4. 401(k) Accounts ### Formulas

# Money Management ## The Basics of Loans ### Learning Objectives After completing this section, you should be able to: 1. Describe various reasons for loans. 2. Describe the terminology associated with loans. 3. Understand how credit scoring works. 4. Calculate the payment necessary to pay off a loan. 5. Read an amortization table. 6. Determine the cost to finance for a loan. New car envy is real. Some people look at a new car and feel that they too should have a new car. The search begins. They find the model they want, in the color they want, with the features they want, and then they look at the price. That’s often the point where the new car fever breaks and the reality of borrowing money to purchase the car enters the picture. This borrowing takes the form of a loan. In this section, we look at the basics of loans, including terminology, credit scores, payments, and the cost of borrowing money. ### Reasons for Loans Even if you want a new car because you need one, or if you need a new computer since your current one no longer runs as fast or smoothly as you would like, or you need a new chimney because the one on your house is crumbling, it’s likely you do not have that cost in cash. Those are very large purchases. How do you buy that if you don ’t have the cash? You borrow the money. And for helping you with your purchase, the company or bank charges you interest. Loans are taken out to pay for goods or services when a person does not have the cash to pay for the goods or services. We are most familiar with loans for the big purchases in our lives, such as cars, homes, and a college education. Loans are also taken out to pay for repairs, smaller purchases, and home goods like furniture and computers. Loans can come from a bank, or from the company selling the goods or providing the service. The borrower agrees to pay back more than the amount borrowed. So there is a cost to borrowing that should be considered when deciding on a purchase bought with credit or a borrowed money. Even using a credit card is a form of a loan. Essentially, a loan can be obtained for just about any purchase, large or small, that has a cost beyond a person’s cash on hand. ### The Terminology of Loans There are many words and acronyms that get used in relation to loans. A few are below. APR is the annual percentage rate. It is the annual interest paid on the money that was borrowed. The principal is the total amount of the loan, or that has been financed. A fixed interest rate loan has an interest rate that does not change during the life of the loan. A variable interest rate loan has an interest rate that may change during the life of the loan. The term of the loan is how long the borrower has to pay the loan back. An installment loan is a loan with a fixed period, and the borrower pays a fixed amount per period until the loan is paid off. The periods are almost uniformly monthly. Loan amortization is the process used to calculate how much of each payment will be applied to principal and how much is applied to interest. Revolving credit, also known as open-end credit, is how most credit cards work but is also a kind of loan account. (We will learn about credit cards in Credit Cards) You can use up to some specified value, called the limit, any way you want, and as long as you pay the issuer of the credit according to their terms, you can keep borrowing from this account. These and other terminologies can be researched further at Forbes. ### Calculating Loan Payments Loan payments are made up of two components. One component is the interest that accrued during the payment period. The other component is part of the principal. This should remind you of partial payments from Simple Interest. Over the course of the loan, the amount of principal remaining to be paid decreases. The interest you pay in a month is based on the remaining principal, just as in the partial payments of Simple Interest. The payment of the loan has to be such that the principal of the loan is paid off with the last payment. In any period, the amount of interest is defined by the formula above, but changes from period to period since the principal is decreasing with each payment. The trick is knowing how much principal should be paid each payment so that the loan is paid off at the stated time. Fortunately, that is found using the following formula. ### Reading Amortization Tables An amortization table or amortization schedule is a table that provides the details of the periodic payments for a loan where the payments are applied to both the principal and the interest. The principal of the loan is paid down over the life of the loan. Typically, the payments each period are equal. Importantly, one of the columns will show how much of each payment is used for interest, another column shows how much is applied to the outstanding principal, and another column shows the remaining principal or balance . ### Cost of Finance There are often costs associated with a loan beyond the interest being paid. The cost of finance of a loan is the sum of all costs, fees, interest, and other charges paid over the life of the loan. ### Check Your Understanding ### Key Terms 1. Fixed interest rate 2. Variable interest rate 3. Installment loan 4. Loan amortization 5. Revolving credit 6. Amortization table 7. Cost of finance ### Key Concepts 1. There are many reasons for a loan, but primarily it is taken out for a large expense when cash is not available. 2. Each payment for an installment loan consists of an interest portion and a principal portion. 3. There is a formula to calculate the payment necessary to pay off a loan in installments. 4. Amortization schedules, or tables, show how each payment is applied to principal and interest. It also includes other details such as remaining balance and total interest paid. 5. Loans often have other fees associated with them such as origination fees or application fees. The total of the interest paid and the fees is the cost of finance. ### Video 1. Credit Scores Explained 2. Reading an Amortization Table ### Formula

# Money Management ## Understanding Student Loans ### Learning Objectives After completing this section, you should be able to: 1. Describe how to obtain a student loan. 2. Distinguish between federal and private student loans and state distinctions. 3. Understand the limits on student loans. 4. Summarize the standard prepayment plan. 5. Understand student loan consolidation. 6. Summarize and describe benefits or drawbacks of other repayment plans. 7. Summarize possible courses of action if a student loan defaults. ### Obtaining a Student Loan All college students are eligible to apply for a loan regardless of their financial situation or credit rating. Federal student loans do not require a co-signer or a credit check. Most students do not have a credit history when they begin college, and the federal government is aware of this. However, private loans will generally require a co-signer as well as a credit check. The co-signer will assume responsibility for paying off the loan if the student cannot make the payments. The first step in applying for student loans is to fill out the FAFSA (Free Application for Student Aid). FAFSA determines financial need and what type of loan the student is qualified to obtain. For students who are still dependents on their parent’s taxes, the parents also fill out the FAFSA, as their wealth and income impacts what the dependent student is eligible for. Students who cannot demonstrate financial need will also be helped by applying with FAFSA, as it will help guide them to the type of loan most appropriate. The FAFSA must be submitted each year. As soon as an offer letter from the college is received, the student should start the application process. The college will determine the loan amount needed. Also, there are limits on the amount a student can borrow. There are both yearly limits and aggregate limits. See the table later in this section that outlines the loan limits per school year and in the aggregate. If the student receives a direct subsidized loan, there is a limit on the eligibility period. The time limit on eligibility depends on the college program into which the student enrolls. The school publishes how long a program is expected to take. The eligibility period is 150% of that published time. For example, if a student is enrolled in a 4-year program, such as a bachelor’s degree program, their eligibility period is 6 years, as 1.50(4) = 6. Therefore, the student may receive direct subsidized loans for a period of 6 years. ### Types and Features of Student Loans Once a tuition statement is received, and all the non-loan awards are analyzed that are applicable to the costs of college (such as scholarships and grants), there still may be quite of bit of an expense to attend college. This difference between what college will cost (including tuition, room and board, books, computers) and the non-loan awards received is the college funding gap. There are several loan types, which basically break down into four broad categories: subsidized loans, unsubsidized loans, PLUS loans, and private loans. These loans are meant to fill the college funding gap. Federal subsidized loans are backed by the U.S. Department of Education. These loans are intended for undergraduate students who can demonstrate financial need. Subsidized federal loans, including Stafford loans, defer payments until the student has graduated. During the deferment, the government pays the interest while the student is enrolled at least half-time. These loans are generally made directly to students. However, there are restrictions on how the money can be used. It can only be used for tuition, room and board, computers, books, fees, and college-related expenses. Interest rates are not based on the financial markets but determined by Congress. Federal loans are backed by the Department of Education. Federal unsubsidized loans, including unsubsidized Stafford loans, are available for undergraduate and graduate students who cannot demonstrate financial need. If the student meets the program requirements, they are automatically approved. The student is not required to pay these loans during their time in college (enrolled at least half-time). However, the interest rate is generally higher and there is no deferment period, as with subsidized loans. Interest begins accruing as soon as the money is disbursed. Parent Loans for Undergraduate Students (PLUS) are federal loans made directly to parents. They are available even if parents are not deemed financially needy. A credit check is performed and approval is not automatic. The limit to what parents can borrow from a PLUS loan each year is still the college funding gap, but the aggregate of the PLUS loans does not have a limit. This means the PLUS loan can cover whatever is left in the funding gap once all other aid and loans are applied. Payments do not begin until the student is out of school, but interest begins to accrue the moment funds are disbursed. Because the parents take out the loan, the parents are responsible for paying back the loan. Private student loans are backed by a bank or credit institution and require a credit check, and interest rates are variable. As private loans are not subsidized by the government, no one pays the interest but the borrower. The student does not have to start repaying the loan until after graduation, but interest starts to accrue immediately. This loan has fewer repayment options, more fees and penalties, and the loan cannot be discharged through bankruptcy. Many students need a co-signer to acquire a private loan. Like PLUS loans, private student loans can cover whatever is left in the funding gap once all other aid and loans are applied Student loans, in general, have a term of 10 years, that is, the loans are paid back over 10 years. This can vary, but 10 years is the standard. ### Limits on Student Loans As mentioned earlier, there are limits to how much a student can borrow, per year and in total. The following table shows a general breakdown of the amounts the federal government and private lenders will lend. Amounts are based on level of need and whether the student is a dependent or an independent student. Independent students include those who are at least 24 years old, married, a professional, a graduate student, a veteran, a member of the armed forces, an emancipated minor, or an orphan. The amounts shown are as of this writing in 2022. Check out this Edvisors page on the limits of student borrowing to learn more! Putting this all together, we have a way to determine the student loans needed for a student to attend college. 1. First, determine the funding gap. If the student or family can cover the gap, then no loans are necessary. 2. Second, determine how much in federal subsidized and unsubsidized student loans can be taken out. If the total federal loans available is more than the funding gap, no other loans are needed. 3. Third, if the federal subsidized and unsubsidized loans do not cover the gap, PLUS and private student loans can be taken out to cover the remainder of the gap. At each step, if the student and family can cover some or all of the gap, they can do so without taking out a loan. ### Student Loan Interest Rates Student loans are first and foremost loans. Students will pay them back and will pay interest. In the fall of 2022, the federal student loan interest rate was 4.99%. Private student loans rates ranged between 3.22% and 13.95%. Finding the lowest interest rate you can helps with the payments, and especially helps if the loan is not federally subsidized. Remember, if the loan is not federally subsidized, the student is on the hook for the interest that is accumulating with the loan. ### Interest Accrual The interest on student loans begins as soon as the loan is disbursed (paid to the borrower). When the loan is federally subsidized, the government pays that interest for the student. This means the loan for a subsidized loan of $3,000 is still a loan for $3,000 when the student graduates. However, if the loan is not federally subsidized, the student is responsible for the interest that accrues on the loan. The $3,000 loan from year 1 of college is now a loan for more due to that added interest. The interest on that loan grew while the student was in college. The formula for growth of the loan’s balance is the same as compound interest formula from Compound Interest, . ### The Standard Repayment Plan There are various repayment plans available. The one most likely to apply to a student loan is the standard repayment plan, which is available to everyone. Borrowers pay a fixed amount monthly so the loan is paid in full within 10 years. Consolidated loans, discussed later in this section, also qualify for the standard repayment plan, and may allow the payoff period to range from 10 to 30 years. Direct subsidized and unsubsidized loans, PLUS loans, and federal Stafford loans are eligible. Since these are loans, they are paid back with interest. As with most installment loans, their payments are due monthly. The formula for paying back these loans is the same as the formula used for paying loans in The Basics of Loans: Using that formula, we can calculate how much the payment is for a student loan. Remember that all loan payments are rounded up to the next penny. ### Student Loan Consolidation When a student graduates, they may have multiple different student loans. Keeping track of them and paying them off separately can be a burden. Instead, these loans can be consolidated into a single loan. If they are federal loans the combination is called federal consolidation. Combining private loans is often referred to as refinancing. Refinancing, or private consolidation, can be used to combine both private and federal student loans. Be aware that consolidated federal loans may still be subject to the rules and protections that govern subsidized loans. Refinancing loans, private or federal, are no longer subject to those rules and guidelines. Check out this Experian article about consolidation and refinancing for more deatil. In consolidation of federal direct student loans, the interest rate is the weighted average of the interest rates on the subsidized loans. This means the interest rate remains the same. However, if the term is extended, then the student will pay back more over time than if they did not extend the loan term. In refinancing, it is possible to obtain a lower interest rate on the student loans, which may lower how much is paid per month and lower the total paid back over time. These monthly payments are calculated using the same formula as for any other loan payment, . The term of the refinanced loan may also be changed, which would also impact the payment per month. In either case, consolidating or refinancing, the monthly financial burden on the student can decrease. However, if the term is extended, the total amount repaid may increase. ### Other Repayment Plans There are various other repayment plans available to students. Plans other than the standard repayment plan typically require the student to meet certain criteria. The following plans are independent of student income, but may make early payments easier. 1. Graduated repayment plans are plans where the amount of payments gradually increases so that the loan is paid off in 10 years, or within 10 to 30 years for consolidated loans. Payments start off small and increase approximately every 2 years. Almost all loan types are eligible, including direct subsidized and unsubsidized loans, Stafford loans, PLUS loans, and consolidated loans. 2. Extended repayment plans are available to the direct loan borrower if the outstanding direct loans are over $30,000. The payments, fixed or graduated, are designed so that the loans are satisfied within 25 years. Eligible loans include both direct subsidized or unsubsidized loans, Stafford loans, PLUS loans, and consolidated loans. If student earnings are such that the standard, graduated, or extended repayment plans are unaffordable, then one can make payments that are based on their discretionary income. Discretionary income is federally defined to be the difference between (adjusted) gross income and 150% of the poverty guideline for location and family size. This discretionary income then depends on where one lives (contiguous United States or Hawaii or Alaska) and how many dependents one has. If married, a spouse’s income will be included in the adjusted gross income. Understanding discretionary income is necessary to understand how income driven payments plans work. The following plans all depend on discretionary income. 1. Pay As You Earn (PAYE) repayment plans have monthly payments that are 10% of discretionary income based on a student’s updated income and family size. If a borrower files a joint tax return, their spouse’s income and debt may also be considered. Eligible loans include direct subsidized and unsubsidized loans, PLUS loans made to students, and some consolidated loans. These loans are forgiven (student does not pay the remaining balance) after 20 years of monthly payments if they were direct federal student loans. 2. Revised Pay As You Earn (REPAYE) repayment plans have payment amounts that are based on income and family size and calculated as 10% of discretionary income. Eligible loans include direct subsidized and unsubsidized loans, direct PLUS loans, and some consolidated loans. These loans are forgiven, that is, the student does not pay more, after 20 or 25 years, provided they were direct federal student loans. 3. Income-Based Repayment (IBR) plans sound like a few of the others and there are similarities in all of them. The payments are either 10% or 15% of discretionary income, but this plan is meant for those with a relatively high debt. Every year, income and family size must be updated, and payments are calculated based on those figures. Eligible loans include direct subsidized and unsubsidized loans, Stafford loans, and PLUS loans made to students, but not PLUS loans made to parents. 4. Income-Contingent Repayment (ICR) plans have payments that are either 20% of discretionary income or whatever would be paid if a student were on a fixed payment plan for more than 12 years, whichever is less. Eligible loans include direct subsidized and unsubsidized, PLUS loans to students, and consolidated loans. There are many similarities among these repayment plans, and it is easy to misunderstand the nuances of each. Therefore, be careful entering into any type of repayment contract until you fully understand all the details and repercussions of the plan you choose. For more detail, see this nerdwallet article "Income-Driven Repayment: Is It Right for You?" to learn more! Using income to determine payments initially seems excellent. However, if there is no forgiveness at the end of the loan, then the income driven payment plans can cause problems. For one, your payment may not be sufficient to cover the interest rate of your loans. In that case, your loan balance actually increases as you make your payments. Eventually, you are paying for not only your original loan balance, but interest that has been growing and compounding over time. Also, if the loan term is extended, you may pay more, perhaps a lot more, money over time. You may find yourself in the position of paying these loans for decades. With those possible drawbacks, great care must be taken to avoid large problems down the line. ### Student Loan Default and Consequences The first day a payment is late, the account becomes delinquent. After 90 days, this delinquency is reported to the credit bureaus, and goes into default. This is serious, as now a credit score is affected, meaning that it will be harder to buy a car, a home, get a credit card, or a cell phone. Even renting an apartment may be a task not easily overcome. The default rate for students who do not complete their degree is three times higher than for students who do. Further, defaulting on a student loan may mean that the borrower loses eligibility for repayment plans, as the balance and any unpaid interest may become due immediately, and any tax refunds may be withheld and applied to the loan, and wages may be garnished. One should immediately contact your loan servicer and try to make other arrangements for repayment if this situation becomes apparent, as different repayment plans are available, if actions are taken quickly. There are several options that may be open to avoid defaulting. One is called rehabilitation, or is the process in which a borrower may bring a student loan out of default by adhering to specified repayment requirements, and the other is consolidation. Certain criteria must be met to enter these programs. Both of these options are detailed, including the criteria required for eligibility, on the studentaid.gov loan management page. Professionals advise hiring an attorney if one of these paths is chosen. ### Check Your Understanding ### Key Terms 1. FAFSA 2. College funding gap 3. Subsidized loan 4. Unsubsidized loan 5. Parent loan for undergraduate students 6. Private student loan 7. School-channel loan 8. Direct-to-consumer loan 9. Standard repayment plan 10. Federal consolidation 11. Refinancing 12. Private consolidation 13. Graduated repayment plan 14. Extended repayment plan 15. Discretionary income 16. Pay as you earn (PAYE) repayment plan 17. Revised pay as you earn (REPAYE) repayment plan 18. Income-based (IBR) repayment plan 19. Income-contingent (ICR) repayment plan 20. Delinquent 21. Default 22. Rehabilitation ### Key Concepts 1. The FAFSA must be filled out each year that a student wishes to borrow for. 2. A student’s funding gap determines how much they need in loans to pay for college. 3. Federal subsidized student loans defer payments until after graduation and interest does not accrue on these loans. 4. Unsubsidized student loans defer payment until after graduation but interest begins accruing as soon as the loan finds are disbursed. 5. There are both yearly and aggregate limits for student loans to prevent over-borrowing, among other reasons. 6. Federal direct loans have a low interest rate set by the government, but other student loans have varying rates of interest set by the banks. 7. The standard repayment plan lasts 10 years and is made up of monthly payments. 8. Consolidating or refinancing student loans merges many student loans into one loan. 9. If only federal loans are consolidated, the interest rate is the same as the individual loans, currently set at 4.99%. 10. If other loans are refinanced together, the interest rate may be lower with the new loan. 11. Other repayment plans are available. Such a plan may have payment that start small and grow as the loan is paid off, or it may have a longer term, or may be based on the discretionary income of the student. 12. Being delinquent on a student loan is a precursor to being in default. Making payments in a timely fashion allows the student to avoid this situation. ### Video 1. Types of Student Loans 2. Repayment Plans ### Formula

# Money Management ## Credit Cards ### Learning Objectives After completing this section, you should be able to: 1. Apply for a credit card armed with basic knowledge. 2. Distinguish between three basic types of credit cards. 3. Compare and contrast the benefits and drawbacks of credit cards. 4. Read and understand the basic parts of a credit card statement. 5. Compute interest, balance due, and minimum payment due for a credit card. It can be difficult to get along these days without at least one credit card. Most hotels and rental car agencies require that a credit card is used. There are even a number of retailers and restaurants that no longer accept cash. They make online purchasing easier. And nothing contributes more to a good credit rating than a solid history of making credit card payments on time. Being granted a credits card is a privilege. Used unwisely that privilege can become a curse and the privilege may be withdrawn. In this section, we will talk about the different types of credit cards and their advantages and disadvantages. The more knowledge a cardholder has about the credit card industry, the better able credit accounts can be managed, and that knowledge may cause major adjustments to a cardholder’s lifestyle. All credit cards are not equal, but they all represent consumers borrowing money, usually from a bank, to pay for needs and “wants.” As such, they are a type of loan, and your repayment may include interest. (You might want to review Section 6.8, which discusses loans and repayment plans.) There are many institutions and credit cards to choose from. Use caution as you shop around for a credit card that suits you. Your top concern is likely the interest rates on purchases and cash advances. But be careful to also read the small print regarding charges for late payments, and other fees such as an annual fee, where the credit card charges you (the cardholder) a fee each year for the privilege of using the cards. Many cards charge no such fee, but there are many that charge modest to heavy fees. Make sure to understand rules for reward programs, where the credit card issuer grants benefits based on one’s spending. Finally, once one applies for and is granted a credit card, pay attention to the credit limit the bank offers. Once a company is owed that much money, use of the card for purchases should be curtailed until some of the debt is paid off. ### Types of Credit Cards There are basically three types of credit cards: bank-issued credit cards, store-issued credit cards, and travel/entertainment credit cards. We will look at all three and explain the good and the bad qualities of each. ### Bank-Issued Credit Cards Perhaps the most widely used credit card type is the bank-issued credit card, like Visa or MasterCard (and even American Express and Discover cards). These types of cards are an example of revolving credit, meaning that additional credit is extended before the previous balance is paid—but only up to the assigned credit limit. Bank-issued cards are considered the most convenient, as they can be used to purchase anything, including apparel, furniture, groceries, fuel for automobiles, meals, hotel bills, and so on, just as if paying with cash. The interest rates on bank-issued credit cards are usually lower than those for other credit cards we’ll discuss, and the credit limits are generally higher. Currently, bank-issued cards have an average 20.09% APR. ### Store-Issued Credit Cards Store-issued credit cards are issued by retailers. One can hardly walk into a store these days without being offered a discount on purchases if one applies for the store credit card. These cards can only be used in that store or family of stores that issues the card. However, if a store credit card is associated with Visa, MasterCard, or American Express, then the card might be used the same way that the bank-issued cards are used. This is called cobranding. The logo of the bank-issued card will be present on the store card. Many stores offer both types. Like other credit cards, they may come with an annual fee. Store credit cards usually charge higher interest rates than bank-issued cards. Currently, store credit cards have an APR (annual percentage rate) of 24.15%. Any rewards offered by store credit cards are usually limited to purchases made in their own store, and it typically takes longer to accumulate enough rewards or points to redeem them, whereas cobranded credit cards offer opportunities to earn rewards on all purchases, regardless of whether purchases are made in the issuing store or not. Store credit cards usually offer lower credit limits, at least in the beginning. After being proved to be a responsible credit card owner, credit limits can be raised. Nevertheless, store credit cards are a good choice for those new to the credit card industry. If on-time payments are consistently made, it is an excellent way to get started building a credit history. ### Travel/Entertainment Cards, or Charge Cards This is the third type of credit card. The travel and entertainment cards, also known as charge cards, first and foremost offer very high limits or unlimited credit, but they must be paid in full every month. They generally charge high annual fees and impose expensive penalties should a payment be late. On the other hand, they typically have longer grace periods and offer many and various kinds of rewards. Check out this nerdwallet article about the differences between a charge card and a credit card. ### Credit Card Statements Cardholders usually receive monthly statements and have 21 days to pay the minimum amount due. The statements itemize and summarize activity on the credit card for that statement’s billing period. The billing period for a credit card is generally a month long, but typically does not start and end on the first and last days of the month. The statement will include the current balance, interest rate, the minimum payment due, and the due date. Be aware, different companies produce statements that are laid out differently. The information will be clearly labeled though. The due date is a top concern. Missing a due date is one of the worst things a cardholder can do financially, and this is by far the biggest downfall of owning a credit card. Not only is the cardholder subject to late fees, but when a payment is late more than once there is a high probability that the cardholder will be negatively reported to the credit bureaus, which can quickly erode a credit score. shows an excerpt from an actual statement from a Chase Bank Visa card, based on the current $668.25 balance. Specifically pay attention to the late payment penalty and minimum payment warning statements. stating that if no other purchases are made and you continue making only the minimum payment, it will take 19 months to pay off the balance and you will pay $754.00. You can’t say you were not warned. It is critical that you examine your statement every month because it is always a possibility that your account may have been compromised. If you should notice fraudulent charges on your statement, notifying the credit card company is often enough to have those charges researched by the company and removed. The card with the fraudulent charges will be canceled and a new card with a new account number will be sent to you. ### Compute Interest, Balance Due, and Minimum Payment Due for a Credit Card Computing all of these values depends on understanding and computing the average daily balance on a credit card. Once that is known, the interest, balance due, and minimum payment can be found. Above all else, if you pay off the entire balance each month, interest is not charged. ### Average Daily Balance Most credit card companies compute interest using the average daily balance method. To find the average daily balance on your credit card, determine the balance on the card each day of the billing period (often that month), and take the average. One process to find that average daily balance follows these steps: 1. Start with a list of transactions with their dates and amounts. 2. For each day that had transactions, find the total of the transactions for the day. Expenditures are treated as positive values, payments are treated as negative values. 3. Create a table containing each day with a different balance. The balance is the previous balance plus the day’s total transactions. 4. Add a column for the number of days those balances until the balance changed. 5. Add a column that contains the balances multiplied by the number of days until the balance changed. 6. Find the sum of that last column. 7. Divide the sum by the number of days in the billing period (often the number of days in the month). This is the average daily balance. ### Calculating the Interest for a Credit Card The interest charged for a credit card is based on the daily interest rate of the card, the number of days in the billing cycle, and the average daily balance on the card. ### Calculating the Balance of a Credit Card The balance, or sometimes balance due, on a credit card is the previous balance, plus all expenses, minus all payments and credits, plus the interest on the card. As stated before, if the card was paid off, there is no interest to be paid. The next example puts all those steps together. ### Minimum Payment Due The minimum payment due is the smallest required amount to be paid on a credit card to avoid late fees and penalties, such as an increased interest rate. The calculations for this may differ from card to card. They also depend in the balance of the credit card. General guidelines for minimum payment due are: 1. For larger balances (usually over $1,000), the minimum payment will be some percentage of the balance due. 2. For moderate balances (between $25 and $1,000), the minimum would be a specified dollar amount. $25 seems to be a common value. 3. If the balance is small (under $25 for instance), then the minimum payment is the balance. Those are just guidelines. Individual cards may vary in these values. Minimum payments should only be paid if money is short in a given month. The length of time to pay off a credit card using minimum payments is quite long, and results in paying a lot of interest. It is strongly discouraged. Check out this nerdwallet article about minimum payments for more! ### Check Your Understanding ### Key Terms 1. Reward program 2. Annual fee 3. Credit limit 4. Bank-issued credit card 5. Store-issued credit card 6. Travel and entertainment cards 7. Charge cards 8. Billing period 9. Balance 10. Minimum payment 11. Average daily balance ### Key Concepts 1. Credit cards can be a flexible way to pay for almost anything, but can become a financial hazard if used unwisely. 2. When deciding which credit card to apply for, evaluate the interest rate, fees (annual and late), reward programs and credit limit. Be sure they meet your criteria. 3. Paying off the balance of your credit card every month will control your spending and will never result in paying interest. 4. Credit card statements hold all important information about your credit card, including payment, balances, charges and billing cycle dates. 5. Although the minimum payment is attractive precisely because it is so small, paying only the minimum results is a long payoff term and higher interest costs. ### Video 1. Choosing a Credit Card 2. Reading Credit Card Statements ### Formula

# Money Management ## Buying or Leasing a Car ### Learning Objectives After completing this section, you should be able to: 1. Evaluate basics of car purchasing. 2. Compute purchase payments and identify the related interest cost. 3. Evaluate the basics of leasing a car. 4. Identify and contrast the pros and cons of purchasing versus leasing a car. 5. Investigate the types of car insurance. 6. Solve application problems involving owning and maintaining a car. There are people who don’t need a car and won’t purchase one. But for many people, whether or not to have a car is not a question. Having a car is a basic necessity for these people. Obtaining a car can be daunting. The models, the features, the additional costs, and finding funding are all steps that need to be taken. One of the big decisions is whether to buy the car or to lease the car. This section will address some of the issues associated with each option. ### The Basics of Car Purchasing The biggest questions you will answer before purchasing a car are, what do you want and what do you need? Does it have to be new? Does it have to be a make and model you are familiar with? Does it have to have assisted driving? What other details are important to you? For a new vehicle, every feature beyond standard features comes with additional cost, which leads to the question that constrains all of your decisions about a car. How much can you afford to spend on a car? What you can afford must include insurance costs (discussed later in this section) and maintenance and upkeep. Once you have this in mind, you can search for a car that matches, as closely as possible, what you want and can afford. Most, if not all, dealers have websites that you can search through to identify the car you want. If new cars are not affordable, used cars cost less but come with the wear and tear of use. The sticker price of the car, called the manufacturer’s suggested retail price (MSRP), or the negotiated price you arrive at, isn’t the end of the cost to buying a car. There are many fees that accompany the purchase of the car, and perhaps even sales tax. These include but aren’t necessarily limited to the following: 1. the title and registration fee, which includes registering your car with the state, getting the license plate, and assigning the title of the car to the lender. This cannot be avoided. 2. a destination fee, which covers the cost of delivering the vehicle to the dealer 3. a documentation fee, sometimes referred to a processing fee of handling fee, is the cost of all the paperwork the dealer did to get you the car 4. a dealer preparation fee, which is for washing the car and other preparation of that sort. You should try to negotiate that out of the cost of the dealer tries to charge for that 5. extended warranties and maintenance plans, which help cover some of the costs of caring for the car. 6. Sales tax. You could pay for these immediately, but they are often added to the financing of the car, meaning they become part of the principal of your loan. One way to bring down payments on a car is to provide a down payment or a trade in. This is money applied to the purchase price before financing happens. Be warned, the sales tax applies to the full purchase price! If you reduce the amount financed, the payments respond by going down. This often becomes part of the negotiating process. When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score (see The Basics of Loans). But you can choose from different lenders. The dealership will likely offer to finance your car loan. Frequently, dealerships offer special financing with very low rates. This is to help move inventory, and may indicate their desire to make sales. This might make negotiating easier. Even if the dealership offers financing, check with your bank or credit union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find. ### Purchase Payments and Interest Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates available for used cards are frequently higher than those for new cars. These loan payments work exactly the same way as other loans do as far as payments are concerned. The payment function comes from The Basics of Loans. The difference between financing a new car or a used car is that financing a new car typically comes with a lower interest rate and a longer term that financing a used car. ### The Basics of Leasing a Car Leasing a car is an alternative to purchasing a car. It is still a loan, and acts like one in many respects. They typically last either 24 months or 36 months, though other terms are available. Leases also come with mileage limits, frequently 10,000, 12,000, or 15,000 miles per year. When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit. There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, These often are paid before the lease is complete. These include: 1. a down payment, which is your initial payment that is applied to the price of the car. It reduces the amount you finance, much the same as when you purchase a car. It is recommended that this be negotiated away. 2. the acquisition fee, sometimes called the bank fee. This is the money charge for the company to set up the lease. It is essentially a paperwork fee. It is not likely that this can be negotiated. 3. a security deposit, which might be required. It is about the same as 1 month’s payment for the lease. The deposit is returned to you if the car is in good shape at the end. This can be negotiated away. 4. disposition fees, which cover the cost the company will incur when they take your car back and are typically between $200 and $450. 5. the title, registration, and license fees, just as with the purchase of a car. 6. sales tax, which will likely be applied. The sales tax only covers the depreciated portion of the car (more on depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine the sales tax rules for where you lease the car. As you can imagine, this can come to a fairly high dollar amount. You have some obligations when you lease a car. You must keep the car in good condition, cleaned, maintained, and free of anything more than minor damage. If the car is in poor condition when the car is returned, you will be responsible for the cost to bring the car to an acceptable condition. You are also expected to keep the mileage under its limit. If you go over, you will pay 10 to 25 cents per mile over. ### Lease Payments Lease payments are similar to regular loan payments, but have some other details. Calculating a lease payment involves knowing the following values: 1. The price of the car. This is the cost you would pay for the car after applying all discounts, incentives, and negotiations. 2. Residual Value. This is the manufacturer's estimate of the car's value after a set period of time. The residual value is expressed as a percentage of the manufacturer’s suggested retail price (MSRP). 3. Months. This is the length of the lease. Most leases are either 24- or 36-month leases, but other terms are available. 4. Monthly Depreciation. The monthly depreciation is the difference between the price of the car and the residual value, divided by the number of months of the lease, and represents the monthly loss of value of the car while it’s being used. 5. Money Factor (MF). This is the interest rate, but expressed in a different way for a lease. Converting from the money factor to the annual percentage rate (APR) is done by multiplying the MF by 2400. Naturally, converting an APR to a MF is done by dividing the APR by 2400. Once the values above are found, the payment for the lease can be calculated. ### Comparing Purchasing and Leasing When deciding to buy or lease a car, the differences between the two options should be carefully evaluated. The following is a list of points of comparison between the two. 1. The payments for a lease are likely less than the payment for purchasing. 2. When leasing, you get a new car after the lease term is over, typically 24 or 36 months. Buying the car means the same car is driven until it is re-sold and a new one bought. Essentially, leasing a car is equivalent to renting a car. 3. The leased car is new, so all warrantees are in force and you drive the car during its best years. When the car is purchased, it may be kept past its warrantees and may be driven until it is quite old. 4. Each time you lease a new car, all the fees and taxes must be paid again. When buying a car, these fees are only paid once. 5. Leasing contracts carry restrictions on the mileage you can drive per year, and going over incurs more cost at the end of the lease. Buying the car means no such mileage limits. 6. When leasing, you are obligated to keep the vehicle in good condition and maintained according to the dealer’s schedule. Some dealerships will even pay for oil changes over the life of the lease. When the car is purchased, the upkeep schedule is the choice of the owner. 7. When a car is purchased and kept for long enough, the warranty expires and the owner is responsible for all maintenance items and repairs. The warrantee for a car won’t expire during the lease term. 8. When a new car is purchased and the loan is paid off the car is still owned by the buyer and may be traded in when a new car is to be purchased. When leasing, the car is returned to the dealer when the lease term is over. When deciding between the two, you are choosing between these features. If you aren’t willing to drive an older car or deal with the upkeep that accompanies an older car, you may want to lease. This means you will need to pay those beginning costs each time the lease is up. If you want to own the car after the payments are over, then you may want to buy a car. This means you are paying for all the upkeep after the warrantees expire, but you have no limits on mileage and own the car at the end. It really depends on your preferences. ### Car Insurance Car insurance is meant to cover costs associated with accidents involving cars. Most states (all except New Hampshire and Virginia) require some insurance. Without insurance, the state may not let you get a license for your car or register your car. Your state’s requirements can be hard to follow. Fortunately, insurance companies and brokers will make sure your insurance is sufficient for your state and will warn you if you try to not meet the requirements. Of course, they may offer more than what is sufficient, so it is your responsibility to determine how much coverage you want, as long as the minimum insurance requirements are met. The cost of insurance should be accounted for when evaluating the affordability of buying or leasing a car. Whether your car is leased or owned, you do need insurance. This contributes to the cost of having the vehicle. Leasing or owning makes no difference to the insurance company you choose, because they are insuring you based on what you are driving, your driving record, and other information about you including where you live and your age. These insurance policies have many components that address different costs that can come from auto accidents. This may make details confusing, and you may not realize what you are paying for until you must use it. Here is a brief outline of the different components of auto insurance, many of which are required by the state that issues your driver’s license. 1. Liability insurance is mandatory coverage in most states. Liability insurance covers property damage and injuries to others should you be found legally responsible for an accident. You are required to have the minimum amount of coverage, as determined by your state, in both areas. 2. Collision insurance is insurance covering the damage caused to your car if involved in an accident with another vehicle. 3. Comprehensive insurance is an extra level of coverage if involved in an accident with another vehicle and covers other things like theft, vandalism, fire, or weather events as outlined in your policy. There is a deductible assigned to each type of insurance, an amount that you pay out of pocket before your comprehensive coverage takes effect. Comprehensive insurance is often required if you lease or finance the purchase of a vehicle. 4. Uninsured or underinsured motorist insurance: If you are hit by an uninsured or underinsured motorist, this insurance will help pay medical bills and damage to your car. 5. Medical payments insurance is mandatory in some states and helps pay for medical costs associated with an accident, regardless of who is at fault. 6. Personal injury protection insurance is coverage for certain medical bills and other expenses due to a car accident. Other covered expenses may include loss of income or childcare, depending on your policy. 7. Gap insurance is designed to cover the gap between what is owed on the car and what the car is worth in the event your car is a total loss. 8. Rental reimbursement insurance is coverage for a rental car while your car is under repair resulting from an accident. You can also purchase other special insurance policies, such as classic car insurance, new car replacement insurance, and sound system replacement insurance, to name a few. It is important that you determine exactly what you need, as insurance policies can be expensive and vary according to your age, driving history, and where you live. ### Maintaining a Car Cars are not a buy it and forget it item. They require upkeep, which adds to the cost of owning the car. Tires, brakes, and wipers need replacing. Oli changes, inspections, so many things other than gasoline. Below is a list of some maintenance requirements for cars, along with cost and roughly how often they should happen. When designing a budget, these expected costs should be accounted for. Extra money per month should be saved in addition to this budget category, to handle unanticipated, and perhaps very costly, repairs. ### Check Your Understanding ### Key Terms 1. Title and registration fees 2. Destination fee 3. Documentation fee 4. Dealer preparation fee 5. Extended warranty 6. Down payment 7. Acquisition fee 8. Security deposit 9. Disposition fees 10. Liability insurance 11. Collision insurance 12. Comprehensive insurance 13. Uninsured or underinsured motorist insurance 14. Medical payment insurance 15. Personal injury insurance 16. Gap insurance 17. Rental reimbursement insurance ### Key Concepts 1. There are many factors to consider when choosing to buy or lease a car. 2. The cost of the car is increased by a number of fees and sales tax. 3. There are advantages to buying a car and advantages to leasing a car. The decision between the two depends on the preference of the buyer. 4. Insurance covers costs associate with accidents. It is made up of various components. 5. The costs of owning a car, including insurance and maintenance, should be a part of the budgeting process. 6. Budgeting for unexpected repairs can ease the stress of encountering large repair bill. ### Formula

# Money Management ## Renting and Homeownership ### Learning Objectives After completing this section, you should be able to: 1. Evaluate advantages and disadvantages of renting. 2. Evaluate advantages and disadvantages of homeownership. 3. Calculate the monthly payment for a mortgage and related interest cost. 4. Read and interpret an amortization schedule. 5. Solve application problems involving affordability of a mortgage. After renting an apartment for 10 years, you realize that it may be time to purchase a home. Your job is stable, and you could use more space. It is time to investigate becoming a homeowner. What are the things that you must consider, and what is the financial benefit of owning as opposed to renting? This section is about the advantages, disadvantages, and costs of homeownership as opposed to renting. ### Advantages and Disadvantages of Renting When renting, you will likely sign a lease, which is a contract between a renter and a landlord. A landlord is the person or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and cannot, do with the property while you occupy the property. Like leasing a car, there are advantages to renting but also some disadvantages. Some advantages are: 1. Lower cost. 2. Short-term commitment. 3. Little to no maintenance cost. The landlord pays for or performs most maintenance. 4. You need not stay at end of lease. Once the lease term is over (the lease is up), you are not obligated to stay. 5. If renting in an apartment complex, there may be a pool, gym, or community room for renters to use. Of course, there are disadvantage too: 1. No tax incentives. 2. Housing cost is not fixed. When the lease is up, the rent can change. 3. No equity. When you are done living in a rental, you have built no value. 4. Restrictions on occupants. There may be a limit on how many can live in the apartment. 5. Restrictions on decorating. The property is not yours, so any decorating or improvements need landlord permission. 6. Limits on pets. Permission for pets, and their number and type, will be set forth in the lease. 7. May not be able to remain when lease term is over. The landlord can, at the end of your lease, invite you to leave. 8. The building may be sold, and the new landlord may institute changes to the lease when the previous lease expires. Renting has fees to be paid at the start of the lease. Typically, when you rent, you will pay first and last months’ rent and a security deposit. A security deposit is a sum of money that the landlord holds until the renter leaves the rental property. The deposit will cover repairs for damage to the apartment during the renter’s stay but may be returned if the apartment is in good condition. If your landlord runs a credit check on you, the landlord may charge you for that. ### Advantages of Buying a Home The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership mirror the advantages of renting. Some advantages to buying a home are: 1. There are tax incentives. The interest you pay for your mortgage (more on that later) is deductible on your federal income tax. 2. There are no restrictions on pets or occupants, unless laws in your area specify limits for homes. 3. You can redecorate any way you wish, limited only by the laws in your area. 4. Once your mortgage is set with a fixed-interest rate, your housing cost is fixed. 5. Your home grows equity, that is, the difference between what you owe and what the house is worth grows. You can use the equity to secure loans, and you recover the equity (and more if you’re fortunate) when you sell the house. 6. As long as you pay your mortgage and maintain the home to the standards of your community, you can stay as long as you wish. Some disadvantages to home ownership are: 1. The cost is higher than renting. Mortgages and associated costs are typically higher than rent for a similar living space. 2. The owner is responsible for upkeep, maintenance, and repairs. These can be extremely costly. 3. The owner cannot walk away from the property. It can be sold, but simply leaving the property, especially if not paid off yet, has serious consequences. The big question of affordability looms large over the decision to rent or buy. Renting, strictly from an affordability viewpoint, comes with much less initial outlay and smaller commitment. If you do not have sufficient income to regularly save for possibly expensive repairs, or your credit isn’t quite as good as it needs to be, then renting may be the best choice. Of course, even if you can afford to buy a home, you may choose to rent based on the comparative advantages. Buying a home really involves two buyers. You and the mortgage company. The mortgage company has interest in the home, as they are providing the funds for the home. They want to protect their investment, and many fees are about the bank as much as the buyer. They fund a mortgage based on the value they assign the property. Not you. This means they will want some certainty that the home is sound, and you are a good investment. In the end, you must weigh your options and carefully consider your priorities in choosing to rent or buy a home. ### Mortgages Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the penalty for late payments. The title of the home belongs to the bank. Since a mortgage is a loan, everything about loans from The Basics of Loans holds true, including the formula for the payments. ### Monthly Mortgage Payments The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate. To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. This can be useful information, but not too many people reach the end of their mortgage. They tend to move before the mortgage is paid off. With the principal of the mortgage and how much total is paid over the life of the mortgage, the cost of financing can be found by subtracting the principal of the mortgage from the total paid over the life of the mortgage. ### Reading and Interpreting Amortization Tables Amortization tables were addressed in The Basics of Loans. They are most frequently encountered when analyzing mortgages. The amortization table for a 30-year mortgage is quite long, containing 360 rows. A full table will not be reproduced here. We can, though, read information from a portion of an amortization table. ### Escrow Payments The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also has you pay into an escrow account, which is a savings account maintained by the mortgage company. Your insurance payments will be set by your insurer and the mortgage company will pay them on time for you from your escrow account. Your property taxes are set by where you live and are typically a percentage of your property’s assessed value. The assessed value is the estimation of the value of your home and does not necessary reflect the purchase or resale value of the home. Your property taxes will also be paid on time by the mortgage company from your escrow account. For example, in Kalamazoo, Michigan, the effective tax rate for property is 1.69% of the assessed value of the home. These escrow payments, which cover bills for the home, can increase the monthly payments for your home well beyond the basic principal and interest payment. ### Check Your Understanding ### Key Terms 1. Lease 2. Landlord 3. Mortgage 4. Escrow account 5. Assessed value ### Key Concepts 1. There are many points of comparison between renting and buying a house. 2. Before deciding to buy a house, you should carefully consider all the responsibilities that come with home ownership. 3. Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from. 4. Owning a house has more costs but has more freedom, plus the owner creates equity. 5. Mortgages are loans, and payments are calculated in the same way as any other loan. 6. Amortization tables help a homeowner understand the mortgage and how the payments are applied to the principal and interest. 7. In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, which covers insurance and taxes. ### Video 1. Rent or Buy 2. Closing Costs ### Formula

# Money Management ## Income Tax ### Learning Objectives After completing this section, you should be able to: 1. Determine gross, adjusted gross, and taxable income. 2. Apply exemptions, deductions, and credits to basic income tax calculations. 3. Compute FICA tax. 4. Solve tax application problems for working students. Before the start of the American Civil War in 1861, most of the country’s revenue came from tariffs on trade and excise taxes. However, this fell far short of the high cost of the war. Because of this, the federal government enacted the nation’s first income tax with the Revenue Act of 1861, which created the Internal Revenue Service as we know it today. No one likes paying income tax, but it is a reality of life. In this section, we will learn about Form 1040, the U.S. Individual Income Tax Return, and ways to prepare for tax time. The U.S. tax code may change from year to year. Because of this, this section includes examples of how taxes, deductions, and exemptions might be computed. The types of income, deductions, and exemptions that are used in the examples are used in the current tax code. ### Gross, Adjusted Gross, and Taxable Income Your income drives how much you pay in taxes. The more you earn, the more you are likely to pay. But your income alone is not the full story. When you add all the money you earned from your job, freelance work, interest from savings, and other sources, you have your gross income. If you are an employee, your income from your job will be reported on a W-2, which is sent to you by your employer. Income from freelance work will be reported on a 1099-MISC form, and is sent by the company that paid you. Income from interest is reported on a 1099-INT form and comes from the entity that paid the interest. Before you determine how much you owe in taxes, you will make certain adjustments to that gross income. You will deduct, or subtract, some of income from the gross income. That’s your adjusted gross income, or AGI. That is still not what you are taxed on. Next, you need to apply exemptions to your income. These are pieces of income that the government does not tax. After that is done, you reach your taxable income. We will look at each of these parts of the taxable income. You will notice that your paycheck already has taxes taken out of it. Your employer will withhold some of your income, sending it directly to the federal, state, and local governments. It is an estimate of how much you will owe in income tax. In the end, it reduces how much you will pay when your taxes are due. If they withhold too much income, you will receive the extra they withheld in the form of a refund. Your adjusted gross income (AGI) is computed before your taxes are determined. It begins with the gross income, and then subtracts from that income any deduction. Deductions are expenditures on your part that the government won’t tax. These deductions include money deposited into tax-deferred investments, and mortgage interest that you paid, charitable contributions if you made any, medical bills over a threshold, medical insurance under certain circumstances, and property taxes. If you add all these up, and they are all legal deductions, the sum is subtracted from your gross income, leaving the AGI. Remember that your AGI is not your taxable income. Exemptions need to be subtracted from the AGI to reach your taxable income. Exemptions are income that the government does not tax. Some examples of exempt income are disbursements from health savings accounts for qualified medical expenses, bond interest, some IRA distributions, and gifts given that are under $16,000. Note that exemptions are different from deductions: exemptions are excused incomes, whereas deductions are excused expenditures. ### Tax Credits Another piece of the tax puzzle is tax credits. This is money subtracted from the tax you owe. Tax credits are very different from deductions or exemptions. Deductions and exemptions are taken away from your gross income before the tax you owe is calculated. A tax credit, is subtracted, dollar for dollar, from your tax bill. Once the tax you owe is calculated, subtract the any tax credits from that calculated tax. Some of the tax credits are refundable. This means that if subtracting them from your tax results in a negative number, you receive a tax refund. For more details, see this article about tax credits. The federal government has placed income limits and restrictions and on those eligible to receive tax credits because their value is so high. Here is a partial list of tax credits that you might qualify for: 1. Earned income credit is a refundable tax credit for low- to moderate-income workers and ranges from $560 to $6,935 depending on dependents and income. This is refundable 2. American opportunity credit is a credit taken by parents who have children enrolled in college at least half time and pursuing a degree. This credit is worth $2,500 per student for the first 4 years of undergraduate school, subject to income limits. This is a refundable tax credit. 3. Lifetime learning credit is a credit is equivalent to 20% of educational expenses, up to $2,000 per year, subject to income limits. There is no cap to how many years you can apply for this credit. 4. Child tax credit is worth $2,000 per child under the age of 17 if that child lives at home at least half the year, subject to income limits. This is a refundable tax credit. 5. Child and dependent care tax credit was designed to help pay for child care while the parent works. The amount of the credit is dependent on your income. However, the maximum amount that can be received is, in 2022, $4,000 for one eligible person, or $8,000 for two or more qualifying people. A dependent qualifies if they are a child under 13 years old, a spouse who is unable to care for themselves, or some other qualifying person. This is a refundable tax credit. 6. Premium tax credit was created by the Affordable Care Act, and it is one that is received by many people throughout the year. In essence it is a health insurance premium subsidy. The amount of the credit is based on your income and the price of health insurance in your area. This is a refundable tax credit. ### Computing FICA Taxes FICA stands for the Federal Insurance Contributions Act of 1935. FICA taxes are used solely to fund Social Security and Medicare and are separate from federal income tax. It amounts to 7.65% of your gross pay, which is withheld from your paycheck automatically. Your employer is required to match the 7.65% amount. Of the 7.65%, 6.2% goes to Social Security (SSI), and 1.45% goes to Medicare. As of 2022, SSI tax only applies to the first $147,000 of earnings. Any gross income above that is not taxed for social security. This limit changes every year. Medicare tax, on the other hand, applies to the entirety of your gross income. ### Calculating Your Income Tax Your income tax bill and your income tax rate are based on your taxable income. The tax system in the United States is progressive, meaning that the tax rates are marginal so the higher your taxable income the higher the tax rate you will pay. Taxable income is broken into brackets, or ranges of income. Each bracket has a different tax rate. The tax brackets and rates for single filers as of 2022 are given below: So if your taxable income is $76,500 and you are filing as a single filer, your tax bill will be 22% of that $76,500, right? Wrong. Your income is split among those brackets and the money in each bracket is taxed at that bracket’s tax rate. Seems confusing. Here is a list of steps to follow to find the tax owed. Step 1: Find the bracket for the income. Step 2: For each bracket below the income bracket, the tax from that bracket is: Step 2a: Find the difference between the upper limit of that bracket and upper limit of the next lower bracket. If this is bracket 1, use 0 as the upper limit of the previous bracket. Step 2b: The tax from that bracket is the bracket tax rate applied to the difference from Step 2a. Step 3: For the bracket that the income belongs to, find the income minus the lower limit for the bracket. Step 4: The tax for the bracket of the income is tax rate for that bracket applied to the difference found in Step 3. Step 5: Add these various tax values to get the total income tax. There are various tax brackets, and the rates may change in any given year. The income limits may also change. For all examples going forward, we will use the single filer tax brackets, even if those brackets are not appropriate (e.g., married or head of household filers). ### Check Your Understanding ### Key Terms 1. Gross income 2. Adjusted gross income 3. Exemption 4. Taxable income 5. Deduction 6. Tax credit 7. Earned income credit 8. American opportunity credit 9. Lifetime learning credit 10. Child tax credit 11. Child and dependent care tax credit 12. Premium tax credit ### Key Concepts 1. Federal income tax is based on income after certain adjustments. 2. Gross income is income from all sources, including gifts and winnings. 3. Before taxes are calculated, the taxable income is found by subtracting deductions and exemptions from gross income. 4. Income tax is progressive, increasing in rate as income increases. 5. Being in the 32% tax bracket means some of your income is taxed at 10%, some at 12%, some at 22%, some at 24%, and the rest at 32%. 6. Income in each tax bracket is taxed at that bracket’s rate, which means in 2022 the first $10,275 earned is taxed at 10% only. 7. Tax credits are subtracted from the taxes that are owed. 8. Some tax credits are refundable, which means they can make the amount you owe negative, which results in a refund. ### Video 1. Deductions Versus Credits 2. Tax Brackets ### Project ### Creating Your Future Budget In this project, you will create a budget based on a job you are likely to have after you graduate. 1. Go online and research the average starting salary for the profession you are studying for. Use at least two sources. Be sure to record the web address from your search. 2. Approximate your monthly take-home pay. You may use the SmartAsset website to estimate this. 3. Use the 50-30-20 budget philosophy to determine how much you should budget for needs, wants and savings, or extra debt reduction. 4. Create a list of likely expenses. This list must include rent/mortgage, utilities, food, and school loan repayment. You may also want to include car payments, gasoline, and other items. 5. Categorize each expense as need, want, or savings. 6. Using the amounts found in step 3, decide how much to allocate to each of your expenses. It may help to quickly research how much rent is where you want to live. 7. Discuss the choices you had to make, and why you prioritized some expenses over others. ### Interest Rate and Time: What Is the Relationship? The interplay between interest rate and time for an annuity is not easily seen. How the amount that must be deposited per compounding period, , changes based on the time and interest rate would be useful to understand. In this project, you will explore this relationship. We will use a fixed future value of = $1,000,000 and a fixed number of periods per year, 12 (monthly compounding). With those, we’ll find various annuity payments that must be made to reach the goal. The annual interest rate for the investment is in the top row. The number of years for the investment is in the left column. In each cell (or box), find the monthly payment necessary to reach the goal of $1,000,000. Describe how the interest rates and number of years impact the payment necessary to reach the goal of $1,000,000. ### Finding a Home In this project you will identify a home you like, and then estimate the costs associated with that home. 1. Find a home in your region that you would like to buy using an online search of listings in your area. Zillow is a good place to begin. 2. Find the asking price for this home. Assume you would pay that price. 3. Find an estimation for closing costs in your area. Assume you finance those costs also. 4. Estimate the taxes to be paid on the home per year. It is likely that the online listing of the home has an estimate for the taxes for the house. 5. Use Google to determine the average homeowner’s insurance cost in your region. 6. Use the internet to determine the average interest rate for a 30-year mortgage. 7. Find how much you would pay per month, based on the answers to the previous questions, including the escrow payments for taxes and insurance. 8. Assume you will pay $50 per $100,000 borrowed in PMI. Add this to the monthly payment. ### Chapter Review ### Understanding Percent ### Discounts, Markups, and Sales Tax ### Simple Interest ### Compound Interest ### Making a Personal Budget ### Methods of Savings ### Investments ### The Basics of Loans ### Understanding Student Loans ### Credit Cards ### Buying or Leasing a Car ### Renting and Homeownership ### Income Tax ### Chapter Test

# Probability ## Introduction Casinos are big business; according to the American Gaming Association, commercial casinos in the United States brought in over $43 billion in revenue in 2019. Casinos must walk a fine line in order to be profitable. Their customers must lose more money than they win, on average, in order to stay in business. But if the chances of a single customer winning more money than they lose is too small, people will stop coming in the door to play the games. In this chapter, we'll study the techniques a casino must use to determine how likely it is that a customer will win a particular game, and then how the casino decides how much money a winner will rake in so that the customers are happy, but the casino also turns a profit in the long run. In order to figure out those likelihoods, we have to be able to somehow consider every possible outcome of these games. For example, in a game that involves players receiving 5 cards from a deck of 52, there are 2,598,960 possibilities for each player. We'll start off this chapter by learning how to count those possible outcomes.

# Probability ## The Multiplication Rule for Counting ### Learning Objectives After completing this section, you should be able to: 1. Apply the Multiplication Rule for Counting to solve problems. One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that method becomes onerous (or, for very large numbers, impossible for humans to accomplish in a typical human lifespan). So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical discipline of combinatorics, which is devoted to counting. ### Multiplication as a Combinatorial Short Cut One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects (): Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them (). If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., ), we can use this grouping to quickly see that there are 24 balls. Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns (Figure 7.5). Since there are 3 rows (cakes) and 4 columns (frostings), we have possible combinations. This is the reasoning behind the Multiplication Rule for Counting, which is also known as the Fundamental Counting Principle. This rule says that if there are ways to accomplish one task and ways to accomplish a second task, then there are ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product. ### Check Your Understanding ### Key Terms 1. combinatorics 2. Multiplication Rule for Counting (Fundamental Counting Principle) ### Key Concepts 1. The Multiplication Rule for Counting is used to count large sets.

# Probability ## Permutations ### Learning Objectives After completing this section, you should be able to: 1. Use the Multiplication Rule for Counting to determine the number of permutations. 2. Compute expressions containing factorials. 3. Compute permutations. 4. Apply permutations to solve problems. Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like this one, which relate to permutations. A permutation is an ordered list of objects taken from a given population. The length of the list is given, and the list cannot contain any repeated items. ### Applying the Multiplication Rule for Counting to Permutations In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are. ### Factorials The pattern we see in occurs commonly enough that we have a name for it: factorial. For any positive whole number , we define the factorial of (denoted and read " factorial") to be the product of every whole number less than or equal to . We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them. ### Permutations As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of objects and we wish to create an ordered list of of the objects (where ), we’ll call the number of those permutations (read “the number of permutations of objects taken at a time”). We formalize the formula we'll use to compute permutations below. If you wondered why we defined earlier, it was to make formulas like this one work; if we have objects and want to order all of them (so, we want the number of permutations of objects taken at a time), we get . Next, we’ll get some practice computing these permutations. ### Check Your Understanding ### Key Terms 1. permutation 2. factorial ### Key Concepts 1. Using the Multiplication Rule for Counting to enumerate permutations. 2. Simplifying and computing expressions involving factorials. 3. Using factorials to count permutations. ### Formulas 1.

# Probability ## Combinations ### Learning Objectives After completing this section, you should be able to: 1. Distinguish between permutation and combination uses. 2. Compute combinations. 3. Apply combinations to solve applications. In Permutations, we studied permutations, which we use to count the number of ways to generate an ordered list of a given length from a group of objects. An important property of permutations is that the order of the list matters: The results of a race and the selection of club officers are examples of lists where the order is important. In other situations, the order is not important. For example, in most card games where a player receives a hand of cards, the order in which the cards are received is irrelevant; in fact, players often rearrange the cards in a way that helps them keep the cards organized. ### Combinations: When Order Doesn’t Matter In situations in which the order of a list of objects doesn’t matter, the lists are no longer permutations. Instead, we call them combinations. ### Counting Combinations Permutations and combinations are certainly related, because they both involve choosing a subset of a large group. Let’s explore that connection, so that we can figure out how to use what we know about permutations to help us count combinations. We’ll take a basic example. How many ways can we select 3 letters from the group A, B, C, D, and E? If order matters, that number is . That’s small enough that we can list them all out in the table below. Now, let’s look back at that list and color-code it so that groupings of the same 3 letters get the same color, as shown in : After color-coding, we see that the 60 cells can be seen as 10 groups (colors) of 6. That’s no coincidence! We’ve already seen how to compute the number of permutations using the formula To compute the number of combinations, let’s count them another way using the Multiplication Rule for Counting. We’ll do this in two steps: Step 1: Choose 3 letters (paying no attention to order). Step 2: Put those letters in order. The number of ways to choose 3 letters from this group of 5 (A, B, C, D, E) is the number of combinations we’re looking for; let’s call that number (read “the number of combinations of 5 objects taken 3 at a time”). We can see from our chart that this is ten (the number of colors used). We can generalize our findings this way: remember that the number of permutations of things taken at a time is . That number is also equal to , and so it must be the case that . Dividing both sides of that equation by gives us the formula below. ### Check Your Understanding ### Key Terms 1. combination ### Key Concepts 1. Permutations are used to count subsets when order matters; combinations work when order doesn't matter. 2. Combinations can also be computed using factorials. ### Formulas 1. The formula for counting combinations is:

# Probability ## Tree Diagrams, Tables, and Outcomes ### Learning Objectives After completing this section, you should be able to: 1. Determine the sample space of single stage experiment. 2. Use tables to list possible outcomes of a multistage experiment. 3. Use tree diagrams to list possible outcomes of a multistage experiment. In the 19th century, an Augustinian friar and scientist named Gregor Mendel used his observations of pea plants to set out his theory of genetic propagation. In his work, he looked at the offspring that resulted from breeding plants with different characteristics together. For applications like this, it is often insufficient to only know in how many ways a process might end; we need to be able to list all of the possibilities. As we’ve seen, the number of possible outcomes can be very large! Thus, it’s important to have a strategy that allows us to systematically list these possibilities to make sure we don’t leave any out. In this section, we’ll look at two of these strategies. ### Single Stage Experiments When we are talking about combinatorics or probability, the word “experiment” has a slightly different meaning than it does in the sciences. Experiments can range from very simple (“flip a coin”) to very complex (“count the number of uranium atoms that undergo nuclear fission in a sample of a given size over the course of an hour”). Experiments have unknown outcomes that generally rely on something random, so that if the experiment is repeated (or replicated) the outcome might be different. No matter what the experiment, though, analysis of the experiment typically begins with identifying its sample space. The sample space of an experiment is the set of all of the possible outcomes of the experiment, so it’s often expressed as a set (i.e., as a list bound by braces; if the experiment is “randomly select a number between 1 and 4,” the sample space would be written ). ### Multistage Experiments Some experiments have more complicated sample spaces because they occur in stages. These stages can occur in succession (like drawing cards one at a time) or simultaneously (rolling 2 dice). Sample spaces get more complicated as the complexity of the experiment increases, so it’s important to choose a systematic method for identifying all of the possible outcomes. The first method we’ll discuss is the table. ### Using Tables to Find Sample Spaces Tables are useful for finding the sample space for experiments that meet two criteria: (1) The experiment must have only two stages, and (2) the outcomes of each stage must have no effect on the outcomes of the other. When the stages do not affect each other, we say the stages are independent. Otherwise, the stages are dependent and so we can’t use tables; we’ll look at a method for analyzing dependent stages soon. If you have a two-stage experiment with independent stages, a table is the most straightforward way to identify the sample space. To build a table, you list the outcomes of one stage of the experiment along the top of the table and the outcomes of the other stage down the side. The cells in the interior of the table are then filled using the outcomes associated with each cell’s row and column. Let’s look at an example. ### Using Tree Diagrams to Identify Sample Spaces In experiments where there are more than two stages, or where the stages are dependent, a tree diagram is a helpful tool for systematically identifying the sample space. Tree diagrams are built by first drawing a single point (or node), then from that node we draw one branch (a short line segment) for each outcome of the first stage. Each branch gets its own node at the other end (which we typically label with the corresponding outcome for that branch); from each of these, we draw another branch for each outcome of the second stage, assuming that the outcome of the first stage matches the branch we were on. If there are other stages, we can continue from there by continuing to add branches and nodes. This sounds really complicated, but it’s easier to understand through an example. ### Check Your Understanding ### Key Terms 1. experiment 2. replication 3. sample space 4. independent/dependent ### Key Concepts 1. We identify the sample space of an experiment by identifying all of its possible outcomes. 2. Tables can help us find a sample space by keeping the possible outcomes organized. 3. Tree diagrams provide a visualization of the sample space of an experiment that involves multiple stages.

# Probability ## Basic Concepts of Probability ### Learning Objectives After completing this section, you should be able to: 1. Define probability including impossible and certain events. 2. Calculate basic theoretical probabilities. 3. Calculate basic empirical probabilities. 4. Distinguish among theoretical, empirical, and subjective probability. 5. Calculate the probability of the complement of an event. It all comes down to this. The game of Monopoly that started hours ago is in the home stretch. Your sister has the dice, and if she rolls a 4, 5, or 7 she’ll land on one of your best spaces and the game will be over. How likely is it that the game will end on the next turn? Is it more likely than not? How can we measure that likelihood? This section addresses this question by introducing a way to measure uncertainty. ### Introducing Probability Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; for example, in the Monopoly example at the beginning of this section, we were interested in the roll of 2 dice that might fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event. In other words, an event is a subset of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter. Once we have defined the event that interests us, we can try to assess the likelihood of that event. We do that by assigning a number to each event () called the probability of that event (). The probability of an event is a number between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event with probability 1 is certain to occur. In general, the higher the probability of an event, the more likely it is that the event will occur. ### Three Ways to Assign Probabilities The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we do about those in-between cases, for events that might or might not occur? There are three methods to assign probabilities that we can choose from. We’ll discuss them here, in order of reliability. ### Method 1: Theoretical Probability The theoretical method gives the most reliable results, but it cannot always be used. If the sample space of an experiment consists of equally likely outcomes, then the theoretical probability of an event is defined to be the ratio of the number of outcomes in the event to the number of outcomes in the sample space. ### Method 2: Empirical Probability Theoretical probabilities are precise, but they can’t be found in every situation. If the outcomes in the sample space are not equally likely, then we’re out of luck. Suppose you’re watching a baseball game, and your favorite player is about to step up to the plate. What is the probability that he will get a hit? In this case, the sample space is {hit, not a hit}. That doesn’t mean that the probability of a hit is , since those outcomes aren’t equally likely. The theoretical method simply can’t be used in this situation. Instead, we might look at the player’s statistics up to this point in the season, and see that he has 122 hits in 531 opportunities. So, we might think that the probability of a hit in the next plate appearance would be about . When we use the outcomes of previous replications of an experiment to assign a probability to the next replication, we’re defining an empirical probability. Empirical probability is assigned using the outcomes of previous replications of an experiment by finding the ratio of the number of times in the previous replications the event occurred to the total number of previous replications. Empirical probabilities aren’t exact, but when the number of previous replications is large, we expect them to be close. Also, if the previous runs of the experiment are not conducted under the exact set of circumstances as the one we’re interested in, the empirical probability is less reliable. For instance, in the case of our favorite baseball player, we might try to get a better estimate of the probability of a hit by looking only at his history against left- or right-handed pitchers (depending on the handedness of the pitcher he’s about to face). ### Method 3: Subjective Probability In cases where theoretical probability can’t be used and we don’t have prior experience to inform an empirical probability, we’re left with one option: using our instincts to guess at a subjective probability. A subjective probability is an assignment of a probability to an event using only one’s instincts. Subjective probabilities are used in cases where an experiment can only be run once, or it hasn’t been run before. Because subjective probabilities may vary widely from person to person and they’re not based on any mathematical theory, we won’t give any examples. However, it’s important that we be able to identify a subjective probability when we see it; they will in general be far less accurate than empirical or theoretical probabilities. ### New Probabilities from Old: Complements One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will come later. Given an event , the complement of (denoted ) is the collection of all of the outcomes that are not in . (This is language that is taken from set theory, which you can learn more about elsewhere in this text.) Since every outcome in the sample space either is or is not in , it follows that . So, if the outcomes in are equally likely, we can compute theoretical probabilities and . Then, adding these last two equations, we get Thus, if we subtract from both sides, we can conclude that . Though we performed this calculation under the assumption that the outcomes in are all equally likely, the last equation is true in every situation. How is this helpful? Sometimes it is easier to compute the probability that an event won’t happen than it is to compute the probability that it will. To apply this principle, it’s helpful to review some tricks for dealing with inequalities. If an event is defined in terms of an inequality, the complement will be defined in terms of the opposite inequality: Both the direction and the inclusivity will be reversed, as shown in the table below. ### Check Your Understanding ### Key Terms 1. event 2. probability 3. theoretical probability 4. empirical probability 5. subjective probability ### Key Concepts 1. The theoretical probability of an event is the ratio of the number of equally likely outcomes in the event to the number of equally likely outcomes in the sample space. 2. Empirical probabilities are computed by repeating the experiment many times, and then dividing the number of replications that result in the event of interest by the total number of replications. 3. Subjective probabilities are assigned based on subjective criteria, usually because the experiment can’t be repeated and the outcomes in the sample space are not equally likely. 4. The probability of the complement of an event is found by subtracting the probability of the event from one. ### Formulas 1. For an experiment whose sample space consists of equally likely outcomes, the theoretical probability of the event is the ratio where and denote the number of outcomes in the event and in the sample space, respectively. 2.

# Probability ## Probability with Permutations and Combinations ### Learning Objectives After completing this section, you should be able to: 1. Calculate probabilities with permutations. 2. Calculate probabilities with combinations. In our earlier discussion of theoretical probabilities, the first step we took was to write out the sample space for the experiment in question. For many experiments, that method just isn’t practical. For example, we might want to find the probability of drawing a particular 5-card poker hand. Since there are 52 cards in a deck and the order of cards doesn’t matter, the sample space for this experiment has possible 5-card hands. Even if we had the patience and space to write them all out, sorting through the results to find the outcomes that fall in our event would be just as tedious. Luckily, the formula for theoretical probabilities doesn’t require us to know every outcome in the sample space; we just need to know how many outcomes there are. In this section, we’ll apply the techniques we learned earlier in the chapter (The Multiplication Rule for Counting, permutations, and combinations) to compute probabilities. ### Using Permutations to Compute Probabilities Recall that we can use permutations to count how many ways there are to put a number of items from a list in order. If we’re looking at an experiment whose sample space looks like an ordered list, then permutations can help us to find the right probabilities. ### Combinations to Computer Probabilities If the sample space of our experiment is one in which order doesn’t matter, then we can use combinations to find the number of outcomes in that sample space. ### Check Your Understanding ### Key Concepts 1. We use permutations and combinations to count the number of equally likely outcomes in an event and in a sample space, which allows us to compute theoretical probabilities.

# Probability ## What Are the Odds? ### Learning Objectives After completing this section, you should be able to: 1. Compute odds. 2. Determine odds from probabilities. 3. Determine probabilities from odds. A particular lottery instant-win game has 2 million tickets available. Of those, 500,000 win a prize. If there are 500,000 winners, then it follows that there are 1,500,000 losing tickets. When we evaluate the risk associated with a game like this, it can be useful to compare the number of ways to win the game to the number of ways to lose. In the case of this game, we would compare the 500,000 wins to the 1,500,000 losses. In other words, there are 3 losing tickets for every winning ticket. Comparisons of this type are the focus of this section. ### Computing Odds The ratio of the number of equally likely outcomes in an event to the number of equally likely outcomes not in the event is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the event to the number in the event to the number in the event is called the odds against the event. ### Odds as a Ratio of Probabilities We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with 500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is , and the probability of losing is . Notice that the ratio of the probability of winning to the probability of losing is , which matches the odds in favor of winning. We can use these formulas to convert probabilities to odds, and vice versa. Now, let’s convert odds to probabilities. Let’s say the odds for an event are . Then, using the formula above, we have . Converting to fractions and solving for , we get: Let’s put this result in a formula we can use. ### Check Your Understanding ### Key Terms 1. odds (for/against) ### Key Concepts 1. Odds are computed as the ratio of the probability of an event to the probability of its compliment. ### Formulas 1. For an event , 2. If the odds in favor of are , then .

# Probability ## The Addition Rule for Probability ### Learning Objectives After completing this section, you should be able to: 1. Identify mutually exclusive events. 2. Apply the Addition Rule to compute probability. 3. Use the Inclusion/Exclusion Principle to compute probability. Up to this point, we have looked at the probabilities of simple events. Simple events are those with a single, simple characterization. Sometimes, though, we want to investigate more complicated situations. For example, if we are choosing a college student at random, we might want to find the probability that the chosen student is a varsity athlete or in a Greek organization. This is a compound event: there are two possible criteria that might be met. We might instead try to identify the probability that the chosen student is both a varsity athlete and in a Greek organization. In this section and the next, we’ll cover probabilities of two types of compound events: those build using “or” and those built using “and.” We’ll deal with the former first. ### Mutual Exclusivity Before we get to the key techniques of this section, we must first introduce some new terminology. Let’s say you’re drawing a card from a standard deck. We’ll consider 3 events: is the event “the card is a ,” is the event “the card is a 10,” and is the event “the card is a .” If the card drawn is , then and didn’t occur, but did. If the card drawn is instead , then didn’t occur, but both and did. We can see from these examples that, if we are interested in several possible events, more than one of them can occur simultaneously (both and , for example). But, if you think about all the possible outcomes, you can see that and can never occur simultaneously; there are no cards in the deck that are both and . Pairs of events that cannot both occur simultaneously are called mutually exclusive. Let’s go through an example to help us better understand this concept. ### The Addition Rule for Mutually Exclusive Events If two events are mutually exclusive, then we can use addition to find the probability that one or the other event occurs. Why does this formula work? Let’s consider a basic example. Suppose we’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. What is the probability of drawing an E or an S? Since 3 of the tiles are marked with E and 2 are marked with S, there are 5 tiles that satisfy the criteria. There are ten tiles in the bag, so the probability is . Notice that the probability of drawing an E is and the probability of drawing an S is ; adding those together, we get . Look at the numerators in the fractions involved in the sum: the 3 represents the number of E tiles and the 2 is the number of S tiles. This is why the Addition Rule works: The total number of outcomes in one event or the other is the sum of the numbers of outcomes in each of the individual events. ### Finding Probabilities When Events Aren’t Mutually Exclusive Let’s return to the example we used to explore the Addition Rule: We’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. Consider these events: is “draw a vowel” and is “draw a letter that comes after L in the alphabet.” Since there are 6 vowels, . There are 4 tiles with letters that come after L alphabetically, so . What is ? If we blindly apply the Addition Rule, we get , which would mean that the compound event or is certain. However, it’s possible to draw a B, in which case neither nor happens. Where’s the error? The events are not mutually exclusive: the outcome U belongs to both events, and so the Addition Rule doesn’t apply. However, there’s a way to extend the Addition Rule to allow us to find this probability anyway; it’s called the Inclusion/Exclusion Principle. In this example, if we just add the two probabilities together, the outcome U is included in the sum twice: It’s one of the 6 outcomes represented in the numerator of , and it’s one of the 4 outcomes represented in the numerator of . So, that particular outcome has been “double counted.” Since it has been included twice, we can get a true accounting by excluding it once: . We can generalize this idea to a formula that we can apply to find the probability of any compound event built using “or.” It’s worth noting that this formula is truly an extension of the Addition Rule. Remember that the Addition Rule requires that the events and are mutually exclusive. In that case, the compound event is impossible, and so . So, in cases where the events in question are mutually exclusive, the Inclusion/Exclusion Principle reduces to the Addition Rule. ### Check Your Understanding ### Key Terms 1. mutually exclusive ### Key Concepts 1. The Addition Rule is used to find the probability that one event or another will occur when those events are mutually exclusive. 2. The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive. ### Formulas 1. If and are mutually exclusive events, then 2. If and are events that contain outcomes of a single experiment, then

# Probability ## Conditional Probability and the Multiplication Rule ### Learning Objectives After completing this section, you should be able to: 1. Calculate conditional probabilities. 2. Apply the Multiplication Rule for Probability to compute probabilities. Back in , we constructed the following table () to help us find the probabilities associated with rolling two standard 6-sided dice: For example, 3 of these 36 equally likely outcomes correspond to rolling a sum of 10, so the probability of rolling a 10 is . However, if you choose to roll the dice one at a time, the probability of rolling a 10 will change after the first die comes to rest. For example, if the first die shows a 5, then the probability of rolling a sum of 10 has jumped to —the event will occur if the second die also shows a 5, which is 1 of 6 equally likely outcomes for the second die. If instead the first die shows a 3, then the probability of rolling a sum of 10 drops to 0—there are no outcomes for the second die that will give us a sum of 10. Understanding how probabilities can shift as we learn new information is critical in the analysis of our second type of compound events: those built with “and.” This section will explain how to compute probabilities of those compound events. ### Conditional Probabilities When we analyze experiments with multiple stages, we often update the probabilities of the possible final outcomes or the later stages of the experiment based on the results of one or more of the initial stages. These updated probabilities are called conditional probabilities. In other words, if is a possible outcome of the first stage in a multistage experiment, then the probability of an event conditional on (denoted , read “the probability of given ”) is the updated probability of under the assumption that occurred. In the example that opened this section, we might consider rolling two dice as a multistage experiment: rolling one, then the other. If we define to be the event “roll a sum of 10,” to be the event “first die shows 5,” and to be the event “first die shows 3,” then we computed , , and . ### Compound Events Using “And” and the Multiplication Rule For multistage experiments, the outcomes of the experiment as a whole are often stated in terms of the outcomes of the individual stages. Commonly, those statements are joined with “and.” For example, in the sock drawer example just above, one outcome might be “the left sock is black and the right sock is blue.” As with “or” compound events, these probabilities can be computed with basic arithmetic. It is often useful to combine the rules we’ve seen so far with the techniques we used for finding sample spaces. In particular, trees can be helpful when we want to identify the probabilities of every possible outcome in a multistage experiment. The next example will illustrate this. ### Check Your Understanding ### Key Terms 1. conditional probability ### Key Concepts 1. Conditional probabilities are computed under the assumption that the condition has already occurred. 2. The Multiplication Rule for Probability is used to find the probability that two events occur in sequence. ### Formulas 1. If and are events associated with the first and second stages of an experiment, then .

# Probability ## The Binomial Distribution ### Learning Objectives After completing this section, you should be able to: 1. Identify binomial experiments. 2. Use the binomial distribution to analyze binomial experiments. It’s time for the World Series, which determines the champion for this season in Major League Baseball. The scrappy Los Angeles Angels are facing the powerhouse Cincinnati Reds. Computer models put the chances of the Reds winning any single game against the Angels at about 65%. The World Series, as its name implies, isn’t just one game, though: it’s what’s known as a “best-of-seven” contest: the teams play each other repeatedly until one team wins 4 games (which could take up to 7 games total, if each team wins three of the first 6 matchups). If the Reds truly have a 65% chance of winning a single game, then the probability that they win the series should be greater than 65%. Exactly how much bigger? If you have the patience for it, you could use a tree diagram like we used in to trace out all of the possible outcomes, find all the related probabilities, and add up the ones that result in the Reds winning the series. Such a tree diagram would have final nodes, though, so the calculations would be very tedious. Fortunately, we have tools at our disposal that allow us to find these probabilities very quickly. This section will introduce those tools and explain their use. ### Binomial Experiments The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this category. Experiments that satisfy each of these criteria are called binomial experiments. A binomial experiment is an experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of success. ### Repeated Binomial Trials The first criterion involves the structure of the stages. Each stage of the experiment should be a replication of every other stage; we call these replications trials. An example of this is flipping a coin 10 times; each of the ten flips is a trial, and they all occur under the same conditions as every other. Further, each trial must have only two possible outcomes. These two outcomes are typically labeled “success” and “failure,” even if there is not a positive or negative connotation associated with those outcomes. Experiments with more than two outcomes in their sample spaces are sometimes reconsidered in a way that forces just two outcomes; all we need to do is completely divide the sample space into two parts that we can label “success” and “failure.” For example, your grade on an exam might be recorded as A, B, C, D, or F, but we could instead think of the grades A, B, C, and D as “success” and a grade of F as “failure.” Trials with only two outcomes are called binomial trials (the word binomial derives from Latin and Greek roots that mean “two parts”). ### Independent Trials The next criterion that we’ll be looking for is independence of trials. Back in Tree Diagrams, Tables, and Outcomes, we said that two stages of an experiment are independent if the outcome of one stage doesn’t affect the other stage. Independence is necessary for the experiments we want to analyze in this section. ### Fixed Number of Trials Next, we require that the number of trials in the experiment be decided before the experiment begins. For example, we might say “flip a coin 10 times.” The number of trials there is fixed at 10. However, if we say “flip a coin until you get 5 heads,” then the number of trials could be as low as 5, but theoretically it could be 50 or a 100 (or more)! We can’t apply the tools from this section in cases where the number of trials is indeterminate. ### Constant Probability The next criterion needed for binomial experiments is related to the independence of the trials. We must make sure that the probability of success in each trial is the same as the probability of success in every other trial. ### The Binomial Formula If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that? Each of these questions is a question about the number of successes in a binomial experiment (flip a coin 100 times, “success” is flipping heads). We could theoretically use the techniques we’ve seen in earlier sections to answer each of these, but the number of calculations we’d have to do is astronomical; just building the tree diagram that represents this situation is more than we could complete in a lifetime; it would have final nodes! To put that number in perspective, if we could draw 1,000 dots every second, and we started at the moment of the Big Bang, we’d currently be about 0.00000003% of the way to drawing out those final nodes. Luckily, there’s a shortcut called the Binomial Formula that allows us to get around doing all those calculations! We can use this formula to answer one of our questions about 100 coin flips. What is the probability of flipping exactly 50 heads? In this case, , , and , so . Unfortunately, many calculators will balk at this calculation; that first factor () is an enormous number, and the other two factors are very close to zero. Even if your calculator can handle numbers that large or small, the arithmetic can create serious errors in rounding off. ### The Binomial Distribution If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a complete list of the possible outcomes in one column and the probability in the other. For example, suppose Kristen is practicing her basketball free throws. Assume Kristen always makes 82% of those shots. If she attempts 5 free throws, then the Binomial Formula gives us these probabilities: A table that lists all possible outcomes of an experiment along with the probabilities of those outcomes is an example of a probability density function (PDF). A PDF may also be a formula that you can use to find the probability of any outcome of an experiment. If we want to know the probability of a range of outcomes, we could add up the corresponding probabilities. Going back to Kristen’s free throws, we can find the probability that she makes 3 or fewer of her 5 attempts by adding up the probabilities associated with the corresponding outcomes (in this case: 0, 1, 2, or 3): The probability that the outcome of an experiment is less than or equal to a given number is called a cumulative probability. A table of the cumulative probabilities of all possible outcomes of an experiment is an example of a cumulative distribution function (CDF). A CDF may also be a formula that you can use to find those cumulative probabilities. Here are the PDF and CDF for Kristen’s free throws: Finally, we can answer the question posed at the beginning of this section. Remember that the Reds are facing the Angels in the World Series, which is won by the team who is first to win 4 games. The Reds have a 65% chance to win any game against the Angels. So, what is the probability that the Reds win the World Series? At first glance, this is not a binomial experiment: The number of games played is not fixed, since the series ends as soon as one team wins 4 games. However, we can extend this situation to a binomial experiment: Let’s assume that 7 games are always played in the World Series, and the winner is the team who wins more games. In a way, this is what happens in reality; it’s as though the first team to lose 4 games (and thus cannot win more than the other team) forfeits the rest of their games. So, we can treat the actual World Series as a binomial experiment with seven trials. If is the number of games won by the Reds, the probability that the Reds win the World Series is . Using the techniques from the last example, we get . ### Check Your Understanding ### Key Terms 1. binomial experiment 2. probability density function (PDF) 3. cumulative distribution function (CDF) ### Key Concepts 1. Binomial experiments result when we count the number of successful outcomes in a fixed number of repeated, independent trials with a constant probability of success. 2. The binomial distribution is used to find probabilities associated with binomial experiments. 3. Probability density functions (PDFs) describe the probabilities of individual outcomes in an experiment; cumulative distribution functions (CDFs) give the probabilities of ranges of outcomes. ### Formulas 1. Suppose we have a binomial experiment with trials and the probability of success in each trial is . Then:

# Probability ## Expected Value ### Learning Objectives After completing this section, you should be able to: 1. Calculate the expected value of an experiment. 2. Interpret the expected value of an experiment. 3. Use expected value to analyze applications. The casino game roulette has dozens of different bets that can be made. These bets have different probabilities of winning but also have different payouts. In general, the lower the probability of winning a bet is, the more money a player wins for that bet. With so many options, is there one bet that’s “smarter” than the rest? What’s the best play to make at a roulette table? In this section, we’ll develop the tools we need to answer these questions. ### Expected Value Many experiments have numbers associated with their outcomes. Some are easy to define; if you roll 2 dice, the sum of the numbers showing is a good example. In some card games, cards have different point values associated with them; for example, in some forms of the game rummy, aces are worth 15 points; 10s, jacks, queens, and kings are worth 10; and all other cards are worth 5. The outcomes of casino and lottery games are all associated with an amount of money won or lost. These outcome values are used to find the expected value of an experiment: the mean of the values associated with the outcomes that we would observe over a large number of repetitions of the experiment. (See Conditional Probability and the Multiplication Rule for more on means.) That definition is a little vague; How many is “a large number?” In practice, it depends on the experiment; the number has to be large enough that every outcome would be expected to appear at least a few times. For example, if we’re talking about rolling a standard 6-sided die and we note the number showing, a few dozen replications should be enough that the mean would be representative. Since the probability of each outcome is , we would expect to see each outcome about 8 times over the course of 48 replications. However, if we’re talking about the Powerball lottery, where the probability of winning the jackpot is about , we would need several billion replications to ensure that every outcome appears a few times. Luckily, we can find the theoretical expected value before we even run the experiment the first time. Let’s make note of some things we can learn from . First, as Exercises 1 and 3 demonstrate, the expected value of an experiment might not be a value that could come up in the experiment. Remember that the expected value is interpreted as a mean, and the mean of a collection of numbers doesn’t have to actually be one of those numbers. Second, looking at Exercise 1, the expected value (3.5) was just the mean of the numbers on the faces of the die: . This is no accident! If we break that fraction up using the addition in the numerator, we get , which we can rewrite as . That’s exactly the computation we did to find the expected value! In fact, expected values can always be treated as a special kind of mean called a weighted mean, where the weights are the probabilities associated with each value. When the probabilities are all equal, the weighted mean is just the regular mean. ### Interpreting Expected Values As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example. ### Using Expected Value Now that we know how to find and interpret expected values, we can turn our attention to using them. Suppose someone offers to play a game with you. If you roll a die and get a 6, you get $10. However, if you get a 5 or below, you lose $1. Is this a game you’d want to play? Let’s look at the expected value: The probability of winning is and the probability of losing is , so the expected value is . That means, on average, you’ll come out ahead by about 83 cents every time you play this game. It’s a great deal! On the other hand, if the winnings for rolling a 6 drop to $3, the expected value becomes , meaning you should expect to lose about 33 cents on average for every time you play. Playing that game is not a good idea! In general, this is how casinos and lottery corporations make money: Every game has a negative expected value for the player. ### Check Your Understanding ### Key Terms 1. expected value ### Key Concepts 1. The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with their corresponding probabilities. 2. The expected value of an experiment is the most likely value of the average of a large number of replications of the experiment. ### Formulas 1. If represents an outcome of an experiment and represents the value of that outcome, then the expected value of the experiment is: where ### Projects 1. The Binomial Distribution is one of many examples of a discrete probability distribution. Other examples include the Geometric, Hypergeometric, Multinomial, Poisson, and Negative Binomial Distributions. Choose one of these distributions, and find out what makes it different from the Binomial Distribution. In what situations can it be applied? How is it used? Once you have an idea of how it’s used, write a series of five questions like the ones in this chapter that can be answered with that distribution, and find the answers. 2. Binomial is a word that also comes up in algebra; the word describes polynomials with two terms. At first glance, there isn’t much to indicate that these two uses of the word are related, but it turns out there is a connection. Explore the connection between the Binomial Distribution and the algebraic concept of binomial expansion, (the process of multiplying out expressions like for a positive whole number ). Search for a connection with the mathematical object known as Pascal’s Triangle. 3. Hazard is a dice game that was mentioned in Chaucer’s Canterbury Tales. It was a popular game of chance played in taverns and coffee houses well into the 18th century; its popularity at the time of the foundation of probability theory means that it was a common example in early texts on finding expected values and probabilities. Find the rules of the game, and get some practice playing it. Then, analyze the choices that the caster gets to make, and decide which is most advantageous, using the language of expected values. ### Chapter Review ### The Multiplication Rule for Counting ### Permutations ### Combinations ### Tree Diagrams, Tables, and Outcomes ### Basic Concepts of Probability ### Probability with Permutations and Combinations ### What Are the Odds? ### The Addition Rule for Probability ### Conditional Probability and the Multiplication Rule ### The Binomial Distribution ### Expected Value ### Chapter Test

# Statistics ## Introduction Before the 2021 WNBA season, professional basketball player Candace Parker signed a contract with the Chicago Sky, which entitled her to a salary of $190,000. This amount was the 23rd highest in the league at the time. How did the team’s management decide on her salary? They likely considered some intangible qualities, like her leadership skills. However, much of their deliberations probably took into account her performance on the court. For example, Parker led the league in rebounds in the 2020 season (214 of them) and scored 14.7 points per game (which ranked her 18th among all WNBA players). Further, Parker brought 13 seasons of experience to the team. All of these factors played a role in deciding the terms of her contract. Estimating the value of one variable (like salary) based on other, measurable variables (points per game, experience, rebounds, etc.) is among the most important applications of statistics, which is the mathematical field devoted to gathering, organizing, summarizing, and making decisions based on data.

# Statistics ## Gathering and Organizing Data ### Learning Objectives After completing this section, you should be able to: 1. Distinguish among sampling techniques. 2. Organize data using an appropriate method. 3. Create frequency distributions. When a polling organization wants to try to establish which candidate will win an upcoming election, the first steps are to write questions for the survey and to choose which people will be asked to respond to the survey. These can seem like simple steps, but they have far-reaching implications in the analysis the pollsters will later carry out. The process by which samples (or groups of units from which we collect data) are chosen can strongly affect the data that are collected. Units are anything that can be measured or surveyed (such as people, animals, objectives, or experiments) and data are observations made on units. One of the most famous failures of good sampling occurred in the first half of the twentieth century. The Literary Digest was among the most respected magazines of the early twentieth century. Despite the name, the Digest was a weekly newsmagazine. Starting in 1916, the Digest conducted a poll to try to predict the winner of each US Presidential election. For the most part, their results were good; they correctly predicted the outcome of all five elections between 1916 and 1932. In 1936, the incumbent President Franklin Delano Roosevelt faced Kansas governor Alf Landon, and once again the Digest ran their famous poll, with results published the week before the election. Their conclusion? Landon would win in a landslide, 57% to 43%. Once the actual votes had been counted, though, Roosevelt ended up with 61% of the popular vote, 18% more than the poll predicted. What went wrong? The short answer is that the people who were chosen to receive the survey (over ten million of them!) were not a good representation of the population of voting adults. The sample was chosen using the Digest's own base of subscribers as well as publicly available lists of people that were likely adults (and therefore eligible to vote), mostly phone books and vehicle registration records. The pollsters then mailed every single person on these lists a survey. Around a quarter of those surveys were returned; this constituted the sample that was used to make the Digest’s disastrously incorrect prediction. However, the Digest made an error in failing to consider that the election was happening during the Great Depression, and only the wealthy had disposable income to spend on telephone lines, automobiles, and magazine subscriptions. Thus, only the wealthy were sent the Digest’s survey. Since Roosevelt was extremely popular among poorer voters, many of Roosevelt’s supporters were excluded from the Digest’s sample. Another more complicated factor was the low response rate; only around 25% of the surveys were returned. This created what’s called a non-response bias. ### Sampling and Gathering Data The Digest's failure highlights the need for what is now considered the most important criterion for sampling: randomness. This randomness can be achieved in several ways. Here we cover some of the most common. A simple random sample is chosen in a way that every unit in the population has an equal chance of being selected, and the chances of a unit being selected do not depend on the units already chosen. An example of this is choosing a group of people by drawing names out of a hat (assuming the names are well-mixed in the hat). A systematic random sample is selected from an ordered list of the population (for example, names sorted alphabetically or students listed by student ID). First, we decide what proportion of the population will be in our sample. We want to express that proportion as a fraction with 1 in the numerator. Let’s call that number D. Next, we’ll choose a random number between one and D. The unit at that position will go into our sample. We’ll find the rest of our sample by choosing every Dth unit in the list, starting with our random number. To walk through an example, let’s say we want to sample 2% of the population: . (Note: If the number in the denominator isn’t a whole number, we can just round it off. This part of the process doesn’t have to be precise.) We can then use a random number generator to find a random number between 1 and 50; let's use 31. In our example, our sample would then be the units in the list at positions 31, 81 (31 + 50), 131 (81 + 50), and so forth. A stratified sample is one chosen so that particular groups in the population are certain to be represented. Let’s say you are studying the population of students in a large high school (where the grades run from 9th to 12th), and you want to choose a sample of 12 students. If you use a simple or systematic random sample, there’s a pretty good chance that you’ll miss one grade completely. In a stratified sample, you would first divide the population into groups (the strata), then take a random sample within each stratum (that’s the singular form of “strata”). In the high school example, we could divide the population into grades, then take a random sample of three students within each grade. That would get us to the 12 students we need while ensuring coverage of each grade. A cluster sample is a sample where clusters of units are chosen at random, instead of choosing individual units. For example, if we need a sample of college students, we may take a list of all the course sections being offered at the college, choose three of them at random (the sections are the clusters), and then survey all the students in those sections. A sample like this one has the advantage of convenience: If the survey needs to be administered in person, many of your sample units will be located in one place at the same time. ### Organizing Data Once data have been collected, we turn our attention to analysis. Before we analyze, though, it’s useful to reorganize the data into a format that makes the analysis easier. For example, if our data were collected using a paper survey, our raw data are all broken down by respondent (represented by an individual response sheet). To perform an analysis on all the responses to an individual question, we need to first group all the responses to each question together. The way we organize the data depends on the type of data we’ve collected. There are two broad types of data: categorical and quantitative. Categorical data classifies the unit into a group (or category). Examples of categorical data include a response to a yes-or-no question, or the color of a person’s eyes. Quantitative data is a numerical measure of a property of a unit. Examples of quantitative data include the time it takes for a rat to run through a maze or a person’s daily calorie intake. We’ll look at each type of data in turn when considering how best to organize. ### Categorical Data Organization The best way to organize categorical data is using a categorical frequency distribution. A categorical frequency distribution is a table with two columns. The first contains all the categories present in the data, each listed once. The second contains the frequencies of each category, which are just a count of how often each category appears in the data. ### Quantitative Data We have a couple of options available for organizing quantitative data. If there are just a few possible responses, we can create a frequency distribution just like the ones we made for categorical data above. For example, if we’re surveying a group of high school students and we ask for each student’s age, we’ll likely only get whole-number responses between 13 and 19. Since there are only around seven (and likely fewer) possible responses, we can treat the data as if they’re categorical and create a frequency distribution as before. If there are many possible responses, a frequency distribution table like the ones we’ve seen so far isn’t really useful; there will likely be many responses with a frequency of one, which means the table will be no better than looking at the raw data. In these cases, we can create a binned frequency distribution. A binned frequency distribution groups the data into ranges of values called bins, then records the number of responses in each bin. For example, if we have height data for individuals measured in centimeters, we might create bins like 150–155 cm, 155–160 cm, and so forth (making sure that every data value falls into a bin). We must be careful, though; in this scenario, it’s not clear which bin would contain a response of 155 cm. Usually, responses on the edge of a bin are placed in the higher bin, but it’s good practice to make that clear. In cases where responses are rounded off, you can avoid this issue by leaving a gap between the bins that couldn’t contain any responses. In our example, if the measurements were all rounded off to the nearest centimeter, we could make bins like 150–154 cm, 155–159 cm, etc. (since a response like 154.2 isn’t possible). We’ll use this method going forward. How do we decide what the boundaries of our bins should be? There’s no one right way to do that, but there are some guidelines that can be helpful. 1. Every data value should fall into exactly one bin. For example, if the lowest value in our data is 42, the lowest bin should not be 45–49. 2. Every bin should have the same width. Note that if we shift the upper limits of our bins down a bit to avoid ambiguity (like described above), we can’t simply subtract the lower limit from the upper limit to get the bin width; instead, we subtract the lower limit of the bin from the lower limit of the next bin. For example, if we’re looking at GPAs rounded to the nearest hundredth, we might choose bins like 2.00–2.24, 2.25–2.49, 2.50–2.74, etc. These bins all have a width of 0.25. 3. If the minimum or maximum value of the data falls right on the boundary between two bins, then it’s OK to bend the rule just a little in order to avoid having an additional bin containing just that one value. We’ll see an example of this in just a moment. 4. If we have too many or too few bins, it can be difficult to get a good sense of the distribution. Seven or eight bins is ideal, but that’s not a firm rule; anything between five and twelve is fine. We often choose the number of bins so that the widths are round numbers. ### Check Your Understanding ### Key Terms 1. sample 2. units 3. data 4. population 5. simple random sample 6. systematic random sample 7. stratified sample 8. cluster sample 9. quantitative data 10. categorical data 11. categorical frequency Distribution 12. binned frequency distribution ### Key Concepts 1. Categorical data places units into groups (categories), while quantitative data is a numerical measure of a property of a unit. 2. The sampling method for a study depends on the way that randomization is used to select units for the sample. 3. Frequency distributions help to summarize data by counting the number of units that fall into a particular category or range of quantitative values.

# Statistics ## Visualizing Data ### Learning Objectives After completing this section, you should be able to: 1. Create charts and graphs to appropriately represent data. 2. Interpret visual representations of data. 3. Determine misleading components in data displayed visually. Summarizing raw data is the first step we must take when we want to communicate the results of a study or experiment to a broad audience. However, even organized data can be difficult to read; for example, if a frequency table is large, it can be tough to compare the first row to the last row. As the old saying goes: a picture is worth a thousand words (or, in this case, summary statistics)! Just as our techniques for organizing data depended on the type of data we were looking at, the methods we’ll use for creating visualizations will vary. Let’s start by considering categorical data. ### Visualizing Categorical Data If the data we’re visualizing is categorical, then we want a quick way to represent graphically the relative numbers of units that fall in each category. When we created the frequency distributions in the last section, all we did was count the number of units in each category and record that number (this was the frequency of that category). Frequencies are nice when we’re organizing and summarizing data; they’re easy to compute, and they’re always whole numbers. But they can be difficult to understand for an outsider who’s being introduced to your data. Let’s consider a quick example. Suppose you surveyed some people and asked for their favorite color. You communicated your results using a frequency distribution. Jerry is interested in data on favorite colors, so he reads your frequency distribution. The first row shows that twelve people indicated green was their favorite color. However, Jerry has no way of knowing if that’s a lot of people without knowing how many people total took your survey. Twelve is a pretty significant number if only twenty-five people took the survey, but it’s next to nothing if you recorded a thousand responses. For that reason, we will often summarize categorical data not with frequencies, but with proportions. The proportion of data that fall into a particular category is computed by dividing the frequency for that category by the total number of units in the data. Proportions can be expressed as fractions, decimals, or percentages. Now that we can compute proportions, let’s turn to visualizations. There are two primary visualizations that we’ll use for categorical data: bar charts and pie charts. Both of these data representations work on the same principle: If proportions are represented as areas, then it’s easy to compare two proportions by assessing the corresponding areas. Let’s look at bar charts first. ### Bar Charts A bar chart is a visualization of categorical data that consists of a series of rectangles arranged side-by-side (but not touching). Each rectangle corresponds to one of the categories. All of the rectangles have the same width. The height of each rectangle corresponds to either the number of units in the corresponding category or the proportion of the total units that fall into the category. In practice, most graphs are now made with computers. You can use Google Sheets, which is available for free from any web browser. Now that we’ve explored how bar graphs are made, let’s get some practice reading bar graphs. ### Pie Charts A pie chart consists of a circle divided into wedges, with each wedge corresponding to a category. The proportion of the area of the entire circle that each wedge represents corresponds to the proportion of the data in that category. Pie charts are difficult to make without technology because they require careful measurements of angles and precise circles, both of which are tasks better left to computers. Pie charts are sometimes embellished with features like labels in the slices (which might be the categories, the frequencies in each category, or the proportions in each category) or a legend that explains which colors correspond to which categories. When making your own pie chart, you can decide which of those to include. The only rule is that there has to be some way to connect the slices to the categories (either through labels or a legend). ### Visualizing Quantitative Data There are several good ways to visualize quantitative data. In this section, we’ll talk about two types: stem-and-leaf plots and histograms. ### Stem-and-Leaf Plots Stem-and-leaf plots are visualization tools that fall somewhere between a list of all the raw data and a graph. A stem-and-leaf plot consists of a list of stems on the left and the corresponding leaves on the right, separated by a line. The stems are the numbers that make up the data only up to the next-to-last digit, and the leaves are the final digits. There is one leaf for every data value (which means that leaves may be repeated), and the leaves should be evenly spaced across all stems. These plots are really nothing more than a fancy way of listing out all the raw data; as a result, they shouldn’t be used to visualize large datasets. This concept can be difficult to understand without referencing an example, so let’s first look at how to read a stem-and-leaf plot. Stem-and-leaf plots are useful in that they give us a sense of the shape of the data. Are the data evenly spread out over the stems, or are some stems “heavier” with leaves? Are the heavy stems on the low side, the high side, or somewhere in the middle? These are questions about the distribution of the data, or how the data are spread out over the range of possible values. Some words we use to describe distributions are uniform (data are equally distributed across the range), symmetric (data are bunched up in the middle, then taper off in the same way above and below the middle), left-skewed (data are bunched up at the high end or larger values, and taper off toward the low end or smaller values), and right-skewed (data are bunched up at the low end, and taper off toward the high end). See below figures. Looking back at the stem-and-leaf plot in the previous example, we can see that the data are bunched up at the low end and taper off toward the high end; that set of data is right-skewed. Knowing the distribution of a set of data gives us useful information about the property that the data are measuring. Now that we have a better idea of how to read a stem-and-leaf plot, we’re ready to create our own. As we mentioned above, stem-and-leaf plots aren’t always going to be useful. For example, if all the data in your dataset are between 20 and 29, then you’ll just have one stem, which isn’t terribly useful. (Although there are methods like stem splitting for addressing that particular problem, we won’t go into those at this time.) On the other end of the spectrum, the data may be so spread out that every stem has only one leaf. (This problem can sometimes be addressed by rounding off the data values to the tens, hundreds, or some other place value, then using that place for the leaves.) Finally, if you have dozens or hundreds (or more) of data values, then a stem-and-leaf plot becomes too unwieldy to be useful. Fortunately, we have other tools we can use. ### Histograms Histograms are visualizations that can be used for any set of quantitative data, no matter how big or spread out. They differ from a categorical bar chart in that the horizontal axis is labeled with numbers (not ranges of numbers), and the bars are drawn so that they touch each other. The heights of the bars reflect the frequencies in each bin. Unlike with stem-and-leaf plots, we cannot recreate the original dataset from a histogram. However, histograms are easy to make with technology and are great for identifying the distribution of our data. Let’s first create one histogram without technology to help us better understand how histograms work. Now that we’ve seen the connection between stem-and-leaf plots and histograms, we are ready to look at how we can use Google Sheets to build histograms. Let’s use Google Sheets to create a histogram for a large dataset. ### Bar Charts for Labeled Data Sometimes we have quantitative data where each value is labeled according to the source of the data. For example, in the Your Turn above, you looked at in-state tuition data. Every value you used to create that histogram was associated with a school; the schools are the labels. In YOUR TURN 8.11, you found a histogram of the wins of every Major League Baseball team in 2019. Each of those win totals had a label: the team. If we’re interested in visualizing differences among the different teams, or schools, or whatever the labels are, we create a different version of the bar graph known as a bar chart for labeled data. These graphs are made in Google Sheets in exactly the same way as regular bar graphs. The only change is that the vertical axis will be labeled with the units for your quantitative data instead of just “Frequency.” ### Misleading Graphs Graphical representations of data can be manipulated in ways that intentionally mislead the reader. There are two primary ways this can be done: by manipulating the scales on the axes and by manipulating or misrepresenting areas of bars. Let’s look at some examples of these. ### Check Your Understanding ### Key Terms 1. proportion 2. bar chart 3. pie chart 4. stem-and-leaf plot 5. distribution (of quantitative data) 6. histogram 7. bar chart for labeled data ### Key Concepts 1. Categorical data can be visualized using pie charts or bar charts; quantitative data can be visualized using stem-and-leaf plots or histograms. 2. Areas in pie charts and bar charts represent proportions of the data falling into a particular category, while areas in histograms represent proportions of the data that fall into a given range of data values (or “bins”). Stem-and-leaf plots are visual representations of entire datasets. 3. By manipulating the axes, changing widths of bars, or making bad choices for bins, we can create data visualizations that misrepresent the distribution of data. ### Videos 1. Make a Simple Bar Graph in Google Sheets 2. Create Pie Charts Using Google Sheets 3. Make a Histogram Using Google Sheets 4. How to Spot a Misleading Graph

# Statistics ## Mean, Median and Mode ### Learning Objectives After completing this section, you should be able to: 1. Calculate the mode of a dataset. 2. Calculate the median of a dataset. 3. Calculate the mean of a dataset. 4. Contrast measures of central tendency to identify the most representative average. 5. Solve application problems involving mean, median, and mode. What exactly do we mean when we describe something as "average"? Is the height of an average person the height that more people share than any other? What if we line up every person in the world, in order from shortest to tallest, and find the person right in the middle: Is that person’s height the average? Or maybe it’s something more complicated. Imagine a game where you and a friend are trying to guess the typical person’s height. Once the guesses are made, you bring in every person and measure their height. You and your friend figure out how far off each of your guesses were from the actual value, then square that number. The result is the number of points you earn for that person. After we check every height and award points accordingly, the person with the lower score wins (because a lower score means that person’s guess was, overall, closer to the actual values). Could we define the average height to be the number that you should guess to give you the smallest possible score? Each of these three methods of determining the “average” is commonly used. They are all methods of measuring centrality (or central tendency). Centrality is just a word that describes the middle of a set of data. All give potentially different results, and all are useful for different reasons. In this section, we’ll explore each of these methods of finding the “average.” ### The Mode In our discussion of average heights, the first possible definition we offered was the height that more people share than any other. This is the mode, or the value that appears most often. If there are two modes, the data are bimodal. Let’s look at some examples. When we have a complete list of the data or a stem-and-leaf plot, it’s pretty straightforward to find the mode; we just need to find the number that appears most often. If we’re given a frequency distribution instead, the technique is different (but just as straightforward): we’re looking for the number with the highest frequency. What happens if there is no number in the data that appears more than once? In that case, by our definition, every data value is a mode. But according to some other definitions, the data would have no mode. In practice, though, it doesn’t really matter; if no data value appears more than once, then the mode is not helpful at all as a measure of centrality. ### The Median Let's revisit our example of trying to identify the height of the “average” person. If we lined everyone up in order by height and found the person right in the middle, that person’s height is called the median, or the value that is greater than no more than half and less than no more than half of the values. Let’s look at a really simple example. Consider the following list of numbers: 11, 12, 13, 13, 14. Is the first number on the list, 11, the median? There are no values less than 11 (that’s 0%), and there are four values greater than 11 (that’s 80%). Since more than 50% of the data are greater than 11, the definition is violated; it’s not the median. Here’s a chart with the rest of the data, with red shading to show where the definition is violated: Only 13 has no violations, so it’s the median according to the definition. In practice, we find the median just like we described in the average height example: by lining up all the data values in order from smallest to largest and picking the value in the middle. For our easy example (with data values 11, 12, 13, 13, 14), that first 13 is right in the middle; there are two values to the left and two values to the right. If there’s not one value right in the middle, we pick the two closest, then choose the number exactly between them. For example, let’s say we have the data 41, 44, 46, 53. Since there are an even number of data values in our list, we can’t pick the one right in the middle. The two closest to the middle are 44 and 46, so we’ll choose the number halfway between those to be the median: 45. As this example shows, the median (unlike the mode) doesn’t have to be a number in our original set of data. In the examples we’ve looked at so far, it’s been pretty easy to identify which number is right in the middle. If we had a very large dataset, though, it might be harder. Fortunately, we have some formulas to help us with that. Let’s put those formulas to work in an example. Now, let’s tackle an example with an even number of values. ### The Mean Recall our example of ways we could identify the “average” height of an individual. The last method we discussed was also the most complicated. It involved a game where the player guesses a height, then figures out how far off that guess is from every single person’s height. Those differences get squared and added together to get a score. Our next measure of centrality gives the lowest possible score: No other guess would beat it in the game. Given a dataset containing n total values, the mean of the dataset is the sum of all the data values, divided by n. This is a computation you have likely done before. In many places, including spreadsheet programs like Microsoft Excel and Google Sheets, this number is called the average. For statisticians, though, the word average has too many possible meanings, so they prefer the one we’ll use: mean. As the number of data values we are considering grows, the computation for the mean gets more and more complicated. That’s why people generally trust technology to perform that computation. Note that a recent update to Google Sheets introduced a new function called “MODE.MULT,” which will find every mode (not just the first one on the list). ### Which Is Better: Mode, Median, or Mean? If the mode, median, and mean all purport to measure the same thing (centrality), why do we need all three? The answer is complicated, as each measure has its own strengths and weaknesses. The mode is simple to compute, but there may be more than one. Further, if no data value appears more than once, the mode is entirely unhelpful. As for the mean and median, the main difference between these two measures is how each is affected by extreme values. Consider this example: the mean and median of 1, 2, 3, 4, 5 are both 3. But what if the dataset is instead 1, 2, 3, 4, 10? The median is still 3, but the mean is now 4. What this example shows is that the mean is sensitive to extreme values, while the median isn’t. This knowledge can help us decide which of the two is more relevant for a given dataset. If it is important that the really high or really low values are reflected in the measure of centrality, then the mean is the better option. If very high or low values are not important, however, then we should stick with the median. The decision between mean and median only really matters if the data are skewed. If the data are symmetric, then the mean and median are going to be approximately equal, and the distinction between them is irrelevant. If the data are skewed, the mean gets pulled in the direction of the skew (i.e., if the data are right-skewed, then the mean will be bigger than the median; if the data are left-skewed, the opposite relation is true). ### Check Your Understanding ### Key Terms 1. mode 2. bimodal 3. median 4. mean ### Key Concepts 1. The mode of a dataset is the value that appears the most frequently. The median is a value that is greater than or equal to no more than 50% of the data and less than or equal to no more than 50% of the data. The mean is the sum of all the data values, divided by the number of units in the dataset. 2. The median of a dataset is not affected by outliers, but the mean will be biased toward outliers. This distinction might affect which measure of centrality is used to summarize a dataset. ### Formulas Suppose we have a set of data with values, ordered from smallest to largest. If is odd, then the median is the data value at position . If is even, then we find the values at positions and . If those values are named and , then the median is defined to be . ### Videos 1. Compute Measures of Centrality Using Google Sheets

# Statistics ## Range and Standard Deviation ### Learning Objectives After completing this section, you should be able to: 1. Calculate the range of a dataset 2. Calculate the standard deviation of a dataset Measures of centrality like the mean can give us only part of the picture that a dataset paints. For example, let’s say you’ve just gotten the results of a standardized test back, and your score was 138. The mean score on the test is 120. So, your score is above average! But how good is it really? If all the scores were between 100 and 140, then you know your score must be among the best. But if the scores ranged from 0 to 200, then maybe 140 is good, but not great (though still above average). Knowing information about how the data are spread out can help us put a particular data value in better context. In this section, we’ll look at two numbers that help us describe the spread in the data: the range and the standard deviation. These numbers are called measures of dispersion. ### The Range Our first measure of dispersion is the range, or the difference between the maximum and minimum values in the set. It’s the measure we used in the standardized test example above. Let’s look at a couple of examples. For large datasets, finding the maximum and minimum values can be daunting. There are two ways to do it in a spreadsheet. First, you can ask the spreadsheet program to sort the data from smallest to largest, then find the first and last numbers on the sorted list. The second method uses built-in functions to find the minimum and maximum. In either method, once you’ve found the maximum and minimum, all you have to do is subtract to find the range. The range is very easy to compute, but it depends only on two of the data values in the entire set. If there happens to be just one unusually high or low data value, then the range might give a distorted measure of dispersion. Our next measure takes every single data value into account, making it more reliable. ### The Standard Deviation The standard deviation is a measure of dispersion that can be interpreted as approximately the average distance of every data value from the mean. (This distance from the mean is the “deviation” in “standard deviation.”) To compute the standard deviation using the formula, we follow the steps below: 1. Compute the mean of all the data values. 2. Subtract the mean from each data value. 3. Square those differences. 4. Add up the results in step 3. 5. Divide the result in step 4 by 6. Take the square root of the result in step 5. Let’s see that process in action. The computation for the standard deviation is complicated, even for just a small dataset. We’d never want to compute it without technology for a large dataset! Luckily, technology makes this calculation easy. ### Check Your Understanding ### Key Terms 1. range 2. standard deviation ### Key Concepts 1. The range of a dataset is the difference between its largest and smallest values. The standard deviation is approximately the mean difference (in absolute value) that individual units fall from the mean of the dataset. ### Formulas Here, s is the standard deviation, represents each data value, is the mean of the data values, is the number of data values, and the capital sigma () indicates that we take a sum. ### Videos 1. Find the Minimum and Maximum Using Google Sheets 2. Find the Standard Deviation Using Google Sheets

# Statistics ## Percentiles ### Learning Objectives After completing this section, you should be able to: 1. Compute percentiles. 2. Solve application problems involving percentiles. A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that are below a given number can help us evaluate differences between individuals in separate populations. These proportions are called percentiles. If of the values in a dataset are less than a number , then we say that is at the th percentile. ### Finding Percentiles There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the quart-), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root quant– means “number”). In Mean, Median, and Mode, we defined the median as a number that is greater than no more than half of the data in a dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: The median is the value at the 50th percentile (or second quartile). Let’s look at some examples. In each of the examples above, the computations were made easier by the fact that the we were looking for percentiles that “came out evenly” with respect to the number of values in our dataset. Things don’t always work out so cleanly. Further, different sources will define the term percentile in different ways. In fact, Google Sheets has three built-in functions for finding percentiles, none of which uses our definition. Some of the definitions you’ll see differ in the inequality that is used. Ours uses “less than or equal to,” while others use “less than” (these correspond roughly to Google Sheets’ ‘PERCENTILE.INC’ and ‘PERCENTILE.EXC’). Some of them use different methods for interpolating values. (This is what we did when we first computed medians without technology; if there were an even number of data values in our dataset, found the mean of the two values in the middle. This is an example of interpolation. Most computerized methods use this technique.) Other definitions don’t interpolate at all, but instead choose the closest actual data value to the theoretical value. Fortunately, for large datasets, the differences among the different techniques become very small. So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute something involving percentiles without technology , use the technique we used in the previous example. In all other cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” definitions). ### Check Your Understanding ### Key Terms 1. percentile 2. quartile 3. quintile 4. quantile ### Key Concepts 1. The percentile rank of a data value is the percentage of all values in the dataset that are less than or equal to the given value. ### Videos 1. Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets

# Statistics ## The Normal Distribution ### Learning Objectives After completing this section, you should be able to: 1. Describe the characteristics of the normal distribution. 2. Apply the 68-95-99.7 percent groups to normal distribution datasets. 3. Use the normal distribution to calculate a -score. 4. Find and interpret percentiles and quartiles. Many datasets that result from natural phenomena tend to have histograms that are symmetric and bell-shaped. Imagine finding yourself with a whole lot of time on your hands, and nothing to keep you entertained but a coin, a pencil, and paper. You decide to flip that coin 100 times and record the number of heads. With nothing else to do, you repeat the experiment ten times total. Using a computer to simulate this series of experiments, here’s a sample for the number of heads in each trial: 54, 51, 40, 42, 53, 50, 52, 52, 47, 54 It makes sense that we’d get somewhere around 50 heads when we flip the coin 100 times, and it makes sense that the result won’t always be exactly 50 heads. In our results, we can see numbers that were generally near 50 and not always 50, like we thought. ### Moving Toward Normality Let’s take a look at a histogram for the dataset in our section opener: This is interesting, but the data seem pretty sparse. There were no trials where you saw between 43 and 47 heads, for example. Those results don’t seem impossible; we just didn’t flip enough times to give them a chance to pop up. So, let’s do it again, but this time we'll perform 100 coin flips 100 times. Rather than review all 100 results, which could be overwhelming, let's instead visualize the resulting histogram. From the histogram, we see that most of the trials resulted in between, say, 44 and 56 heads. There were some more unusual results: one trial resulted in 70 heads, which seems really unlikely (though still possible!). But we’re starting to maybe get a sense of the distribution. More data would help, though. Let’s simulate another 900 trials and add them to the histogram! We can still see that 70 is a really unusual observation, though we came close in another trial (one that had 68 heads). Now, the distribution is coming more into focus: It looks quite symmetric and bell-shaped. Let’s just go ahead and take this thought experiment to an extreme conclusion: 10,000 trials. The distribution is pretty clear now. Distributions that are symmetric and bell-shaped like this pop up in all sorts of natural phenomena, such as the heights of people in a population, the circumferences of eggs of a particular bird species, and the numbers of leaves on mature trees of a particular species. All of these have bell-shaped distributions. Additionally, the results of many types of repeated experiments generally follow this same pattern, as we saw with the coin-flipping example; this fact is the basis for much of the work done by statisticians. It’s a fact that’s important enough to have its own name: the Central Limit Theorem. ### The Normal Distribution In the coin flipping example above, the distribution of the number of heads for 10,000 trials was close to perfectly symmetric and bell-shaped: Because distributions with this shape appear so often, we have a special name for them: normal distributions. Normal distributions can be completely described using two numbers we’ve seen before: the mean of the data and the standard deviation of the data. You may remember that we described the mean as a measure of centrality; for a normal distribution, the mean tells us exactly where the center of the distribution falls. The peak of the distribution happens at the mean (and, because the distribution is symmetric, it’s also the median). The standard deviation is a measure of dispersion; for a normal distribution, it tells us how spread out the histogram looks. To illustrate these points, let’s look at some examples. Let’s put it all together to identify a completely unknown normal distribution. ### Properties of Normal Distributions: The 68-95-99.7 Rule The most important property of normal distributions is tied to its standard deviation. If a dataset is perfectly normally distributed, then 68% of the data values will fall within one standard deviation of the mean. For example, suppose we have a set of data that follows the normal distribution with mean 400 and standard deviation 100. This means 68% of the data would fall between the values of 300 (one standard deviation below the mean: ) and 500 (one standard deviation above the mean: ). Looking at the histogram below, the shaded area represents 68% of the total area under the graph and above the axis: Since 68% of the area is in the shaded region, this means that of the area is found in the unshaded regions. We know that the distribution is symmetric, so that 32% must be divided evenly into the two unshaded tails: 16% in each. Of course, datasets in the real world are never perfect; when dealing with actual data that seem to follow a symmetric, bell-shaped distribution, we’ll give ourselves a little bit of wiggle room and say that approximately 68% of the data fall within one standard deviation of the mean. The rule for one standard deviation can be extended to two standard deviations. Approximately 95% of a normally distributed dataset will fall within 2 standard deviations of the mean. If the mean is 400 and the standard deviation is 100, that means 95% calculation describes the way we compute standardized scores. (two standard deviations below the mean: ) and 600 (two standard deviations above the mean: ). We can visualize this in the following histogram: As before, since 95% of the data are in the shaded area, that leaves 5% of the data to go into the unshaded tails. Since the histogram is symmetric, half of the 5% (that’s 2.5%) is in each. We can even take this one step further: 99.7% of normally distributed data fall within 3 standard deviations of the mean. In this example, we’d see 99.7% of the data between 100 (calculated as ) and 700 (calculated as ). We can see this in the histogram below, although you may need to squint to find the unshaded bits in the tails! This observation is formally known as the 68-95-99.7 Rule. There are more problems we can solve using the 68-95-99.7 Rule. but first we must understand what the rule implies. Remember, the rule says that 68% of the data falls within one standard deviation of the mean. Thus, with normally distributed data with mean 100 and standard deviation 10, we have this distribution: Since we know that 68% of the data lie within one standard deviation of the mean, the implication is that 32% of the data must fall beyond one standard deviation away from the mean. Since the histogram is symmetric, we can conclude that half of the 32% (or 16%) is more than one standard deviation above the mean and the other half is more than one standard deviation below the mean: Further, we know that the middle 68% can be split in half at the peak of the histogram, leaving 34% on either side: So, just the “68” part of the 68-95-99.7 Rule gives us four other proportions in addition to the 68% in the rule. Similarly, the “95” and “99.7” parts each give us four more proportions: We can put all these together to find even more complicated proportions. For example, since the proportion between 100 and 120 is 47.5% and the proportion between 100 and 110 is 34%, we can subtract to find that the proportion between 110 and 120 is : ### Standardized Scores When we want to apply the 68-95-99.7 Rule, we must first figure out how many standard deviations above or below the mean our data fall. This calculation is common enough that it has its own name: the standardized score. Values above the mean have positive standardized scores, while those below the mean have negative standardized scores. Since it's common to use the letter to represent a standard score, this value is also often referred to as a -score. So far, we’ve only really considered -scores that are whole numbers, but in general they can be any number at all. For example, if we have data that are normally distributed with mean 80 and standard deviation 6, the value 85 is five units above the mean, which is less than one standard deviation. Dividing by the standard deviation, we get . Since 85 is of one standard deviation above the mean, we’d say that the standardized score for 85 is (which is positive, since ). This calculation describes the way we compute standardized scores. ### Using Google Sheets to Find Normal Percentiles The 68-95-99.7 Rule is great when we’re dealing with whole-number -scores. However, if the -score is not a whole number, the Rule isn’t going to help us. Luckily, we can use technology to help us out. We’ll talk here about the built-in functions in Google Sheets, but other tools work similarly. Let’s say we’re working with normally distributed data with mean 40 and standard deviation 7, and we want to know at what percentile a data value of 50 would fall. That corresponds to finding the proportion of the data that are less than 50. If we create our histogram and mark off whole-number multiples of the standard deviation like we did before, we’ll see why the 68-95-99.7 Rule isn’t going to help: Since 50 doesn’t line up with one of our lines, the 68-95-99.7 Rule fails us. Looking back at and , the best we can say is that 50 is between the 84th and 99.5th percentiles, but that’s a pretty wide range. Google Sheets has a function that can help; it’s called NORM.DIST. Here’s how to use it: 1. Click in an empty cell in your worksheet. 2. Type “=NORM.DIST(“ 3. Inside the parentheses, we must enter a list of four things, separated by commas: the data value, the mean, the standard deviation, and the word “TRUE”. These have to be entered in this order! 4. Close the parentheses, and hit Enter. The result is then displayed in the cell; convert it to a percent to get the percentile. So, for our example, we should type “=NORM.DIST(50, 40, 7, TRUE)” into an empty cell, and hit Enter. The result is 0.9234362745; converting to a percent and rounding, we can conclude that 50 is at the 92nd percentile. Let’s walk through a few more examples. Google Sheets can also help us go the other direction: If we want to find the data value that corresponds to a given percentile, we can use the NORM.INV function. For example, if we have normally distributed data with mean 150 and standard deviation 25, we can find the data value at the 30th percentile as follows: 1. Click on an empty cell in your worksheet. 2. Type “=NORM.INV(“ 3. Inside the parentheses, we’ll enter a list of three numbers, separated by commas: the percentile in question expressed as a decimal, the mean, and the standard deviation. These must be entered in this order! 4. Close the parentheses and hit Enter. The desired data value will be in the cell! In our example, we want the 30th percentile; converting 30% to a decimal gives us 0.3. So, we’ll type “=NORM.INV(0.3, 150, 25)” to get 136.8899872; let’s round that off to 137. ### Check Your Understanding 1. normal distribution 2. 68-95-99.7 Rule 3. standardized score (-score) ### Key Concepts 1. Normally distributed data follow a bell-shaped, symmetrical distribution. 2. The mean of normally distributed data falls at the peak of the distribution. The standard deviation of normally-distributed data is the distance from the peak to either of the inflection points. 3. Data that are normally distributed follow the 68-95-99.7 Rule, which says that approximately 68% of the data fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. 4. The -score for a data value is the number of standard deviations that value falls above (or below, if the -score is negative) the mean. 5. We can use the normal distribution to estimate percentiles. ### Formulas If is a member of a normally distributed dataset with mean and standard deviation , then the standardized score for is If you know a -score but not the original data value , you can find it by solving the previous equation for :

# Statistics ## Applications of the Normal Distribution ### Learning Objectives After completing this section, you should be able to: 1. Apply the normal distribution to real-world scenarios. As we saw in The Normal Distribution, the word “standardized” is closely associated with the normal distribution. This is why tests like college entrance exams, state achievement tests for K–12 students, and Advanced Placement tests are often called “standardized tests”: scores are assigned in a way that forces them to follow a normal distribution, with a mean and standard deviation that are consistent from year to year. Standardization also allows people like college admissions officers to directly compare an applicant who took the ACT (a college entrance exam) to an applicant who instead chose to take the SAT (a different college entrance exam). Standardization allows us to compare individuals from different groups; this is among the most important applications of the normal distribution. We’ll explore this and other real-world uses of the normal distribution in this section. ### College Entrance Exams There are two good ways to compare two data values from different groups: using -scores and using percentiles. The two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better using -scores but the second value is better using percentiles), so use whichever method is more comfortable for you. ### Coin flipping In the opening of The Normal Distribution, we saw that the number of heads we get when we flip a coin 100 times is distributed normally. It can be shown that if is the number of flips, then the mean of that distribution is and the standard deviation is (as long as ). So, for 100 flips, the mean of the distribution is 50 and the standard deviation is 5. In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see 70 heads (or more) only of the time. That’s really unlikely! Analysis like this is related to hypothesis testing, an important application of statistics in the sciences and social sciences. ### Analyzing Data That Are Normally Distributed Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use techniques associated with the normal distribution to analyze the data. ### Check Your Understanding ### Key Concepts 1. We can use -scores to compare data values from different datasets.

# Statistics ## Scatter Plots, Correlation, and Regression Lines ### Learning Objectives After completing this section, you should be able to: 1. Construct a scatter plot for a dataset. 2. Interpret a scatter plot. 3. Distinguish among positive, negative and no correlation. 4. Compute the correlation coefficient. 5. Estimate and interpret regression lines. One of the most powerful tools statistics gives us is the ability to explore relationships between two datasets containing quantitative values, and then use that relationship to make predictions. For example, a student who wants to know how well they can expect to score on an upcoming final exam may consider reviewing the data on midterm and final exam scores for students who have previously taken the class. It seems reasonable to expect that there is a relationship between those two datasets: If a student did well on the midterm, they were probably more likely to do well on the final than the average student. Similarly, if a student did poorly on the midterm, they probably also did poorly on the final exam. Of course, that relationship isn’t set in stone; a student’s performance on a midterm exam doesn’t cement their performance on the final! A student might use a poor result on the midterm as motivation to study more for the final. A student with a really good grade on the midterm might be overconfident going into the final, and as a result doesn’t prepare adequately. The statistical method of regression can find a formula that does the best job of predicting a score on the final exam based on the student’s score on the midterm, as well as give a measure of the confidence of that prediction! In this section, we’ll discover how to use regression to make these predictions. First, though, we need to lay some graphical groundwork. ### Relationships Between Quantitative Datasets Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm, rather than the other way around: if the midterm depended on the final, then we’d need to know the final score first, which doesn’t make sense. Here’s another example: if we collected data on home purchases in a certain area, and noted both the sale price of the house and the annual household income of the purchaser, we might expect a relationship between those two. Which depends on the other? In this case, sale price depends on income: people who have a higher income can afford a more expensive house. If it were the other way around, people could buy a new, more expensive house and then expect a raise! (This is very bad advice.) It's worth noting that not every pair of related datasets has clear dependence. For example, consider the percent of a country’s budget devoted to the military and the percent earmarked for public health. These datasets are generally related: as one goes up, the other goes down. However, in this case, there’s not a preferred choice for dependence, as each could be seen as depending on the other. When exploring the relationship between two datasets, if one set seems to depend on the other, we’ll say that dataset contains values of the response variable (or dependent variable). The dataset that the response variable depends on contains values of what we call the explanatory variable (or independent variable). If no dependence relationship can be identified, then we can assign either dataset to either role. Once we’ve assigned roles to our two datasets, we can take the first step in visualizing the relationship between them: creating a scatter plot. ### Creating Scatter Plots A scatter plot is a visualization of the relationship between two quantitative sets of data. The scatter plot is created by turning the datasets into ordered pairs: the first coordinate contains data values from the explanatory dataset, and the second coordinate contains the corresponding data values from the response dataset. These ordered pairs are then plotted in the -plane. Let's return to our exam example to put this into practice. For large datasets, it’s impractical to create scatter plots manually. Luckily, Google Sheets automates this process for us. ### Reading and Interpreting Scatter Plots Scatter plots give us information about the existence and strength of a relationship between two datasets. To break that information down, there are a series of questions we might ask to help us. First: Is there a curved pattern in the data? If the answer is “yes,” then we can stop; none of the linear regression techniques from here to the end of this section are appropriate. and show several examples of scatter plots that can help us identify these curved patterns. Once we have confirmed that there is no curved pattern in our data, we can move to the next question: Is there a linear relationship? To answer this, we must look at different values of the explanatory variable and determine whether the corresponding response values are different, on average. It's important to look at the values “on average” because, in general, our scatter plots won’t include just one corresponding response point for each value of the explanatory variable (i.e., there may be multiple response values for each explanatory value). So, we try to look for the center of those points. Let’s look again at , but consider some different values for the explanatory variable. Let’s highlight the points whose -values are around 50 and those that are around 80: Now, we can estimate the middle of each group of points. Let's add our estimated averages to the plot as starred points: Since those two starred points occur at different heights, we can conclude that there’s likely a relationship worth exploring. Here’s another example using a different set of data: Let’s look again at the points near 50 and near 80, and estimate the middles of those clusters: Notice that there’s not much vertical distance between our two starred points. This tells us that there’s not a strong relationship between these two datasets. ### Positive and Negative Linear Relationships Another way to assess whether there is a relationship between two datasets in a scatter plot is to see if the points seem to be clustered around a line (specifically, a line that’s not horizontal). The stronger the clustering around that line is, the stronger the relationship. Once we’ve established that there’s a relationship worth exploring, it’s time to start quantifying that relationship. Two datasets have a positive linear relationship if the values of the response tend to increase, on average, as the values of the explanatory variable increase. If the values of the response decrease with increasing values of the explanatory variable, then there is a negative linear relationship between the two datasets. The strength of the relationship is determined by how closely the scatter plot follows a single straight line: the closer the points are to that line, the stronger the relationship. The scatter plots in to depict varying strengths and directions of linear relationships. The strength and direction (positive or negative) of a linear relationship can also be measured with a statistic called the correlation coefficient (denoted ). Positive values of indicate a positive relationship, while negative values of indicate a negative relationship. Values of close to 0 indicate a weak relationship, while values close to correspond to a very strong relationship. Looking again at to , the correlation coefficients for each, in sequential order, are: ‒1, ‒0.97, ‒0.55, ‒0.03, 0.61, 0.97, and 1. There’s no firm rule that establishes a cutoff value of to divide strong relationships from weak ones, but is often given as the dividing line (i.e., if or the relationship is strong, and if the relationship is weak). The formula for computing is very complicated; it’s almost never done without technology. Google Sheets will do the computation for you using the CORREL function. The syntax works like this: if your explanatory values are in cells A2 to A50 and the corresponding response values are in B2 to B50, then you can find the correlation coefficient by entering “=CORREL(A2:A50, B2:B50)”. (Note that the order doesn’t matter for correlation coefficients; “=CORREL(B2:B50, A2:A50)” will give the same result.) Let’s put all of this together in an example. ### Linear Regression The final step in our analysis of the relationship between two datasets is to find and use the equation of the regression line. For a given set of explanatory and response data, the regression line (also called the least-squares line or line of best fit) is the line that does the best job of approximating the data. What does it mean to say that a particular line does the “best job” of approximating the data? The way that statisticians characterize this “best line” is rather technical, but we’ll include it for the sake of satisfying your curiosity (and backing up the claim of "best"). Imagine drawing a line that looks like it does a pretty good job of approximating the data. Most of the points in the scatter plot will probably not fall exactly on the line; the distance above or below the line a given point falls is called that point’s residual. We could compute the residuals for every point in the scatter plot. If you take all those residuals and square them, then add the results together, you get a statistic called the sum of squared errors for the line (the name tells you what it is: “sum” because we’re adding, “squared” because we’re squaring, and “errors” is another word for “residuals”). The line that we choose to be the “best” is the one that has the smallest possible sum of squared errors. The implied minimization (“smallest”) is where the “least” in “least squares” comes from; the “squares” comes from the fact that we’re minimizing the sum of squared errors. This is very similar to the process we outlined in the "game" that we used to introduce the mean. Both the regression line and the mean are designed to minimize a sum of squared errors. Here ends the super technical part. ### Finding the Equation of the Regression Line So, how do we find the equation of the regression line? Recall the point-slope form of the equation of a line: The regression line has two properties that we can use to find its equation. First, it always passes through the point of means. If and are the means of the explanatory and response datasets, respectively, then the point of means is . We’ll use that as the point in the point-slope form of the equation. Second, if and are the standard deviations of the explanatory and response datasets, respectively, and if is the correlation coefficient, then the slope is . Putting all that together with the point-slope formula gives us this: Let's walk through an example. As you can see, finding the equation of the regression line involves a lot of steps if you have to find all of the values of the needed quantities yourself. But, as usual, technology comes to our rescue. This video (which you actually watched earlier when learning how to create scatter plots) covers the regression line at around the 3:30 mark. Note that Google Sheets calls it the "trendline." Let's put this into practice. ### Using the Equation of the Regression Line Once we’ve found the equation of the regression line, what do we do with it? We’ll look at two possible applications: making predictions and interpreting the slope. We can use the equation of the regression line to predict the response value for a given explanatory value . All we have to do is plug that explanatory value into the formula and see what response value results. This is useful in two ways: first, it can be used to make a guess about an unknown data value (like one that hasn’t been observed yet). Second, it can be used to evaluate performance (meaning, we can predict an outcome given a particular event). In , we created a scatter plot of final exam scores vs. midterm exam scores using this data: The equation of the regression line is , where is the final exam score and is the midterm exam score. If Frank scored 85 on the midterm, then our prediction for his final exam score is . To use the regression line to evaluate performance, we use a data value we’ve already observed. For example, Allison scored 88 on the midterm. The regression line predicts that someone who scores an 88 on the midterm will get on the final. Allison actually scored 84 on the final, meaning she underperformed expectations by almost 4 points . The second application of the equation of the regression line is interpreting the slope of the line to describe the relationship between the explanatory and response datasets. For the exam data in the previous paragraph, the slope of the regression line is 0.687. Recall that the slope of a line can be computed by finding two points on the line and dividing the difference in the -values of those points by the difference in the -values. Keeping that in mind, we can interpret our slope as . Multiplying both sides of that equation by the denominator of the fraction, we get . Thus, a one-point increase in the midterm score would result in a predicted increase in the final score of 0.687 points. A ten-point drop in the midterm score would give us a decrease in the predicted final score of 6.87 points. In general, the slope gives us the predicted change in the response that corresponds to a one unit increase in the explanatory variable. ### Extrapolation A very common misuse of regression techniques involves extrapolation, which involves making a prediction about something that doesn't belong in the dataset. ### Correlation Does Not Imply Causation One of the most common fallacies about statistics has to do with the relationship between two datasets. In the dataset “Public”, we find that the correlation coefficient between the 75th percentile math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.” Does the increasing math score cause the increase in the verbal score? Probably not. What’s really going on is that there’s a third variable that’s affecting them both: To raise the SAT math score by 10 points, a school will recruit students who do better on the SAT in general; these students will also naturally have higher SAT verbal scores. This third variable is sometimes called a lurking variable or a confounding variable. Unless all possible lurking variables are ruled out, we cannot conclude that one thing causes another. ### Check Your Understanding ### Key Terms 1. response variable (dependent variable) 2. explanatory variable (independent variable) 3. scatter plot 4. positive linear relationship 5. negative linear relationship 6. correlation coefficient 7. regression line (least-squares line, line of best fit) ### Key Concepts 1. If one variable affects the value of another variable, we say the first is an explanatory variable and the second is a response variable. 2. Scatter plots place a point in the -plane for each unit in the dataset. The -value is the value of the explanatory variable, and the -value is the value of the response variable. 3. The correlation coefficient gives us information about the strength and direction of the relationship between two variables. If is positive, the relationship is positive: an increase in the value of the explanatory variable tends to correspond to an increase in the value of the response variable. If is negative, the relationship is negative: an increase in the value of the explanatory variable tends to correspond to a decrease in the value of the response variable. Values of that are close to 0 indicate weak relationships, while values close to –1 or indicate strong relationships. 4. The regression line for a relationship between two variables is the line that best represents the data. It can be used to predict values of the response variable for a given value of the explanatory variable. ### Formulas If a line has slope and passes through a point , then the point-slope form of the equation of the line is: Suppose and are explanatory and response datasets that have a linear relationship. If their means are and respectively, their standard deviations are and respectively, and their correlation coefficient is , then the equation of the regression line is: ### Videos 1. Making Scatter Plots in Google Sheets ### Projects 1. Browse through some news websites to find five stories that report on data and include data visualizations. Can you tell from the report how the data were collected? Was randomization used? Are the visualizations appropriate for the data? Are the visualizations presented in a way that might bias the reader? 2. We discussed three measures of centrality in this chapter: the mode, the median, and the mean. In a broader context, the mean as we discussed it is more properly called the arithmetic mean, to distinguish it from other types of means. Examples of these include the geometric mean, harmonic mean, truncated mean, and weighted mean. How are these computed? How do they compare to the arithmetic mean? In what situations would each of these be preferred to the arithmetic mean? 3. Simpson’s Paradox is a statistical phenomenon that can sometimes appear when we observe a relationship within several subgroups of a population, but when the data for all thegroups are analyzed all together, the opposite relationship appears. Find some examples of Simpson’s Paradox in real-world situations, and write a paragraph or two that would explain the concept to someone who had never studied statistics before. ### Chapter Review ### Gathering and Organizing Data ### Visualizing Data ### Mean, Median and Mode ### Range and Standard Deviation ### Percentiles ### The Normal Distribution ### Applications of the Normal Distribution ### Scatter Plots, Correlation, and Regression Lines ### Chapter Test

# Metric Measurement ## Introduction You are planning a road trip from your home state to the sunny beaches of Mexico and need to prepare a budget. While in the United States, gasoline is sold in gallons and distances are measured in miles, but in almost any other country you will find that gasoline is sold in liters and distance is measured in kilometers. Whether you’re traveling, baking, watching an international sporting event, working with machine tools, or using scientific equipment, it's important to understand the metric system, or the International System of Units (SI). The metric system is a decimal measuring system that uses meters, liters, and grams to quantify length, capacity, and mass. It is used in all but three countries in the world, including the United States.

# Metric Measurement ## The Metric System ### Learning Objectives After completing this section, you should be able to: 1. Identify units of measurement in the metric system and their uses. 2. Order the six common prefixes of the metric system. 3. Convert between like unit values. Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For example, if you want to make authentic tamales you might visit the nearest Hispanic grocery store. While shopping, you discover that there are two brands of masa for the same price, but one bag is marked 1,200 g and the other 1 kg. Which one is the better deal? Understanding the metric system allows you to understand that 1,200 grams is equivalent to 1.2 kg, so the 1,200 g bag is the better deal. ### Units of Measurement in the Metric System Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system: 1. The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch. 2. The metric unit for area is the square meter (m2). The area of a professional soccer field is 7,140 square meters (7,140 m2). 3. The metric base unit for volume is the cubic meter (m3). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles. 4. The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g). 5. The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C. While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the relationship between smaller or larger units. While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature. ### Metric Prefixes Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is , or 100 times smaller than a meter, while the kilometer is 103, or 1,000 times larger than a meter. The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The most used prefixes are shown in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died From Drinking Chocolate Milk. ### Converting Metric Units of Measure Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams. The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the measure is of a meter. One meter is equivalent to 100 centimeters. A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number that when used with multiplication or division converts from one metric unit to another, both having the same base unit. In the metric system, these conversion factors are directly related to the powers of 10. The most common used conversion factors are shown in . ### Check Your Understanding ### Key Terms 1. metric system 2. meter (m) 3. gram (g) 4. liter (L) 5. square meter (m2) 6. cubic meter (m3) 7. degrees Celsius (°C) 8. conversion factor ### Key Concepts 1. The metric system is decimal system of weights and measures based on base units of meter, liter, and gram. The system was first proposed in 1670 and has since been adopted as the International System of Units and used in nearly every country in the world. 2. Each successive unit on the metric scale is 10 times larger than the previous one. To convert between units with the same base unit, you must either multiply or divide by a power of 10. 3. The most used prefixes are listed in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died from Drinking Chocolate Milk. ### Videos 1. U.S. Office of Education: Metric Education 2. Neil deGrasse Tyson Explains the Metric System ### Formulas You can convert between unit sizes with the same base unit using the conversion factors shown in .

# Metric Measurement ## Measuring Area ### Learning Objectives After completing this section, you should be able to: 1. Identify reasonable values for area applications. 2. Convert units of measures of area. 3. Solve application problems involving area. Area is the size of a surface. It could be a piece of land, a rug, a wall, or any other two-dimensional surface with attributes that can be measured in the metric unit for distance-meters. Determining the area of a surface is important to many everyday activities. For example, when purchasing paint, you’ll need to know how many square units of surface area need to be painted to determine how much paint to buy. Square units indicate that two measures in the same units have been multiplied together. For example, to find the area of a rectangle, multiply the length units and the width units to determine the area in square units. Note that to accurately calculate area, each of the measures being multiplied must be of the same units. For example, to find an area in square centimeters, both length measures (length and width) must be in centimeters. ### Reasonable Values for Area Because area is determined by multiplying two lengths, the magnitude of difference between different square units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter is times greater in area than a square centimeter . The relationships between benchmark metric area units are shown in the following table. An essential understanding of metric area is to identify reasonable values for area. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given area is reasonable for the situation. ### Converting Units of Measures for Area Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are shown in . ### Solving Application Problems Involving Area While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of the surface that are not relevant to your calculation. When calculating area, you must ensure that both distance measurements are expressed in terms of the same distance units. Sometimes you must convert one measurement before using the area formula. When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not relevant. ### Check Your Understanding ### Key Terms 1. area 2. square units ### Key Concepts 1. Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two-dimensional space. 2. Area is measured in square meters units; in the metric system the base unit for area is square meters (). ### Videos 1. Converting Metric Units of Area 2. Why the Metric System Matters ### Formulas To determine the area of rectangular-shaped objects: You can convert between metric area units using the conversion factors shown in .

# Metric Measurement ## Measuring Volume ### Learning Objectives After completing this section, you should be able to: 1. Identify reasonable values for volume applications. 2. Convert between like units of measures of volume. 3. Convert between different unit values. 4. Solve application problems involving volume. Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the SUV can hold. Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine the volume in square units: Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, to find a volume in cubic centimeters, each of the measures must be in centimeters. ### Reasonable Values for Volume Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m3, is times greater in area than a cubic centimeter, cm3. This relationship between benchmark metric volume units is shown in the following table. To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given volume is reasonable for the situation. ### Converting Like Units of Measures for Volume Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in . ### Understanding Other Metric Units of Volume When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of saline to be administered to a patient, they are referring to 10 mL of saline. The most frequently used factors for converting from cubic meters to liters are listed in . ### Solving Application Problems Involving Volume Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding volume is essential to many every day and professional tasks. ### Check Your Understanding ### Key Terms 1. volume 2. cubic units ### Key Concepts 1. Volume is a measure of the space contained within or occupied by three-dimensional objects. 2. Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m3) 3. The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to one cubic decimeter, which is the volume of a cube measuring . ### Formulas To determine the volume of a rectangular prism: You can convert between metric volume units and metric capacity units using the relationships shown in . ### Videos 1. How to Convert Cubic Centimeters to Cubic Meters 2. Converting Metric Units of Volume

# Metric Measurement ## Measuring Weight ### Learning Objectives After completing this section, you should be able to: 1. Identify reasonable values for weight applications. 2. Convert units of measures of weight. 3. Solve application problems involving weight. In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car weighs about 2 metric tons. Any discussion about metric weight must also include a conversation about mass. Scientifically, mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects with the same mass will weigh less on the moon than on Earth. Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well. ### Reasonable Values for Weight To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation. ### Converting Like Units of Measures for Weight Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in . ### Solving Application Problems Involving Weight From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating. ### Check Your Understanding ### Key Terms 1. mass ### Key Concepts 1. Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of an object never changes; the weight of an object changes depending on the force of gravity. An object with the same mass would weigh less on the moon than on Earth because the moon’s gravity is less than that of Earth. 2. In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or kilograms. ### Videos 1. Metric System: Units of Weight 2. Using Drones to Weigh Whales? 3. Metric Units of Mass: Convert mg, g, and kg ### Formulas You can convert between metric weight units using the conversion factors shown in .

# Metric Measurement ## Measuring Temperature ### Learning Objectives After completing this section, you should be able to: 1. Convert between Fahrenheit and Celsius. 2. Identify reasonable values for temperature applications. 3. Solve application problems involving temperature. When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor). ### Converting Between Fahrenheit and Celsius Temperatures Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember. But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at? lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression. ### Reasonable Values for Temperature While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in . ### Solving Application Problems Involving Temperature Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems. ### Check Your Understanding ### Key Terms 1. temperature ### Key Concepts 1. Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. 2. In the metric system, temperature is measured using the Celsius (°C) scale. 3. The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor). ### Videos 1. Misconceptions About Temperature 2. Temperature Conversion Trick 3. Learn the Metric System in 5 Minutes ### Formulas To convert temperature from Fahrenheit to Celsius: To convert temperature from Celsius to Fahrenheit: To estimate temperature from Fahrenheit to Celsius: To estimate temperature from Celsius to Fahrenheit: ### Projects ### Cooking 1. Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the metric system. 2. Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary System of Measurement, using cups, tablespoons, or teaspoons as required. 3. What did you observe? Was it easier to convert from one system to another? Which system allows for more precise measurements? What kitchen tools would you need in your kitchen if you used the metric system? ### Shopping 1. Compare the average gas price in California to the average gas price in Puerto Rico. 2. What conversions did you need to make to do the comparison? 3. Do you think that the price of the gasoline is affected by the units in which it is sold? ### Sports 1. What system of measurement is used for track and field events? Why do you think this system is used? 2. What system of measurement is used for football? Why do you think this system is used? 3. Research various sports records. Which units of measurement is used? What do you think influenced the unit of measure used? ### Chapter Review ### The Metric System ### Measuring Area ### Measuring Volume ### Measuring Weight ### Measuring Temperature ### Chapter Test

# Geometry ## Introduction The painting The School of Athens presents great figures in history such as Plato, Aristotle, Socrates, Euclid, Archimedes, and Pythagoras. Other scientists are also represented in the painting. To the ancient Greeks, the study of mathematics meant the study of geometry above all other subjects. The Greeks looked for the beauty in geometry and did not allow their geometrical constructions to be “polluted” by the use of anything as practical as a ruler. They permitted the use of only two tools—a compass for drawing circles and arcs, and an unmarked straightedge to draw line segments. They would mark off units as needed. However, they never could be sure of what the units meant. For instance, how long is an inch? These mathematicians defined many concepts. The Greeks absorbed much from the Egyptians and the Babylonians (around 3000 BCE), including knowledge about congruence and similarity, area and volume, angles and triangles, and made it their task to introduce proofs for everything they learned. All of this historical wisdom culminated with Euclid in 300 BCE. Euclid (325–265 BC) is known as the father of geometry, and his most famous work is the 13-volume collection known as The Elements, which are said to be “the most studied books apart from the Bible.” Euclid brought together everything offered by the Babylonians, the Egyptians, and the more refined contributions by the Greeks, and set out, successfully, to organize and prove these concepts as he methodically developed formal theorems. This chapter begins with a discussion of the most basic geometric tools: the point, the line, and the plane. All other topics flow from there. Throughout the eight sections, we will talk about how to determine angle measurement and learn how to recognize properties of special angles, such as right angles and supplementary angles. We will look at the relationship of angles formed by a transversal, a line running through a set of parallel lines. We will explore the concepts of area and perimeter, surface area and volume, and transformational geometry as used in the patterns and rigid motions of tessellations. Finally, we will introduce right-angle trigonometry and explore the Pythagorean Theorem.

# Geometry ## Points, Lines, and Planes ### Learning Objectives After completing this section, you should be able to: 1. Identify and describe points, lines, and planes. 2. Express points and lines using proper notation. 3. Determine union and intersection of sets. In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. These definitions form the foundation of the geometric theories that are applied in everyday life. In The Elements, Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a system composed of postulates. A postulate is another term for axiom, which is a statement that is accepted as truth without the need for proof or verification. There were no formal geometric definitions before Euclid, and when terms could not be defined, they could be described. In order to write his postulates, Euclid had to describe the terms he needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that have been proved and can be proved. ### Points and Lines The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic: 1. Postulate 1: A straight line segment can be drawn joining any two points. 2. Postulate 2: Any straight line segment can be extended indefinitely in a straight line. Before we go further, we will define some of the symbols used in geometry in : From , we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in both directions. Regular lines are typically drawn with arrows on each end. There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line segments representing the roads. See . ### Parallel Lines Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line and the line are parallel we often use the symbol The distance between parallel lines remains constant as the lines extend infinitely in both directions. See . ### Perpendicular Lines Two lines that intersect at a angle are perpendicular lines and are symbolized by . If and are perpendicular, we write When two lines form a right angle, a angle, we symbolize it with a little square See . ### Defining Union and Intersection of Sets Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: When it involves sets of lines, line segments, or points, we are usually referring to the union or intersection of set. The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, and is written by placing this symbol in between each of the sets. For example, let set and let set Then, the union of sets A and B is The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol in between each of the sets referenced. For example, let’s say that set and let set Then, the intersection of sets and is ### Planes A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area. We usually label planes with a single capital letter, such as Plane , as shown in , or by all points that determine the edges of a plane. In the following figure, Plane contains points and , which are on the same line, and point , which is not on that line. By definition, is a plane. We can move laterally in any direction on a plane. One way to think of a plane is the Cartesian coordinate system with the -axis marked off in horizontal units, and -axis marked off in vertical units. In the Cartesian plane, we can identify the different types of lines as they are positioned in the system, as well as their locations. See . This plane contains points , , and . Points and are colinear and form a line segment. Point is not on that line segment. Therefore, this represents a plane. To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the horizontal move and the second number is the vertical move. Point is located at point is located at and point is located at We can also identify the line segment as Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The equation of that line of intersection is left to a study of three-dimensional space. See . To summarize, some of the properties of planes include: 1. Three points including at least one noncollinear point determine a plane. 2. A line and a point not on the line determine a plane. 3. The intersection of two distinct planes is a straight line. ### Check Your Understanding ### Key Terms 1. line segment 2. plane 3. union 4. intersection 5. parallel 6. perpendicular ### Key Concepts 1. Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles associated with the line, the point, and the plane. 2. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. 3. The union of two sets, and , contains all points that are in both sets and is symbolized as 4. The intersection of two sets and includes only the points common to both sets and is symbolized as

# Geometry ## Angles ### Learning Objectives After completing this section, you should be able to: 1. Identify and express angles using proper notation. 2. Classify angles by their measurement. 3. Solve application problems involving angles. 4. Compute angles formed by transversals to parallel lines. 5. Solve application problems involving angles formed by parallel lines. Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, landscaping, space planning, and so on. This is the topic of this section. We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called the vertex. We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is vertices. In , let the ray stay put. Rotate the second ray in a counterclockwise direction to the size of the angle you want. The angle is formed by the amount of rotation of the second ray. When the ray continues to rotate in a counterclockwise direction back to its original position coinciding with ray the ray will have swept out We call the rays the “sides” of the angle. ### Classifying Angles Angles are measured in radians or degrees. For example, an angle that measures radians, or 3.14159 radians, is equal to the angle measuring An angle measuring radians, or 1.570796 radians, measures To translate degrees to radians, we multiply the angle measure in degrees by For example, to write in radians, we have To translate radians to degrees, we multiply by For example, to write radians in degrees, we have Another example of translating radians to degrees and degrees to radians is To write in degrees, we have To write in radians, we have . However, we will use degrees throughout this chapter. Several angles are referred to so often that they have been given special names. A straight angle measures ; a right angle measures an acute angle is any angle whose measure is less than and an obtuse angle is any angle whose measure is between and See . An easy way to measure angles is with a protractor (). A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here. With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of the angle. ### Notation Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols or before the points. When we are referring to the measure of the angle, we use the symbol . See . We can name this angle , or , or ### Adjacent Angles Two angles with the same starting point or vertex and one common side are called adjacent angles. In , angle is adjacent to . Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle. ### Supplementary Angles Two angles are supplementary if the sum of their measures equals In , we are given that so what is These are supplementary angles. Therefore, because , and as we have ### Complementary Angles Two angles are complementary if the sum of their measures equals In , we have and What is the These are complementary angles. Therefore, because the ### Vertical Angles When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well. ### Transversals When two parallel lines are crossed by a straight line or transversal, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See . Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles. ### Alternate Interior Angles Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example, and are alternate interior angles and have equal measure; and are alternate interior angles and have equal measure as well. See . ### Alternate Exterior Angles Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in , and are alternate exterior angles and have equal measures; and are alternate exterior angles and have equal measures as well. ### Corresponding Angles Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In , and are corresponding angles and have equal measures; and are corresponding angles and have equal measures; and are corresponding angles and have equal measures; and are corresponding angles and have equal measures as well. ### Check Your Understanding ### Key Terms 1. vertex 2. right angle 3. acute angle 4. obtuse angle 5. straight angle 6. complementary 7. supplementary ### Key Concepts 1. Angles are classified as acute if they measure less than obtuse if they measure greater than and less than right if they measure exactly and straight if they measure exactly 2. If the sum of angles equals , they are complimentary angles. If the sum of angles equals , they are supplementary. 3. A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles ### Formula To translate an angle measured in degrees to radians, multiply by To translate an angle measured in radians to degrees, multiply by

# Geometry ## Triangles ### Learning Objectives After completing this section, you should be able to: 1. Identify triangles by their sides. 2. Identify triangles by their angles. 3. Determine if triangles are congruent. 4. Determine if triangles are similar. 5. Find the missing side of similar triangles. How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the concept of similar triangles has countless applications in the real world, and we shall explore some of those applications in this section. Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore congruence theorems and similarity. ### Identifying Triangles Joining any three noncollinear points with line segments produces a triangle. For example, given points , , and , connected by the line segments and we have a triangle, as shown in . Triangles are classified by their angles and their sides. All angles in an acute triangle measure One of the angles in a right triangle measures symbolized by □. One angle in an obtuse triangle measures between and Sides that have equal length are indicated by the same hash marks. illustrates the shapes of the basic triangles, their names, and their properties. A few other facts to remember as we move forward: 1. The points where the line segments meet are called the vertices (plural for vertex). 2. We often refer to sides of a triangle by the angle they are opposite. In other words, side is opposite angle , side is opposite angle , and side is opposite angle . We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the hypotenuse, and the sides adjacent to the right angle are called the legs. One of the most important properties of triangles is that the sum of the interior angles equals Euclid discovered and proved this property using parallel lines. The completed sketch is shown in . This is how the proof goes: Step 1: Start with a straight line and a point not on the line. Step 2: Draw a line through point parallel to the line . Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines. Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that Step 5: Notice that by the straight angle property. Step 6: Therefore, by substitution, and we have that Therefore, the sum of the interior angles of a ### Congruence If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent. ### The Congruence Theorems The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol to define congruence. For example, . Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent. See . We have that , and then Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. See . We see that and , , then . Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See . Notice that , and , , then Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent. See . We see that , , and , then . ### Similarity If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction is called a ratio, but two fractions equal to each other is called a proportion, such as This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling factor, which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image. For example, why are the two images in are similar? These two images have the same proportions between elements. Therefore, they are similar. ### Check Your Understanding ### Key Terms 1. acute 2. obtuse 3. isosceles 4. equilateral 5. hypotenuse 6. congruence 7. similarity 8. scaling factor ### Key Concepts 1. The sum of the interior angles of a triangle equals 2. Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal. 3. The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle. 4. Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.

# Geometry ## Polygons, Perimeter, and Circumference ### Learning Objectives After completing this section, you should be able to: 1. Identify polygons by their sides. 2. Identify polygons by their characteristics. 3. Calculate the perimeter of a polygon. 4. Calculate the sum of the measures of a polygon’s interior angles. 5. Calculate the sum of the measures of a polygon’s exterior angles. 6. Calculate the circumference of a circle. 7. Solve application problems involving perimeter and circumference. In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose. Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design. When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle. ### Identifying Polygons A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See for some examples. We show only up to eight-sided polygons, but there are many, many more. If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons. However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in and , are both pentagons because they each have five sides and five vertices. The third shape is a hexagon because it has six sides and six vertices. We should note here that the hexagon in is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out. While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See . ### Perimeter Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths. A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula: For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus, The length is 8 units. The perimeter of a regular polygon with sides is given as . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is . ### Sum of Interior and Exterior Angles To find the sum of the measurements of interior angles of a regular polygon, we have the following formula. For example, if we want to find the sum of the interior angles in a parallelogram, we have Similarly, to find the sum of the interior angles inside a regular heptagon, we have To find the measure of each interior angle of a regular polygon with sides, we have the following formula. For example, find the measure of an interior angle of a regular heptagon, as shown in . We have An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with sides is found using the following formula: In , we have a regular hexagon . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, . Now, an important point is that the sum of the exterior angles of a regular polygon with sides equals This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get For the example in , we multiply the measure of each exterior angle, , by the number of sides, six. Thus, the sum of the exterior angles is ### Circles and Circumference The perimeter of a circle is called the circumference. To find the circumference, we use the formula where is the diameter, the distance across the center, or where is the radius. The radius is ½ of the diameter of a circle. The symbol is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See . Let the radius be equal to 3.5 inches. Then, the circumference is ### Check Your Understanding ### Key Terms 1. perimeter 2. polygon 3. pentagon 4. hexagon 5. heptagon 6. octagon 7. quadrilateral 8. trapezoid 9. parallelogram 10. circumference ### Key Concepts 1. Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides. 2. The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides. 3. The sum of the interior angles of a regular polygon with sides is found using the formula The measure of a single interior angle of a regular polygon with sides is determined using the formula 4. The sum of the exterior angles of a regular polygon is The measure of a single exterior angle of a regular polygon with sides is found using the formula 5. The circumference of a circle is where is the radius, or and is the diameter. ### Formulas The formula for the perimeter of a rectangle is , twice the length plus twice the width . The sum of the interior angles of a polygon with sides is given by The measure of each interior angle of a regular polygon with sides is given by To find the measure of an exterior angle of a regular polygon with sides we use the formula The circumference of a circle is found using the formula where is the diameter of the circle, or where is the radius.

# Geometry ## Tessellations ### Learning Objectives After completing this section, you should be able to: 1. Apply translations, rotations, and reflections. 2. Determine if a shape tessellates. The illustration shown above () is an unusual pattern called a Penrose tiling. Notice that there are two types of shapes used throughout the pattern: smaller green parallelograms and larger blue parallelograms. What's interesting about this design is that although it uses only two shapes over and over, there is no repeating pattern. In this section, we will focus on patterns that do repeat. Repeated patterns are found in architecture, fabric, floor tiles, wall patterns, rug patterns, and many unexpected places as well. It may be a simple hexagon-shaped floor tile, or a complex pattern composed of several different motifs. These two-dimensional designs are called regular (or periodic) tessellations. There are countless designs that may be classified as regular tessellations, and they all have one thing in common—their patterns repeat and cover the plane. We will explore how tessellations are created and experiment with making some of our own as well. The topic of tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects can be moved while retaining the same shape and size. These movements are termed rigid motions and symmetries. ### Tessellation Properties and Transformations A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including some type of movement; that is, some type of transformation or symmetry. Here we consider the rigid motions of translations, rotations, reflections, or glide reflections. A plane of tessellations has the following properties: In , the tessellation is made up of squares. There are four squares meeting at a vertex. An interior angle of a square is and the sum of four interior angles is In , the tessellation is made up of regular hexagons. There are three hexagons meeting at each vertex. The interior angle of a hexagon is and the sum of three interior angles is Both tessellations will fill the plane, there are no gaps, the sum of the interior angle meeting at the vertex is and both are achieved by translation transformations. These tessellations work because all the properties of a tessellation are present. The movements or rigid motions of the shapes that define tessellations are classified as translations, rotations, reflections, or glide reflections. Let’s first define these movements and then look at some examples showing how these transformations are revealed. ### Translation A translation is a movement that shifts the shape vertically, horizontally, or on the diagonal. Consider the trapezoid in . We have translated it 3 units to the right and 3 units up. That means every corner is moved by the number of units and in the direction specified. Mathematicians will indicate this movement with a vector, an arrow that is drawn to illustrate the criteria and the magnitude of the translation. The location of the translated trapezoid is marked with the vertices, but it is still the exact same shape and size as the original trapezoid . ### Rotation The rotation transformation occurs when you rotate a shape about a point and at a predetermined angle. In , the triangle is rotated around the rotation point by and then translated 7 units up and 4 units over to the right. That means that each corner is translated to the new location by the same number of units and in the same direction. We can see that is mapped to by a rotation of up and to the right. If rotated again by , the triangle would be upside down. ### Reflection A reflection is the third transformation. A shape is reflected about a line and the new shape becomes a mirror image. You can reflect the shape vertically, horizontally, or on the diagonal. There are two shapes in . The quadrilateral is reflected horizontally; the arrow shape is reflected vertically. ### Glide Reflection The glide reflection is the fourth transformation. It is a combination of a reflection and a translation. This can occur by first reflecting the shape and then gliding or translating it to its new location, or by translating first and then reflecting. The example in shows a trapezoid, which is reflected over the dashed line, so it appears upside down. Then, we shifted the shape horizontally by 6 units to the right. Whether we use the glide first or the reflection first, the end result is the same in most cases. However, the tessellation shown in the next example can only be achieved by a reflection first and then a translation. ### Interior Angles The sum of the interior angles of a tessellation is . In , the tessellation is made of six triangles formed into the shape of a hexagon. Each angle inside a triangle equals , and the six vertices meet the sum of those interior angles, . In , the tessellation is made up of trapezoids, such that two of the interior angles of each trapezoid equals and the other two angles equal . Thus, the sum of the interior angles where the vertices of four trapezoids meet equals . These tessellations illustrate the property that the shapes meet at a vertex where the interior angles sum to . ### Tessellating Shapes We might think that all regular polygons will tessellate the plane by themselves. We have seen that squares do and hexagons do. The pattern of squares in is a translation of the shape horizontally and vertically. The hexagonal pattern in , is translated horizontally, and then on the diagonal, either to the right or to the left. This particular pattern can also be formed by rotations. Both tessellations are made up of congruent shapes and each shape fits in perfectly as the pattern repeats. We have also seen that equilateral triangles will tessellate the plane without gaps or overlaps, as shown in . The pattern is made by a reflection and a translation. The darker side is the face of the triangle and the lighter side is the back of the triangle, shown by the reflection. Each triangle is reflected and then translated on the diagonal. Escher experimented with all regular polygons and found that only the ones mentioned, the equilateral triangle, the square, and the hexagon, will tessellate the plane by themselves. Let’s try a few other regular polygons to observe what Escher found. Just because regular pentagons do not tessellate the plane by themselves does not mean that there are no pentagons that tessellate the plane, as we see in . Another example of an irregular polygon that tessellates the plane is by using the obtuse irregular triangle from a previous example. What transformations should be performed to produce the tessellation shown in ? First, the triangle is reflected over the tip at point , and then translated to the right and joined with the original triangle to form a parallelogram. The parallelogram is then translated on the diagonal and to the right and to the left. ### Naming A tessellation of squares is named by choosing a vertex and then counting the number of sides of each shape touching the vertex. Each square in the tessellation shown in has four sides, so starting with square , the first number is 4, moving around counterclockwise to the next square meeting the vertex, square , we have another 4, square adds another 4, and finally square adds a fourth 4. So, we would name this tessellation a 4.4.4.4. The hexagon tessellation, shown in has six sides to the shape and three hexagons meet at the vertex. Thus, we would name this a 6.6.6. The triangle tessellation, shown in has six triangles meeting the vertex. Each triangle has three sides. Thus, we name this a 3.3.3.3.3.3. ### Check Your Understanding ### Key Terms 1. tessellation 2. translation 3. reflection 4. rotation 5. glide reflection ### Key Concepts 1. A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps. 2. Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern. ### Videos 1. M.C. Escher: How to Create a Tessellation 2. The Mathematical Art of M.C. Escher

# Geometry ## Area ### Learning Objectives After completing this section, you should be able to: 1. Calculate the area of triangles. 2. Calculate the area of quadrilaterals. 3. Calculate the area of other polygons. 4. Calculate the area of circles. Some areas carry more importance than other areas. Did you know that in a baseball game, when the player hits the ball and runs to first base that he must run within a 6-foot wide path? If he veers off slightly to the right, he is out. In other words, a few inches can be the difference in winning or losing a game. Another example is real estate. On Manhattan Island, one square foot of real estate is worth far more than real estate in practically any other area of the country. In other words, we place a value on area. As the context changes, so does the value. Area refers to a region measured in square units, like a square mile or a square foot. For example, to purchase tile for a kitchen floor, you would need to know how many square feet are needed because tile is sold by the square foot. Carpeting is sold by the square yard. As opposed to linear measurements like perimeter, which in in linear units. For example, fencing is sold in linear units, a linear foot or yard. Linear dimensions refer to an outline or a boundary. Square units refer to the area within that boundary. Different items may have different units, but either way, you must know the linear dimensions to calculate the area. Many geometric shapes have formulas for calculating areas, such as triangles, regular polygons, and circles. To calculate areas for many irregular curves or shapes, we need calculus. However, in this section, we will only look at geometric shapes that have known area formulas. The notation for area, as mentioned, is in square units and we write sq in or sq cm, or use an exponent, such as or Note that linear measurements have no exponent above the units or we can say that the exponent is 1. ### Area of Triangles The formula for the area of a triangle is given as follows. For example, consider the triangle in . The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area: In , the triangle has a base equal to 7 cm and a height equal to 3.5 cm. Notice that we can only find the height by dropping a perpendicular to the base. The area is then ### Area of Quadrilaterals A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The most common quadrilaterals are the square and the rectangle. ### Square In , a grid is represented with twelve squares across each row, and twelve squares down each column. If you count the little squares, the sum equals 144 squares. Of course, you do not have to count little squares to find area—we have a formula. Thus, the formula for the area of a square, where , is . The area of the square in is ### Rectangle Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in has width equal to 5 in and length equal to 12 in. The area is Many everyday applications require the use of the perimeter and area formulas. Suppose you are remodeling your home and you want to replace all the flooring. You need to know how to calculate the area of the floor to purchase the correct amount of tile, or hardwood, or carpet. If you want to paint the rooms, you need to calculate the area of the walls and the ceiling to know how much paint to buy, and the list goes on. Can you think of other situations where you might need to calculate area? ### Parallelogram The area of a parallelogram can be found using the formula for the area of a triangle. Notice in , if we cut a diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area of a triangle by 2, we have the area of a parallelogram: For example, if we have a parallelogram with the base be equal to 10 inches and the height equal to 5 inches, the area will be ### Trapezoid Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a trapezoid with parallel bases and and height is given here. For example, find the area of the trapezoid in that has base equal to 8 cm, base equal to 6 cm, and height equal to 6 cm. The area is . ### Rhombus The rhombus has two sets of parallel sides. To find the area of a rhombus, there are two formulas we can use. One involves determining the measurement of the diagonals. For our purposes here, we will use the formula that uses diagonals. For example, if the area of a rhombus is and the measure of find the measure of To solve this problem, we input the known values into the formula and solve for the unknown. See . We have that ### Area of Polygons To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem of a regular polygon is a line segment that starts at the center and is perpendicular to a side. The radius of a regular polygon is also a line segment that starts at the center but extends to a vertex. See . For example, consider the regular hexagon shown in with a side length of 4 cm, and the apothem measures We have the perimeter, We have the apothem as Then, the area is: ### Changing Units Often, we have the need to change the units of one or more items in a problem to find a solution. For example, suppose you are purchasing new carpet for a room measured in feet, but carpeting is sold in terms of yards. You will have to convert feet to yards to purchase the correct amount of carpeting. Or, you may need to convert centimeters to inches, or feet to meters. In each case, it is essential to use the correct equivalency. ### Area of Circles Just as the circumference of a circle includes the number so does the formula for the area of a circle. Recall that is a non-terminating, non-repeating decimal number: . It represents the ratio of the circumference to the diameter, so it is a critical number in the calculation of circumference and area. For example, to find the of the circle with radius equal to 3 cm, as shown in , is found using the formula We have ### Area within Area Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space for it. These are common issues people face every day. So, let’s take a look at how we solve these problems. ### Check Your Understanding ### Key Terms 1. triangle 2. square 3. rectangle 4. rhombus 5. apothem 6. radius 7. circle ### Key Concepts 1. The area of a triangle is found with the formula where is the base and is the height. 2. The area of a parallelogram is found using the formula where is the base and is the height. 3. The area of a rectangle is found using the formula where is the length and is the width. 4. The area of a trapezoid is found using the formula where is the height, is the length of one base, and is the length of the other base. 5. The area of a rhombus is found using the formula where is the length of one diagonal and is the length of the other diagonal. 6. The area of a regular polygon is found using the formula where is the apothem and is the perimeter. 7. The area of a circle is found using the formula where is the radius. ### Formulas The area of a triangle is given as where represents the base and represents the height. The formula for the area of a square is or The area of a rectangle is given as The area of a parallelogram is The formula for the area of a trapezoid is given as The area of a rhombus is found using one of these formulas: The area of a regular polygon is found with the formula where is the apothem and is the perimeter. The area of a circle is given as where is the radius.

# Geometry ## Volume and Surface Area ### Learning Objectives After completing this section, you should be able to: 1. Calculate the surface area of right prisms and cylinders. 2. Calculate the volume of right prisms and cylinders. 3. Solve application problems involving surface area and volume. Volume and surface area are two measurements that are part of our daily lives. We use volume every day, even though we do not focus on it. When you purchase groceries, volume is the key to pricing. Judging how much paint to buy or how many square feet of siding to purchase is based on surface area. The list goes on. An example is a three-dimensional rendering of a floor plan. These types of drawings make building layouts far easier to understand for the client. It allows the viewer a realistic idea of the product at completion; you can see the natural space, the volume of the rooms. This section gives you practical information you will use consistently. You may not remember every formula, but you will remember the concepts, and you will know where to go should you want to calculate volume or surface area in the future. We will concentrate on a few particular types of three-dimensional objects: right prisms and right cylinders. The adjective “right” refers to objects such that the sides form a right angle with the base. We will look at right rectangular prisms, right triangular prisms, right hexagonal prisms, right octagonal prisms, and right cylinders. Although, the principles learned here apply to all right prisms. ### Three-Dimensional Objects In geometry, three-dimensional objects are called geometric solids. Surface area refers to the flat surfaces that surround the solid and is measured in square units. Volume refers to the space inside the solid and is measured in cubic units. Imagine that you have a square flat surface with width and length. Adding the third dimension adds depth or height, depending on your viewpoint, and now you have a box. One way to view this concept is in the Cartesian coordinate three-dimensional space. The -axis and the -axis are, as you would expect, two dimensions and suitable for plotting two-dimensional graphs and shapes. Adding the -axis, which shoots through the origin perpendicular to the -plane, and we have a third dimension. See . Here is another view taking the two-dimensional square to a third dimension. See . To study objects in three dimensions, we need to consider the formulas for surface area and volume. For example, suppose you have a box () with a hinged lid that you want to use for keeping photos, and you want to cover the box with a decorative paper. You would need to find the surface area to calculate how much paper is needed. Suppose you need to know how much water will fill your swimming pool. In that case, you would need to calculate the volume of the pool. These are just a couple of examples of why these concepts should be understood, and familiarity with the formulas will allow you to make use of these ideas as related to right prisms and right cylinders. ### Right Prisms A right prism is a particular type of three-dimensional object. It has a polygon-shaped base and a congruent, regular polygon-shaped top, which are connected by the height of its lateral sides, as shown in . The lateral sides form a right angle with the base and the top. There are rectangular prisms, hexagonal prisms, octagonal prisms, triangular prisms, and so on. Generally, to calculate surface area, we find the area of each side of the object and add the areas together. To calculate volume, we calculate the space inside the solid bounded by its sides. In , we have three views of a right hexagonal prism. The regular hexagon is the base and top, and the lateral faces are the rectangular regions perpendicular to the base. We call it a right prism because the angle formed by the lateral sides to the base is See . The first image is a view of the figure straight on with no rotation in any direction. The middle figure is the base or the top. The last figure shows you the solid in three dimensions. To calculate the surface area of the right prism shown in , we first determine the area of the hexagonal base and multiply that by 2, and then add the perimeter of the base times the height. Recall the area of a regular polygon is given as where is the apothem and is the perimeter. We have that Then, the surface area of the hexagonal prism is To find the volume of the right hexagonal prism, we multiply the area of the base by the height using the formula The base is and the height is 20 . Thus, ### Right Cylinders There are similarities between a prism and a cylinder. While a prism has parallel congruent polygons as the top and the base, a right cylinder is a three-dimensional object with congruent circles as the top and the base. The lateral sides of a right prism make a angle with the polygonal base, and the side of a cylinder, which unwraps as a rectangle, makes a angle with the circular base. Right cylinders are very common in everyday life. Think about soup cans, juice cans, soft drink cans, pipes, air hoses, and the list goes on. In , imagine that the cylinder is cut down the 12-inch side and rolled out. We can see that the cylinder side when flat forms a rectangle. The formula includes the area of the circular base, the circular top, and the area of the rectangular side. The length of the rectangular side is the circumference of the circular base. Thus, we have the formula for total surface area of a right cylinder. To find the volume of the cylinder, we multiply the area of the base with the height. ### Applications of Surface Area and Volume The following are just a small handful of the types of applications in which surface area and volume are critical factors. Give this a little thought and you will realize many more practical uses for these procedures. ### Optimization Problems that involve optimization are ones that look for the best solution to a situation under some given conditions. Generally, one looks to calculus to solve these problems. However, many geometric applications can be solved with the tools learned in this section. Suppose you want to make some throw pillows for your sofa, but you have a limited amount of fabric. You want to make the largest pillows you can from the fabric you have, so you would need to figure out the dimensions of the pillows that will fit these criteria. Another situation might be that you want to fence off an area in your backyard for a garden. You have a limited amount of fencing available, but you would like the garden to be as large as possible. How would you determine the shape and size of the garden? Perhaps you are looking for maximum volume or minimum surface area. Minimum cost is also a popular application of optimization. Let’s explore a few examples. ### Check Your Understanding ### Key Terms 1. surface area 2. volume 3. right prism 4. right cylinder ### Key Concepts 1. A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that lateral sides form a angle with the base and top. The surface area of a right prism is found using the formula where is the area of the base, is the perimeter of the base, and is the height. The volume of a right prism is found using the formula where is the area of the base and is the height. 2. A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side forms a angle to the base and top. The surface area of a right cylinder is found using the formula where is the radius and is the height. The volume is found using the formula where is the radius and is the height. ### Formulas The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base times the height, where is equal to the area of the base and top, is the perimeter of the base, and is the height. The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the height, where is the area of the base and is the height. The surface area of a right cylinder is given as The volume of a right cylinder is given as

# Geometry ## Right Triangle Trigonometry ### Learning Objectives After completing this section, you should be able to: 1. Apply the Pythagorean Theorem to find the missing sides of a right triangle. 2. Apply the and right triangle relationships to find the missing sides of a triangle. 3. Apply trigonometric ratios to find missing parts of a right triangle. 4. Solve application problems involving trigonometric ratios. This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the Pythagorean Theorem. However, there is evidence that the theorem was known as early as 1900–1100 BC by the Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles. Born in Greece, Pythagoras lived from 569–500 BC. He initiated a cult-like group called the Pythagoreans, which was a secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, and many other concepts that left an immeasurable influence on future mathematicians and scientists. The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only unknown sides but unknown angles, as well, with little information. ### Pythagorean Theorem The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle). For example, given that side and side we can find the measure of side using the Pythagorean Theorem. Thus, ### Distance The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In fact, the distance formula stems directly from the theorem. It works like this: In , the problem is to find the distance between the points and We call the length from point to point side , and the length from point to point side . To find side , we use the distance formula and we will explain it relative to the Pythagorean Theorem. The distance formula is such that is a substitute for in the Pythagorean Theorem and is equal to and is a substitute for in the Pythagorean Theorem and is equal to When we plug in these numbers to the distance formula, we have Thus, , the hypotenuse, in the Pythagorean Theorem. ### Triangles In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the perfect square rule in algebra. When expanding an expression like we do not have to expand it the long way: If we know the perfect square formula, given as we can skip the middle step and just start writing down the answer. This may seem trivial with problems like However, what if you have a problem like That is a different story. Nevertheless, we use the same perfect square formula. The same idea applies in geometry. There are special formulas and procedures to apply in certain types of problems. What is needed is to remember the formula and remember the kind of problems that fit. Sometimes we believe that because a formula is labeled special, we will rarely have use for it. That assumption is incorrect. So, let us identify the triangle and find out why it is special. See . We see that the shortest side is opposite the smallest angle, and the longest side, the hypotenuse, will always be opposite the right angle. There is a set ratio of one side to another side for the triangle given as or Thus, you only need to know the length of one side to find the other two sides in a triangle. ### Triangles The triangle is another special triangle such that with the measure of one side we can find the measures of all the sides. The two angles adjacent to the angle are equal, and each measures If two angles are equal, so are their opposite sides. The ratio among sides is or as shown in . ### Trigonometry Functions Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in other triangles and many applications. Trigonometry is based on three functions. We title these functions using the following abbreviations: Letting which is the hypotenuse of a right triangle, we have . The functions are given in terms of , , and , and in terms of sides relative to the angle, like opposite, adjacent, and the hypotenuse. We will be applying the sine function, cosine function, and tangent function to find side lengths and angle measurements for triangles we cannot solve using any of the techniques we have studied to this point. In , we have an illustration mainly to identify and the sides labeled and . An angle sweeps out in a counterclockwise direction from the positive -axis and stops when the angle reaches the desired measurement. That ray extending from the origin that marks is called the terminal side because that is where the angle terminates. Regardless of the information given in the triangle, we can find all missing sides and angles using the trigonometric functions. For example, in , we will solve for the missing sides. Let’s use the trigonometric functions to find the sides and . As long as your calculator mode is set to degrees, you do not have to enter the degree symbol. First, let’s solve for . We have and Then, Next, let’s find . This is the cosine function. We have Then, Now we have all sides, You can also check the sides using the ratio of is a list of common angles, which you should find helpful. To find angle measurements when we have two side measurements, we use the inverse trigonometric functions symbolized as or The –1 looks like an exponent, but it means inverse. For example, in the previous example, we had and To find what angle has these values, enter the values for the inverse cosine function in your calculator: You can also use the inverse sine function and enter the values of in your calculator given and We have Finally, we can also use the inverse tangent function. Recall We have ### Angle of Elevation and Angle of Depression Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object at a lower level. ### Check Your Understanding ### Key Terms 1. right triangle 2. sine 3. cosine 4. tangent ### Key Concepts 1. The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formula where c is the hypotenuse. 2. To find the measure of the sides of a special angle, such as a triangle, use the ratio where each of the three sides is associated with the opposite angle and 2 is associated with the hypotenuse, opposite the angle. 3. To find the measure of the sides of the second special triangle, the triangle, use the ratio where each of the three sides is associated with the opposite angle and is associated with the hypotenuse, opposite the angle. 4. The primary trigonometric functions are and 5. Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles. ### Formula The Pythagorean Theorem states where and are two sides (legs) of a right triangle and is the hypotenuse. ### Projects 1. One of the reasons so many formulas in geometry were discovered was because of the importance in finding measurements of lengths, areas, perimeter, and angles. Find at least five examples of how geometry can be used in practical applications today. 2. Who were the Pythagoreans? Why did this society exist? Explore what they did and discuss some of their beliefs. ### Chapter Review ### Points, Lines, and Planes ### Angles ### Triangles ### Polygons, Perimeter, and Circumference ### Tessellations ### Area ### Volume and Surface Area ### Right Triangle Trigonometry ### Chapter Test

# Voting and Apportionment ## Introduction Suppose a friend asked you, “When did you last vote?” What would your answer be? Maybe you would tell your friend that the last time you voted was during the last presidential election, or perhaps you would tell your friend that you prefer not to vote. When thinking about voting, presidential campaigns or advertisements for reelections may come to mind, but you can cast your vote in many ways. Have you liked a post, followed a creator, friended a stranger, or clicked a heart online today? In the digital age, it's possible to vote several times a day. Voting systems are not only the machines that drive every democracy on Earth, but they are also the engines driving social media and many other aspects of life. A deeper understanding of these voting systems will enhance your ability to successfully engage with the world in which we live. In this chapter, you will become one of the founders of the new democratic country of Imaginaria. You have a great responsibility to the people of this fledgling democracy. You have been tasked with writing the portion of the constitution that lays out voting procedures. In preparation for this important task, you will explore the various types of voting systems, from school board elections to Twitter wars. You will see how these types are alike, how they differ, and how they might be applied in Imaginaria. Most importantly, you will learn about the mathematically inherent advantages and disadvantages of various voting systems so that you can make informed choices to better the lives of the Imaginarians.

# Voting and Apportionment ## Voting Methods ### Learning Objectives After completing this section, you should be able to: 1. Apply plurality voting to determine a winner. 2. Apply runoff voting to determine a winner. 3. Apply ranked-choice voting to determine a winner. 4. Apply Borda count voting to determine a winner. 5. Apply pairwise comparison and Condorcet voting to determine a winner. 6. Apply approval voting to determine a winner. 7. Compare and contrast voting methods to identify flaws. Today is the day that you begin your quest to collaborate on the constitution of Imaginaria! Let’s begin by thinking about the selection of a leader who can serve as president. It seems straightforward; if the majority of citizens prefer a particular candidate, that candidate should win. But not all votes are decided by a simple majority. Why not? What are the options? ### Majority versus Plurality Voting When an election involves only two options, a simple majority is a reasonable way to determine a winner. A majority is a number equaling more than half, or greater than 50 percent of the total. Let’s take a look at the outcomes of U.S. presidential elections to understand more. displays the results of the 2000 U.S. presidential election. Like most presidential elections, this election involved more than two options. If that is the case, is it possible that no single candidate will receive more than half of the votes cast? Unlike in the 2000 U.S. presidential election, a candidate won the majority of votes in the 2020 election (see ). It is a common occurrence for no single candidate to receive a majority of the votes in an election with more than two candidates. When this occurs, the candidate with the largest portion of the votes is said to have a plurality. Your plans for Imaginarian elections will likely include primary elections, or preliminary elections to select candidates for a principal or general election. displays the results of the 2018 U.S. Senate Republican primary for Maryland. Consider how election by plurality, not majority, is the most common method of selecting candidates for public office. ### Runoff Voting Has your family ever debated what to have for dinner? Suppose your family is deciding on a restaurant and exactly half of you want to have pizza but the other half want hamburgers. How do you decide when the result is a tie? You need a tiebreaker! Will the new democracy of Imaginaria need tiebreakers? When no candidate satisfies the requirements to win the election, a runoff election, or second election, is held to determine a winner. How would runoff voting work in Imaginaria? There are many types of runoff voting systems, which are voting systems that utilize a runoff election when the first round does not result in a winner. The method for implementing a runoff election can vary widely, particularly in the criteria used to determine whether a candidate will be on the ballot in the second election. For example, a two-round system is a runoff voting system in which only the top candidates advance to the runoff election. In some two-round systems, only the top two candidates are on the second ballot, or it may be any candidate who secures a certain percentage of the vote will advance. The Hare Method is another runoff voting system in which only the candidate(s) with the very least votes are eliminated. This can potentially result in several rounds of runoff elections. ### Steps to Determine Winner by Plurality or Majority Election with Runoff To determine the winner by plurality or when a majority election with runoff occurs, we take these three steps: Step 1: If a majority is required to win the election, determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes. If a majority is not required, move to Step 2. Step 2: Count the number of votes for each candidate in the current round of voting. If a single candidate has enough votes to win a plurality, or a majority as appropriate, then you are done! Otherwise, eliminate a predetermined number of candidates based on the rules of the election. Elimination conditions may vary. For example, the rules may state that the candidate(s) with the fewest votes will be eliminated (as in the Hare method), or that only the candidates meeting a certain threshold will move on (as in a two-round system). Once the appropriate candidates are eliminated, move on to Step 3. Step 3: Hold a runoff election. If the runoff is simulated using a list of voter’s preferences, renumber the preferences to reflect the remaining number of options in such a way that the original order of preference is retained. Then repeat Step 2. Note: The second and third steps may be repeated as many times as necessary for voting procedures that allow multiple runoffs. ### Ranked-Choice Voting In and Your Turn 11.4, you were given a list that ordered each voter’s preferences. This ordering is called a preference ranking. A ballot in which a voter is required to give an ordering of their preferences is a ranked ballot, and any voting system in which a voter uses a ranked ballot is referred to as ranked voting. The vote for the Academy Awards uses a ranked ballot. The table below provides an example of a ranked ballot for the 2020 Academy Award nominees for Best Director. As you decide on the voting methods that will be used in your new democracy, budget must be a consideration. You might consider a particular type of ranked voting called ranked-choice voting (RCV), which simulates a series of runoff elections without the usual time and expense involved when voters must repeatedly return to the polls, like we did in . The method of ranked-choice voting (RCV), also called instant runoff voting (IRV), is a version of the Hare Method, using preference ranking so that, if no single candidate receives a majority, the least popular selections can be eliminated and the results can be recounted, without the need for more elections. As we explore examples of ranked voting, we will summarize the voters’ preference rankings using a table in which the top row shows the number of ballots that ranked the options in the same order. Let’s practice interpreting the information in this type of table. Now that we’ve covered how to read a summary of preference rankings, let’s practice using the ranked-choice method to determine the winner of an election. Recall that ranked-choice voting is still the Hare Method where the candidate with the very least number of votes is eliminated each round until a majority is attained. The difference here is that the voters have completed a ranked ballot, so they don't have to visit the polls multiple times. Here are the steps for ranked-choice voting. ### Steps to Determine Winner by Ranked-Choice Voting To determine the winner when ranked-choice voting occurs, we take these three steps: Step 1: Determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes. Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election and we are done! Otherwise, eliminate the candidate(s) with the fewest votes and complete Step 3. Step 3: Reallocate the votes to the remaining candidates, and repeat Step 2. ### Borda Count Voting Ranked-choice voting is one type of ranked voting that simulates multiple runoffs based on ranked ballots. Another type of ranked voting is the Borda count method, which uses ranked ballots that award candidates points corresponding to the number of candidates ranked lower on each ballot. To understand how this works, let’s review the favorite colors of our kindergarten class from the table below. Let’s focus on the votes represented by the first column of the preference summary. Each student had six options. This first column tells us that four students ranked blue as their first choice, red as their second choice, pink as their third choice, purple as their fourth choice, yellow as their fifth choice, and green as their sixth choice. Blue was ranked higher than other colors. For each of the four students who completed their ballot in this way, blue would receive five points. Since there were four ballots with this ordering, blue would receive points from the first column. To determine the total points for each candidate, we have to find the sum of the points they received in each column. To determine the winner of a contest using the Borda count method, we must compare total number of points earned by each candidate. The candidate with the most points is the winner. Each row of the preference summary corresponds to a single candidate. To find the number of points received by a particular candidate in the preference summary, or their Borda score, we will need to focus on the row in which that candidate appears. Before we practice determining the winner of a Borda count election, let’s examine how to find the Borda score for a single candidate. Now let’s determine the winner of an election by comparing the Borda scores for each of the candidates. The Borda count method may seem too complicated to even consider using for Imaginaria, but each voting method has its own pros and cons. The Borda count method, for example, favors compromise candidates over divisive candidates. A compromise candidate is not the first choice of most of the voters, but is more acceptable to the population as a whole than the other candidates. A divisive candidate is simultaneously the first choice of a large portion of the voters and the last choice of another large portion of the voters. In , Candidate A was ranked first by 225 voters, but was ranked last by 185 voters. No voters ranked Candidate A as second or third. It appears that, although Candidate A had the majority of first place votes, there was a significant minority who strongly disliked them. Candidate A was a divisive candidate. Candidate B, on the other hand, was the second choice of every voter, making Candidate B a good compromise. The Borda count method chose Candidate B, a compromise candidate, that was more acceptable to the population as a whole. This scenario is cited by both opponents and proponents of the Borda count method. ### Pairwise Comparison and Condorcet Voting We have discussed two kinds of ranked voting methods so far: ranked-choice and Borda count. A third type of ranked voting is the pairwise comparison method, in which the candidates receive a point for each candidate they would beat in a one-on-one election and half a point for each candidate they would tie. If one candidate earns more points than the others, then that candidate wins. This method is one of several Condorcet voting methods, which are methods in which candidates are ranked and then compared pairwise to each other, a candidate having to beat all others in order to win. These methods vary in the way candidates are scored, and there is not always a clear winner. A candidate who wins each possible pairing is known as a Condorcet candidate. These terms are named after the Marquis de Condorcet, a French philosopher and mathematician who preferred the pairwise comparison method to the Hare method and made public arguments in its favor. If you include a Condorcet voting method in the constitution of Imaginaria, the election supervisors may want to use a pairwise comparison matrix like the one in . It’s a tool used to list the number of wins associated with each pairing of two candidates. Each candidate will receive a point for each win and a half a point for each tie. Each pairing is listed twice, once for the number of wins of a candidate over a particular challenger and once for the number of wins of the challenger over that candidate. ### Steps to Determine a Winner by Pairwise Comparison Method Using a Matrix To determine the winner when the pairwise comparison method is used, we take these three steps: Step 1: On the matrix, indicate a losing matchup by crossing out a box, , and tie match ups by drawing a slash through the box, . Step 2: Award each candidate 1 point for a win, half a point for a tie, and 0 points for a loss. Step 3: Identify the winner, which is the candidate with the most points. Before you decide on the pairwise comparison method for Imaginaria, review what’s involved in constructing a pairwise comparison matrix from a summary of ranked ballots. Then we can use the matrix to determine the winner of the election. Does the winner using the Borda method still win? ### Three Key Questions Before you decide if you want to use the pairwise comparison method for Imaginarian elections, let’s consider three questions that might affect your decision. 1. Is there always a winner? 2. If there is a winner, is the winner always a Condorcet candidate? 3. If there is a Condorcet candidate, does that candidate always win? Let’s think about why these questions might be important to you if you chose the pairwise comparison method. First, if no candidate meets the criteria to win an election, you will need a backup plan such as a runoff election. Second, if the winner is not a Condorcet candidate, then there is at least one candidate who beat the winner in a pairwise matchup and the supporters of that candidate might question the validity of the election. Finally, if there is a Condorcet candidate who beat every other candidate in a pairwise matchup, it is reasonable to conclude that it would be unfair for anyone else to win. The rest of the examples in this section should illustrate these key concepts. illustrates the answer to the first key question. The pairwise comparison method does not always result in a winner. For example, much like the game of Rock, Paper, Scissors, it is possible for a cyclic pattern to emerge in which each candidate beats the next until the last candidate who beats the first. Now, you have the answer to the second key question. The pairwise comparison matrix in YOUR TURN 11.10 is an example of a scenario where a winner is not a Condorcet candidate. The answer to the third question is not as clear. If there is a Condorcet candidate, does that candidate win? So far, we have not come across a contradictory example where the Condorcet candidate didn't win, but we cannot know with certainty that it is not possible by looking at examples. Instead, we will need to use some reasoning. Let’s review some particular cases of elections with a certain number of candidates, and then we will try to generalize the scenario to an election with candidates. Let’s consider a general case where there are candidates. One of the candidates is a Condorcet candidate. Since the Condorcet candidate wins all matchups, the Condorcet candidate wins points. Since each of the other candidates lost to the Condorcet candidate, the most a single candidate could win is . Since the Condorcet candidate won points and each other candidate won points or fewer, the Condorcet candidate is the winner. You have your answer to the third key question! If there is a Condorcet candidate, that candidate is always the winner. ### Approval Voting The last type of voting system you will consider for your budding democracy is an approval voting system. In this system, each voter may approve any number of candidates without rank or preference for one over another (among the approved candidates), and the candidate approved by the most voters wins. This voting system has aspects in common with plurality voting and Condorcet voting methods, but it has characteristics that distinguish it from both. An approval voting ballot lists the candidates and provides the option to approve or not approve each candidate. The term “approval voting” was not used until the 1970s Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, Springer-Verlag, p. xv, ISBN 978-0-387-49895-9, although its use has been documented as early as the 13th century (Brams, Steven J. (April 1, 2006). The Normative Turn in Public Choice (PDF) (Speech). Presidential Address to Public Choice Society. New Orleans, Louisiana.) Approval voting has the appeal of being simpler than ranked voting methods. It also allows an individual voter to support more than one candidate equally. This has appeal for those who do not want a split vote among a few mainstream candidates to lead to the election of a fringe candidate. It also has appeal for those who want an underdog to have a chance of success because voters will not worry about wasting their vote on a candidate who is not believed likely to win. ### Compare and Contrast Voting Methods to Identify Flaws Wow! We have covered a lot of options for the voting methods. Now, you need to decide which one is best for Imaginaria. Imaginarians might consider characteristics of certain voting systems desirable and others undesirable. In some cases, voters may consider these undesirable traits to be flaws in a voting system that are significant enough to motivate them to reject that system. If you are feeling a bit overwhelmed by this decision, maybe it would help to read about the experiences of others who have faced similar questions. Consider the 2000 U.S. presidential election in which Green Party candidate Ralph Nader and Reform Party candidate Pat Buchanan were on the ballet running against the mainstream candidates, Democrat Al Gore and Republican George W. Bush. The voting results for Florida are given in . In more than one state, Buchanan was able to split the Republican vote enough to allow Gore to win that state. Nader split the Democrat vote in Florida and New Hampshire by enough votes to prevent Gore from winning those states. Had Gore won either state, he would have had enough electoral votes to win the election. Instead, Bush won. This is an example of a flaw in the plurality system of voting: the spoiler. A spoiler is a less popular candidate who takes votes from a more popular candidate with similar positions, swinging the race to another candidate with vastly different views that they would not support. This encourages voters not to vote for the candidate that they perceive to be the best, but instead for the candidate they can live with who they perceive to have a better chance of winning. Some voters may prefer a method such as approval voting, which does not have this trait in common with plurality voting. The results in and Your Turn 11.14 highlight one of the characteristics of approval voting. Ralph Nader moved up from a distant third place finish to a close second place finish when Al Gore’s supporters approved him on their ballots. In this way, fringe candidates have a better chance of winning, which some voters consider a flaw but others consider a benefit. Another aspect of approval voting systems that is a concern to many voters is that candidates in approval elections might encourage their loyal supporters to approve them and only them to avoid giving support to any other candidate. If this occurred, the election in effect becomes a traditional plurality election. This is a flaw that cannot occur in an instant runoff system since all candidates are ranked. The election in involves a scenario in which there are two extreme candidates, Planet A and Planet B, and a moderate candidate, Planet C. The supporters of the extreme candidates prefer the moderate candidate to the other extremist ones. This makes Planet C a compromise candidate. In this case, both the plurality method and ranked-choice voting resulted in the election of one of the extreme candidates, but the Borda count method elected the compromise candidate in this scenario. Depending on a person’s perspective, this may be perceived as a flaw in either ranked-choice and plurality systems, or the Borda count method. In Fairness in Voting Methods, we will analyze the fairness of each voting system in greater detail using objective measures of fairness. ### Check Your Understanding ### Key Terms 1. majority 2. plurality 3. runoff election 4. runoff voting system 5. two-round system 6. Hare Method 7. preference ranking 8. ranked ballot 9. ranked-choice voting (RCV) 10. instant runoff voting (IRV) 11. Borda count method 12. Borda score 13. Compromise candidate 14. divisive candidate 15. pairwise comparison method 16. Condorcet voting methods 17. Condorcet candidate 18. approval voting system 19. approval voting ballot 20. spoiler ### Key Concepts 1. In plurality voting, the candidate with the most votes wins. 2. When a voting method does not result in a winner, runoff voting can be used to do so. 3. Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system. 4. The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on the number of candidates ranked lower than them on each ballot. 5. When pairwise comparison is used, the winner will be the Condorcet candidate if one exists. 6. Approval voting allows voters to give equally weighted votes to multiple candidates. 7. When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic a flaw in the voting method and look for an alternative method that does not have that characteristic. ### Videos 1. How Does Ranked-Choice Voting Work? 2. Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff) 3. Determine Winner of Election by Borda Count Method 4. Determine Winner of Election by Pairwise Comparison Method

# Voting and Apportionment ## Fairness in Voting Methods ### Learning Objectives After completing this section, you should be able to: 1. Compare and contrast fairness of voting using majority criterion. 2. Compare and contrast fairness of voting using head-to-head criterion. 3. Compare and contrast fairness of voting using monotonicity criterion. 4. Compare and contrast fairness of voting using irrelevant alternatives criterion. 5. Apply Arrow’s Impossibility Theorem when evaluating voting fairness. Now that we’ve covered a variety of voting methods and discussed their differences and similarities, you might be leaning toward one method over another. You will need to convince the other founders of Imaginaria that your preference will be the best for the country. Before your collaborators approve the inclusion of a voting method in the constitution, they will want to know that the voting method is a fair method. In this section, we will formally define the characteristics of a fair system. We will analyze each voting previously discussed to determine which characteristics of fairness they have, and which they do not. In order to guarantee one ideal, we must often sacrifice others. ### The Majority Criterion One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a candidate to win, and that a candidate should not win without majority support. This concept is known as the majority criterion. With respect to the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—we will explore two important questions: 1. Which of these voting systems satisfy the majority criterion and which do not? 2. Is it always “fair” for a voting system to satisfy the majority criterion? Keep in mind that this criterion only applies when one of the candidates has a majority. So, the examples we will analyze will be based on scenarios in which a single candidate has more than 50 percent of the vote. From , it appears that the plurality and ranked-choice voting methods satisfy the majority criterion. In general, the majority candidate always wins in a plurality election because the candidate that has more than half of the votes has more votes than any other candidate. The same is true for ranked-choice voting; and there will never be a need for a second round when there is a majority candidate. Let’s examine how some of the other voting methods stand up to the majority criterion. demonstrates a concept that we also saw in Borda Count Voting—the Borda method frequently favors the compromise candidate over the divisive candidate. This can happen even when the divisive candidate has a majority, as it did in this example. Although a majority of the voters were in favor of McDonald’s, a significant minority was strongly opposed to McDonald’s, ranking it last. Since the Borda score includes all rankings, this strong opposition has an impact on the outcome of the election. Pairwise comparison will always satisfy the majority criterion because the candidate with the majority of first-place votes wins each pairwise matchup. While it is possible for the majority candidate to win by the Borda count method, it is not guaranteed. So, the Borda method fails the majority criterion. A summary of each voting method as it relates to the majority criterion is found in the following table. If you prefer the Borda method, you might argue that its failure to satisfy the majority criterion is actually one of its strengths. As we saw in and , the majority have the power to vote for their own benefit at the expense of the minority. While four students were very enthusiastic about McDonald’s, three students were strongly opposed to McDonald’s. It is reasonable to say that the better option would be Burger King, the compromise candidate, which everyone ranked highly and no one strongly opposed. The Borda method is designed to favor a candidate that is acceptable to the population as a whole. In this way, the Borda method avoids a downfall of strict majority rule known as the tyranny of the majority, which occurs when a minority of a population is treated unfairly because their situation is different from the situation of the majority. The people of Imaginaria should know that the power of the majority to vote their will has serious implications for other groups. For example, according to the UCLA School of Law Williams Institute, the LGBTQ+ community in the United States makes up approximately 4.5 percent of the population. When elections occur that include issues that affect the LGBTQ+ community, members of the LGBTQ+ community depend on the 95.5 percent of the population who do not identify as LGBTQ+ to consider their perspectives when voting on issues such as same-sex marriage, the use of public restrooms by transgender people, and adoption by same-sex couples. ### Head-to-Head Criterion Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion, also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting method that satisfies the Condorcet criterion. Recall from Three Key Questions that not every election has a Condorcet candidate; the Condorcet criterion will not apply to every election. Also recall that a Condorcet candidate cannot lose an election by pairwise comparison. So, the pairwise comparison voting method is said to satisfy the Condorcet criterion. As we have seen, the plurality method, ranked-choice voting, and the Borda count method each fail the Condorcet criterion in some circumstances. Of the four main ranked voting methods we have discussed, only the pairwise comparison method satisfies the Condorcet criterion every time. A summary of each voting method as it relates to the Condorcet criterion is found in the following table. ### Monotonicity Criterion The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of a violation of the fairness criterion called the monotonicity criterion, which is satisfied when no candidate is harmed by up-ranking nor helped by down-ranking, provided all other votes remain the same. Consider a scenario in which voters are permitted a first round that is not binding, and then they may change their vote before the second round. Such a first round can be called a “straw poll.” Now, let’s suppose that a particular candidate won the straw poll. After that, several voters are convinced to increase their support, or up-rank, that winning candidate and no voters decrease that support. It is reasonable to expect that the winner of the first round will also win the second. Similarly, if some of the voters decide to decrease their support, or down-rank, a losing candidate, it is reasonable to expect that candidate will still lose in the second round. You might be wondering why it’s called the monotonicity criterion. In mathematics, the term monotonicity refers to the quality of always increasing or always decreasing. For example, a person’s age is monotonic because it always increases, whereas a person’s weight is not monotonic because it can increase or decrease. If the only changes to the votes for a particular candidate after a straw poll are in one direction, this change is considered monotonic. If you are going to make an informed decision about which voting method to use in Imaginaria, you need to know which of the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—satisfy the monotonicity criterion. The last few examples illustrate that the plurality method, pairwise comparison voting, and the Borda count method each satisfy the monotonicity criterion. Of the four main ranked voting methods we have discussed, only the ranked-choice method violates the monotonicity criterion. A summary of each voting method as it relates to the Condorcet criterion is found in the table below. ### Irrelevant Alternatives Criterion We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney Morgenbesser: A man is told by his waiter that the dessert options this evening are blueberry pie or apple pie. The man orders the apple pie. The waiter returns and tells him that there is also a third option, cherry pie. The man says, “In that case, I would like the blueberry pie.” (Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It), William Pound stone, p. 50, ISBN 0-8090-4893-0) This story illustrates the concept of the Irrelevant Alternatives Criterion, also known as the Independence of Irrelevant Alternatives Criterion (IIA), which means that the introduction or removal of a third candidate should not change or reverse the rankings of the original two candidates relative to one another. In particular, if a losing candidate is removed from the race or if a new candidate is added, the winner of the race should not change. We have seen that all four of the main voting systems we are working with fail the Irrelevant Alternatives Criterion (IIA). A summary of each voting method as it relates to the IIA criterion is found in the table below. You might be wondering if there is a voting system you could recommend for Imaginaria that satisfies all the fairness criteria. If there is one, it remains to be discovered and it is not a voting system that is based solely on preference rankings. In 1972, Harvard Professor of Economics Kenneth J. Arrow received the Nobel Prize in Economics for proving Arrow’s Impossibility Theorem, which states that any voting system, either existing or yet to be created, in which the only information available is the preference rankings of the candidates, will fail to satisfy at least one of the following fairness criteria: the majority criterion, the Condorcet criterion, the monotonicity criterion, and the independence of irrelevant alternatives criterion. This theorem only applies to a specific category of voting systems—those for which the preference ranking is the only information collected. There are other types of voting systems to which the Impossibility Theorem does not apply. For example, there is a class of voting systems called Cardinal voting systems that allow for rating the candidates in some way. “Rating” is different from “ranking” because a voter can give different candidates the same rating. Consider the five-star rating systems used by various industries, or the thumbs up/thumbs down rating system used on YouTube. Could there be a Cardinal voting system that does not violate any of the fairness criteria we have discussed? It’s possible, but more research must be done in order to prove it. ### Check Your Understanding ### Key Terms 1. majority criterion 2. tyranny of the majority 3. Condorcet criterion 4. Condorcet method 5. monotonicity criterion 6. up-rank 7. down-rank 8. independence of irrelevant alternatives criterion (IIA) 9. Arrow’s Impossibility Theorem 10. cardinal voting system ### Key Concepts 1. There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion. 2. According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria. ### Videos 1. Separation of Powers and Checks and Balances

# Voting and Apportionment ## Standard Divisors, Standard Quotas, and the Apportionment Problem ### Learning Objectives After completing this section, you should be able to: 1. Analyze the apportionment problem and applications to representation. 2. Evaluate applications of standard divisors. 3. Evaluate applications of standard quotas. ### The Apportionment Problem In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting method your citizens will use to elect representatives. However, before that process can even begin, you must decided on how many representatives each state will receive. This decision will present its own challenges. When sharing your birthday cake, it’s only fair that everyone gets the same portion size, right? You were portioning the cake by dividing it up equally and giving everyone a slice. A great thing about cake is that you can slice it any way you want, but how do you apportion, or divide and distribute, items that can't be sliced? Suppose that you have a box of 16 Ring Pops™, gem-shaped lollipops on a plastic ring. You are going to share the box with four other kids. Dividing the 16 Ring Pops™ among the group of five leads to a problem; after each person in the group gets three Ring Pops™, there is still one left! Who gets the last one? The apportionment problem is how to fairly divide, or apportion, available resources that must be distributed to the recipients in whole, not fractional, parts. The apportionment problem applies to many aspects of life, including the representatives in the Imaginarian legislature. The table below provides a short list of examples of resources that must be apportioned in whole parts, and the recipients of those resources. Fair division of a resource is not necessarily equal division of the resource like when distributing cake slices. When distributing airport terminals amongst airlines, there are many factors to consider such as the size of the airline, the number and types of aircraft they have, and the demand for the service. In most cases, fairness is defined as being proportional; two quantities are proportional if they have the same relative size. In the case of the Covid-19 vaccine, the expectation would be that countries with larger populations get more doses of the vaccine. In the Imaginarian legislature, the expectation may be that the states with larger populations will receive the larger number of representatives. This concept is referred to as a part-to-part ratio. Suppose that a supermarket has a special on pies, two for $5. The first customer purchases four pies for $10, and the second customer purchases eight pies for $20. The dollar to pie ratio for the first customer is and the dollar to pie ratio for the second customer is . So, the dollar to pie ratio is constant. Although the customers do not spend the same amount of money, the amount each spent was proportional to the number of pies purchased. Now suppose that the supermarket changed the special to $5 for the first pie, and $2 for each additional pie. In that case, four pies would cost , while 8 pies would cost . The dollar to pie ratios would be and , respectively. This special does not result in a constant part to part ratio. The dollars spent are not proportional to the number of pies purchased. There are some useful relationships between quantities that are proportional to each other. When there is a constant ratio between two quantities, the one quantity can be found by multiplying the other by that ratio. Remember the supermarket special on pies, 2 pies for $5? The ratio of dollars to pies is and the ratio of pies to dollars is. These two values are reciprocals of each other, and . This means that multiplying by one has the same effect as dividing by the other. This also means that knowing either constant ratio allows us to calculate the price given the number of pies. To find the cost of 20 pies, multiply by the ratio of dollars to pies or divide by the ratio of pies to dollars. 1. 2. These patterns are true in general. The apportionment application that will be important to the founders of Imaginaria occurs in representative democracies in which elected persons represent a group. The United Kingdom, France, and India each have a parliament, and the United States has a Congress, just as Imaginaria will have a legislature! The citizens of a country must decide what portion of the representatives each group, such as a state or province or even a political party, will have. A larger portion of representatives means greater influence over policy. You might be wondering why the ratio doesn't appear to be quite the same depending on the rounding of the values. We will see that the key to this variation lies in the fractions. Just like the five children sharing 16 Ring Pops™, there are going to be leftovers and there are many methods for deciding what to do with those leftovers. ### The Standard Divisor There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two senators, but the number of representatives depends on the population of the state. The number of representative seats in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total population to the house size is called the standard divisor, and it is the number of members of the total population represented by one seat. Although apportionment applies to many other scenarios, such as the pencil distribution during the SAT, the terminology of apportionment is based on the House of Representatives scenario. Thus, several government-related terms take on a more general meaning. The states are the recipients of the apportioned resource, the seats are the units of the resource being apportioned, the house size is the total number of seats to be apportioned, the state population is the measurement of the state's size, and the total population is the sum of the state populations. Whether the standard divisor is less than, equal to, or greater than 1 depends on the ratio of the population to the number of seats. 1. The standard divisor will be equal to 1 if the total population is equal to the number of seats. This would mean that each member of the population is allocated their own personal seat. 2. The standard divisor will be a number between 0 and 1 when the total population is less than the number of seats. This means that each member of the population is allocated more than one seat. 3. The standard divisor will be a number greater than 1when the total population is greater than the number of seats. This means that a certain number of members of the population will share 1 seat. If the population is five children and the house consists of five pieces of candy, the standard divisor is meaning each child gets one candy. If the population is five children and ten pieces of candy, the standard divisor is meaning that each child gets more than one candy. If the population is five children and four pieces of candy, the standard divisor is meaning that each child gets less than one candy. If the seats in the Imaginarian legislature are distributed to the states based on population, then the house size will be less than the population and we should expect the standard divisor to be a number greater than 1. ### The Standard Quota Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats that each state should receive, which is referred to as the state’s standard quota. Unless all the states have the same population, each state will receive a different number of seats because the quantities will be proportionate to the state populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units of item is proportionate to the number of units of item , we have: In this case, we are trying to calculate the number of seats a state should be apportioned, the state’s standard quota. So So would refer to seats allocated to a particular state, while would refer to the state population. This means that the ratio of to is the ratio of the total population to house size, which is the standard divisor. So in apportionment terms, we have the following formula. ### Check Your Understanding ### Key Terms 1. apportion 2. apportionment problem 3. proportional 4. part-to-part ratio 5. representative democracies 6. standard divisor 7. states 8. seats 9. house size 10. state population 11. total population 12. standard quota ### Key Concepts 1. The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not fractional, parts. 2. To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state population, total population, and house size, or the total number of seats to be apportioned. 3. The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of seats that each state should receive. ### Formulas Let be a particular item and another such that there is a constant ratio of to . 1. and 2. 3. ### Videos 1. What Is a Ratio? 2. What Are the Different Types of Ratios? 3. Math Antics – Rounding

# Voting and Apportionment ## Apportionment Methods ### Learning Objectives After completing this section, you should be able to: 1. Describe and interpret the apportionment problem. 2. Apply Hamilton’s Method. 3. Describe and interpret the quota rule. 4. Apply Jefferson’s Method. 5. Apply Adams’s Method. 6. Apply Webster’s Method. 7. Compare and contrast apportionment methods. 8. Identify and contrast flaws in various apportionment methods. ### A Closer Look at the Apportionment Problem In Standard Divisors, Standard Quotas, and the Apportionment Problem we calculated the standard divisor and the standard quotas in various apportionment scenarios. The results of those calculations routinely led to fractions and decimals of units. However, the seats in the House of Representatives, laptops in a classroom, or a variety of other resources, are indivisible, meaning they cannot be divided up into fractional parts. This leaves a decision to be made. For example, if the standard quota for the number of laptops to be distributed to a classroom is 12.44 units, how do we deal with the fractional part of 0.44? It is unclear if the classroom should receive 12 units, 13 units, or some other value. Let’s try traditional rounding to the nearest whole number value. demonstrates that we cannot successfully apportion indivisible resources by rounding off each standard quota using traditional rounding. This leaves us with a problem. What is a fair way to distribute the fractional parts of the standard quotas? We will refer to this as the apportionment problem. Several methods for making this decision will be discussed. ### Hamilton's Method of Apportionment One of the problems encountered when standard quotas are transformed into whole numbers using traditional rounding is that it is possible for the sum of the values to be greater than the number of seats available. A reasonable way to avoid this is to always round down, even when the first decimal place is five or greater. For example, a standard quota of 12.33 and a standard quota of 12.99 would both round down to 12. This is called the lower quota. If the standard quotas are all rounded down, their sum will always be less than or equal to the house size. Then, it would only remain to find a fair way to distribute any remaining seats. Alexander Hamilton, who was a general in the American Revolution, author of the Federalist Papers, and the first secretary of the treasury, took this approach to apportionment. ### Steps for Hamilton’s Method of Apportionment There are five steps we follow when applying Hamilton’s Method of apportionment: 1. Find the standard divisor. 2. Find each state’s standard quota. 3. Give each state the state’s lower quota (with each state receiving at least 1 seat). 4. Give each remaining seat one at a time to the states with the largest fractional parts of their standard quotas until no seats remain. 5. Check the solution by confirming that the sum of the modified quotas equals the house size. ### The Quota Rule A characteristic of an apportionment that is considered favorable is when the final quota values all either result from rounding down or rounding up from the standard quotas. The value that results from rounding down is called the lower quota, and the value that results from rounding up is called the upper quota. As we explore more methods of apportionment, we will consider whether they satisfy the quota rule. If a scenario exists in which a particular apportionment allocates a value greater than the upper quota or less than the lower quota, then that apportionment violates the quota rule and the apportionment method that was used violates the quota rule. It is possible for an apportionment method to satisfy the quota rule in some scenarios but violate it in others. However, because the Hamilton method always begins with the lower quota and either adds one to it or keeps it the same, the final Hamilton quota will always consist of values that are either lower quota values or upper quota values. When an apportionment method has this characteristic, it is said to satisfy the quota rule. So, we can say: The Hamilton method of apportionment satisfies the quota rule. Although the Hamilton method of apportionment satisfies the quota rule, it can result in some unexpected outcomes, which has caused it to pass in and out of favor of the U.S. government over the years. There are several apportionment methods that have been popular alternatives, such as Jefferson’s method of apportionment that the founders of Imaginaria should consider. ### Jefferson’s Method of Apportionment Another approach to dealing with the fractional parts of the standard quotas is to modify the standard divisor so that the total of the resulting modified lower quotas is the necessary number of seats. This is the approach used by Jefferson. In Jefferson’s method, the change to the standard divisor is made so that the total of the modified lower quotas equals the house size. The change in the standard divisor to get the modified divisor is relatively small. There is not a formula for this. The modified divisor is found by “guess and check.” It is important to remember that increasing the divisor decreases the quotas, but decreasing the divisor increases the quotas. So, if you need a larger quota, try reducing the divisor, and if you need a smaller quota, try increasing the divisor. When you use Jefferson’s method, you might have to adjust the divisor several times find modified lower quotas that sum to the house size. First, guess what the divisor should be based on the sum of the lower quotas and then increase or decrease it from there based on whether the sum needs to be smaller or larger respectively. If the result still does not produce lower quotas that sum to the house size, adjust again. Keep a record of the values that didn't work to help you narrow your search. ### Steps for Jefferson’s Method of Apportionment We take four steps to apply Jefferson’s Method of apportionment: Step 1: Find the standard divisor. Step 2: Find each state’s quota. This will be the standard quota the first time Step 2 is completed and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas. Step 3: Find the states’ lower quotas (with each state receiving at least one seat), and their sum. Step 4: If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the lower quotas is less than the number of seats, reduce the standard divisor. If the sum of the lower quotas is greater than the number of seats, increase the standard divisor. Return to Step 2 using the modified divisor. Notice that, in this apportionment, Mythbury received more than the upper quota. Since this apportionment of representatives to Imaginarian states by Jefferson’s method does not satisfy the quota rule, we say that: Jefferson’s method violates the quota rule. We have discussed two apportionment methods: one that satisfies the quota rule and one that does not. Before you decide which method to use in Imaginaria, there are a couple more options to consider. ### Adams’s Method of Apportionment Adams’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did, Adams used the upper quotas. To apply Adams’s Method of apportionment, there are four steps we follow: 1. Find the standard divisor. 2. Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resultingin modified quotas. 3. Find the states’ upper quotas and their sum. 4. If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the upper quotas is less than the number of seats, reduce the standard divisor. If the sum of the upper quotas is greater than the number of seats, increase the standard divisor. Return to Step 2 using the modified divisor. In this apportionment, Mythbury received less than the state’s lower quota. So, this apportionment is an example of a scenario in which the Adams’s method violates the quota rule. Adams’s method of apportionment violates the quota rule. So far, only Hamilton’s method satisfies the quota rule, but there is one more apportionment method you should consider for Imaginaria. ### Webster’s Method of Apportionment Webster’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did or the sum of the upper quotas as Adams did, Webster used traditional rounding. To apply Webster’s method of apportionment, there are four steps we take: 1. Find the standard divisor. 2. Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas. 3. Round each state’s quota to the nearest whole number and find the sum of these values. 4. If the sum of the rounded quotas equals the number of seats, the apportionment is complete. If the sum of the rounded quotas is less than the number of seats, reduce the divisor. If the sum of the rounded quotas is greater than the number of seats, increase the divisor. Return to Step 2 using the modified divisor. When using Webster’s method, just as with Jefferson’s method, the modified divisors you use may be different from what another person chooses, but final apportionment values will be the same. So far, we know that the Hamilton method satisfies the quota rule, while the Jefferson and Adams methods do not. The apportionments in the Example and Your Turn above are both scenarios in which the Webster method satisfies the quota rule. Does it always? We have a little more work to do to find out. However, one thing is clear. Not all apportionment methods have the same results. Before you make such an important decision for Imaginaria, it’s important to think about the differences in the apportionments that result from these four methods. How will the differences affect the citizens of Imaginaria? ### Comparing Apportionment Methods Recall that the four apportionment methods discussed in this chapter differ in two main ways: 1. Whether or not a modified divisor is used 2. The type of rounding of the quotas that is used How might these differences affect Imaginarians? In the next two examples, we will compare the results when different apportionment methods are applied to the same scenario. The Adams method favored the smaller states and the Jefferson method favored the larger states in the previous example, but is this the case in general? Since the Jefferson method begins with the lower quotas, any adjustment to the quotas will be an increase. As you have seen, this is accomplished by using a modified divisor that is smaller than the standard divisor. The next example compares the impact of a decreasing divisor on the modified quotas of large states to the impact of the same size decrease on small states. This example demonstrates that the Jefferson method is biased toward states with larger populations because the modified divisor is smaller than the standard divisor. On the other hand, the Adams’s method, which begins with the upper quotas, must increase the standard divisor in order to reduce the quotas. Once again, the effect on the number of seats is greater for the larger states, but this time they are decreased. This means that the Adams’s method favors states with smaller populations. ### Flaws in Apportionment Methods As we have seen, different apportionment methods can have the same results in some scenarios but different results in others. Citizens of states which receive fewer seats with a particular apportionment method will view the apportionment method as flawed and argue in favor of a different method. This inevitably creates debates regarding the use of one method over another. Methods that favor larger states are likely to be challenged by smaller states, methods that favor smaller states are likely to be challenged by larger states, and methods that violate the quota rule are likely to be challenged by states of any size depending on the circumstances. Suppose that the State of Hawaii House of Representatives had 51 representatives, each with their own district. Imagine that redistricting were underway, and the representative districts were to be apportioned to each of five counties based on population. The following table shows the apportionment that would result from the use of the Jefferson, Adams, and Webster methods of apportionment. From the table, you can see that Hawaii, Kalawao, and Maui receive the same number of seats regardless of the method used. However, citizens of Honolulu would likely reject the Adams and Webster methods arguing that they violate the quota rule. Similarly, citizens of Kauai would probably reject the Jefferson method based on the argument that it unfairly favors the larger states. This scenario demonstrates that the Adams and Webster methods violate the quota rule, but the Jefferson method also violates the quota rule at times. The Hamilton method is the only method that satisfies the quota rule in all scenarios. It also consistently favors neither larger nor smaller states. Unfortunately, it can have some strange and results in certain circumstances, which you will see in the next section. ### Check Your Understanding ### Key Terms 1. apportionment problem 2. lower quota 3. upper quota ### Key Concepts 1. Hamilton’s method of apportionment uses the standard divisor and standard lower quotas, and it distributes any remaining seats based on the size of the fractional parts of the standard lower quota. Hamilton’s method satisfies the quota rule and favors neither larger nor smaller states. 2. Jefferson’s method of apportionment uses a modified divisor that is adjusted so that the modified lower quotas sum to the house size. Jefferson’s method violates the quota rule and favors larger states. 3. Adams’s method of apportionment uses a modified divisor that is adjusted so that the modified upper quotas, sum to the house size. Adams’s method violates the quota rule and favors smaller states. 4. Webster’s method of apportionment uses a modified divisor that is adjusted so that the modified state quotas, rounded using traditional rounding, sum to the house size. Webster’s method violates the quota rule but favors neither larger nor smaller states. ### Videos 1. Hamilton Method of Apportionment 2. Jefferson Apportionment Method 3. Adams Method Apportionment Calculator

# Voting and Apportionment ## Fairness in Apportionment Methods ### Learning Objectives After completing this section, you should be able to: 1. Describe and illustrate the Alabama paradox. 2. Describe and illustrate the population paradox. 3. Describe and illustrate the new-states paradox. 4. Identify ways to promote fairness in apportionment methods. ### Apportionment Paradoxes The citizens of Imaginaria will want the apportionment method to be as fair as possible. There are certain characteristics that they would reasonably expect from a fair apportionment. 1. If the house size is increased, the state quotas should all increase or remain the same but never decrease. 2. If one state’s population is growing more rapidly than another state’s population, the faster growing state should not lose a seat while a slower growing state maintains or gains a seat. 3. If there is a fixed number of seats, adding a new state should not cause an existing state to gain seats while others lose them. However, apportionment methods are known to contradict these expectations. Before you decide on the right apportionment for Imaginarians, let’s explore the apportionment paradox, a situation that occurs when an apportionment method produces results that seem to contradict reasonable expectations of fairness. There is a lot that the founders of Imaginaria can learn from U.S. history. The constitution of the United States requires that the seats in the House of Representatives be apportioned according to the results of the census that occurs every decade, but the number of seats and the apportionment method is not stipulated. Over the years, several different apportionment methods and house sizes have been used and scrutinized for fairness. This scrutiny has led to the discovery of several of these apportionment paradoxes. ### The Alabama Paradox At the time of the 1880 U.S. Census, the Hamilton method of apportionment had replaced the Jefferson method. The number of seats in the House of Representatives was not fixed. To achieve the fairest apportionment possible, the house sizes were chosen so that the Hamilton and Webster methods would result in the same apportionment. The chief clerk of the Census Bureau calculated the apportionments for house sizes between 275 and 350. There was a surprising result that became known as the Alabama paradox, which is said to occur when an increase in house size reduces a state’s quota. Alabama would receive eight seats with a house size of 299, but only receive seven seats if the house size increased to 300. (Michael J. Caulfield (Gannon University), "Apportioning Representatives in the United States Congress - Paradoxes of Apportionment," Convergence (November 2010), DOI:10.4169/loci003163) After the 1900 census, the Census Bureau again calculated the apportionment based on various house sizes. It was determined that Colorado would receive three seats with a house size of 356, but only two seats with a house size of 357. ### The Population Paradox It is important for the founders of Imaginaria to keep in mind that the populations of states change as time passes. Some populations grow and some shrink. Some populations increase by a large amount while others increase by a small amount. These changes may necessitate a reapportionment of seats, or the recalculation of state quotas due to a change in population. It would be reasonable for Imaginarians to expect that the state with a population that has grown more than others will gain a seat before the other states. Once again, this is not always the case with the Hamilton method of apportionment. The population growth rate of a state is the ratio of the change in the population to the original size of the population, often expressed as a percentage. This value is positive if the population is increasing and negative if the population is decreasing. The population paradox occurs when a state with an increasing population loses a seat while a state with a decreasing population either retains or gains seats. More generally, the population paradox occurs when a state with a higher population growth rate loses seats while a state with a lower population growth rate retains or gains seats. Notice that the population paradox definitions has two parts. If either part applies, then the population paradox has occurred. The first part of the definition only applies when one state has a decreasing population. The second part of the definition applies in all situations, whether there is a state with a decreasing population or not. It will be easier to identify situations that involve a decreasing population. The other situations requires the calculation of a growth rate. The reason that we don't have to calculate a growth rate when one state has a decreasing population and the other has an increasing population is that increasing population has a positive growth rate which is always greater than the negative growth rate of a decreasing population. ### The New-States Paradox As a founder of Imaginaria, you might also consider the possibility that Imaginaria could annex nearby lands and increase the number of states. This occurred several times in the United States such as when Oklahoma became a state in 1907. The House size was increased from 386 to 391 to accommodate Oklahoma’s quota of five seats. When the seats were reapportioned using Hamilton’s method, New York lost a seat to Maine despite the fact that their populations had not changed. This is an example of the new-state paradox, which occurs when the addition of a new state is accompanied by an increase in seats to maintain the standard ratio of population to seats, but one of the existing states loses a seat in the resulting reapportionment. When a new state is added, it is necessary to determine the amount that the house size must be increased to retain the original ratio of population to seats, in other words to keep the original standard divisor. To calculate the new house size, divide the new population by the original standard divisor, and round to the nearest whole number. ### The Search for the Perfect Apportionment Method The ideal apportionment method would simultaneously satisfy the following four fairness criteria. 1. Satisfy the quota rule 2. Avoid the Alabama paradox 3. Avoid the population paradox 4. Avoid the new-states paradox We have seen that the Hamilton method allows the Alabama paradox, the population paradox, and the new-states paradox in some apportionment scenarios. Let’s explore the results of the other methods of apportionment we have discussed in some of the same scenarios. We have seen in our examples that neither the population paradox nor the new-states paradox occurred when using the Jefferson, Adams, and Webster methods. It turns out that, although all three of these divisor methods violate the quota rule, none of them ever causes the population paradox, new-states paradox, or even the Alabama paradox. On the other hand, the Hamilton method satisfies the quota rule, but will cause the population paradox, the new-states paradox, and the Alabama paradox in some scenarios. In 1983, mathematicians Michel Balinski and Peyton Young proved that no method of apportionment can simultaneously satisfy all four fairness criteria. There are other apportionment methods that satisfy different subsets of these fairness criteria. For example, the mathematicians, Balinski and Young who proved the Balinski-Young Impossibility Theorem created a method that both satisfies the quota rule and is free of the Alabama paradox. (Balinski, Michel L.; Young, H. Peyton (November 1974). “A New Method for Congressional Apportionment.” Proceedings of the National Academy of Sciences. 71 (11): 4602–4606.) However, no method may always follow the quota rule and simultaneously be free of the population paradox. (Balinski, Michel L.; Young, H. Peyton (September 1980). "The Theory of Apportionment" (PDF). Working Papers. International Institute for Applied Systems Analysis. WP-80-131.) So, as you and your fellow founders of Imaginaria make the important decision about the right apportionment method for Imaginaria, do not look for a perfect apportionment method. Instead, look for an apportionment method that best meets the needs and concerns of Imaginarians. ### Check Your Understanding ### Key Terms 1. apportionment paradox 2. Alabama paradox 3. reapportionment 4. population growth rate 5. population paradox 6. new-state paradox ### Key Concepts 1. Several surprising outcomes can occur when apportioning seats that voters may find unfair: Alabama paradox, population paradox, and new-state paradox. 2. Apportionment methods are susceptible to apportionment paradoxes and may violate the quota rule. 3. The Balinsky-Young Impossibility Theorem indicates that no apportionment can satisfy all fairness criteria. ### Formulas ### Projects ### The First Census: Challenges of Collecting Population Data As you and your fellow founders write the Imaginarian constitution, you must also design systems to collect accurate information about the population of Imaginaria. Why is this important and what are the challenges you will face? Let’s find out! Research and answer each question. (Questions adapted from “Authorizing the First Census–The Significance of Population Data,” Census.gov, United States Census Bureau) 1. The First Congress laid out a plan for collecting this data in Chapter II, An Act Providing for the Enumeration of the Inhabitants of the United States, which was approved March 1, 1790. What group did Congress select to carry out the first enumeration? Why did they choose this group? What might be the advantages and disadvantages to this approach? 2. Choose at least two challenges faced by the U.S. government during the first enumeration and explain how the information gathered might have helped address them. 3. President James Madison was a U.S. Representative who participated in the census debate in the first congress. What were the main points of his remarks and how were they relevant to the overall debate about the first enumeration? 4. What issues do you think the U.S. Census Bureau encounters today as it continues to collect and process data about the U.S. population that might be significant to you and the other founders of Imaginaria? ### The Party System: How Many Political Parties Are Enough? The electoral system of Imaginaria will likely involve multiple political parties. The way these parties interact with the system may be determined by the founders of the new democracy. Let’s explore the ways in which political parties interact with the governments of various democracies by researching and answering each of the following questions. 1. What concerns did the founders of the United States have about political parties? Have any of their concerns become a reality? Were political parties addressed in the U.S. Constitution? How did they become such a large part of politics in the U.S.? As a founder of Imaginaria, would you address political parties in your constitution? 2. What is frontloading? Does our current system of frontloading impact fair representation? Why do two small, racially homogeneous states hold their primaries first? Do you think this impacts the final results? 3. How does the interaction political parties and the electoral system in the United States differ from that of other countries? Give at least three specific examples. Of these three, which would you be most likely to use as a model for Imaginaria? ### Technology and Voting: How Has the Digital Age Impacted Elections? Unlike most other countries, Imaginaria will be founded in the digital age. Let’s explore the impact this might have on how you choose to set up your electoral system. Research and answer each question. 1. Participation in an electoral system if very important. In what ways does the Internet positively affect participation? In what ways does it negatively affect participation. What roll, if any, should government of Imaginaria play in tempering or promoting the Impact of the Internet? 2. Find at least three examples of governments that utilize Internet voting around the world. What concerns have slowed the spread of this technology? What are the advantages and disadvantages of Internet voting? Would you be in favor of Internet voting for Imaginaria? Why or why not? ### Chapter Review ### Voting Methods ### Fairness in Voting Methods ### Standard Divisors, Standard Quotas, and the Apportionment Problem ### Apportionment Methods ### Fairness in Apportionment Methods ### Chapter Test

# Graph Theory ## Introduction In this chapter, you will learn the fundamental skills needed to work with graphs used in an area of mathematics known as graph theory. You can think of these graphs as a kind of map. Maps have served many purposes over the course of history. You probably use GPS maps to navigate to various destinations. A scientist from ancient Greece named Ptolemy wanted an accurate map of the world to make more accurate astrological predications. In recent years, neurobiologists have mapped the cerebral cortex to better understand the human brain. Social network analysts map online interactions to assist advertisers in reaching target audiences. Like other maps, the graphs you will study in this chapter can serve many purposes, but they do not have a lot of the details you might expect in a map such as size, shape, and distance between objects. All of that is stripped away so that we can focus on one element of maps, the connections between objects.

# Graph Theory ## Graph Basics ### Learning Objectives After completing this section, you should be able to: 1. Identify parts of a graph. 2. Model applications of graph basics. When you hear the word, graph, what comes to mind? You might think of the -coordinate system you learned about earlier in this course, or you might think of the line graphs and bar charts that are used to display data in news reports. The graphs we discuss in this chapter are probably very different from what you think of as a graph. They look like a bunch of dots connected by short line segments. The dots represent a group of objects and the line segments represent the connections, or relationships, between them. The objects might be bus stops, computers, Snapchat accounts, family members, or any other objects that have direct connections to each other. The connections can be physical or virtual, formal or casual, scientific or social. Regardless of the kind of connections, the purpose of the graph is to help us visualize the structure of the entire network to better understand the interactions of the objects within it. ### Parts of a Graph In a graph, the objects are represented with dots and their connections are represented with lines like those in . displays a simple graph labeled G and a multigraph labeled H. The dots are called vertices; an individual dot is a vertex, which is one object of a set of objects, some of which may be connected. We often label vertices with letters. For example, Graph G has vertices a, b, c, and d, and Multigraph H has vertices, e, f, g, and h. Each line segment or connection joining two vertices is referred to as an edge. H is considered a multigraph because it has a double edge between f and h, and a double edge between h and g. Another reason H is called a multigraph is that it has a loop connecting vertex e to itself; a loop is an edge that joins a vertex to itself. Loops and double edges are not allowed in a simple graph. To sum up, a simple graph is a collection of vertices and any edges that may connect them, such that every edge connects two vertices with no loops and no two vertices are joined by more than one edge. A multigraph is a graph in which there may be loops or pairs of vertices that are joined by more than one edge. In this chapter, most of our work will be with simple graphs, which we will call graphs for convenience. It is not necessary for the edges in a graph to be straight. In fact, you can draw an edge any way you want. In graph theory, the focus is on which vertices are connected, not how the connections are drawn (see ). In a graph, each edge can be named by the two letters of the associated vertices. The four edges in Graph X in are ab, ac, ad, and ae. The order of the letters is not important when you name the edge of a graph. For example, ab refers to the same edge as ba. Since the purpose of a graph is to represent the connections between objects, it is very important to know if two vertices share a common edge. The two vertices at either end of a given edge are referred to as neighboring, or adjacent. For example, in , vertices x and w are adjacent, but vertices y and w are not. ### Analyzing Geographical Maps with Graphs When graphs are used to model and analyze real-world applications, the number of edges that meet at a particular vertex is important. For example, a graph may represent the direct flight connections for a particular airport as in . Representing the connections with a graph rather than a map shifts the focus away from the relative positions and toward which airports are connected. In , the vertices are the airports, and the edges are the direct flight paths. The number of flight connections between a particular airport and other South Florida airports is the number of edges meeting at a particular vertex. For example, Key West has direct flights to three of the five airports on the graph. In graph theory terms, we would say that vertex FYW has degree 3. The degree of a vertex is the number of edges that connect to that vertex. Graphs are also used to analyze regional boundaries. The states of Utah, Colorado, Arizona, and New Mexico all meet at a single point known as the “Four Corners,” which is shown in the map in . In , each vertex represents one of these states, and each edge represents a shared border. States like Utah and New Mexico that meet at only a single point are not considered to have a shared border. By representing this map as a graph, where the connections are shared borders, we shift our perspective from physical attributes such as shape, size and distance, toward the existence of the relationship of having a shared boundary. ### Graphs of Social Interactions Geographical maps are just one of many real-world scenarios which graphs can depict. Any scenario in which objects are connected to each other can be represented with a graph, and the connections don’t have to be physical. Just think about all the connections you have to people around the world through social media! Who is in your network of Twitter followers? Whose Snapchat network are you connected to? ### Check Your Understanding ### Key Terms 1. vertex 2. edge 3. loop 4. graph (simple graph) 5. multigraph 6. adjacent (neighboring) 7. degree ### Key Concepts 1. Graphs and multigraphs represent objects as vertices and the relationships between the objects as edges. 2. The degree of a vertex is the number of edges that meet it and the degree can be zero. 3. An edge must have a vertex at each end. 4. Multigraphs may contain loops and double edges, but simple graphs may not. ### Videos 1. Graph Theory: Create a Graph to Represent Common Boundaries on a Map

# Graph Theory ## Graph Structures ### Learning Objectives After completing this section, you should be able to: 1. Describe and interpret relationships in graphs. 2. Model relationships with graphs. Graph theory is used in neuroscience to study how different parts of the brain connect. Neurobiologists use functional magnetic resonance imaging (fMRI) to measure levels of blood in different parts of the brain, called nodes. When nodes are active at the same time, it suggests there is a functional connection between them so they form a network. This network can be represented as a graph where the vertices are the nodes and the functional connections are the edges between them. (Mikey Taylor, "Graph Theory & Machine Learning in Neuroscience," Medium.com, June 24, 2020. ### Importance of the Degrees of Vertices One reason scientists study these networks is to determine how successful the communication within a network continues to be when it experiences failures in nodes and connections. Graphs can be used to study the resilience of these networks. (Mikey Taylor, "Graph Theory & Machine Learning in Neuroscience," Medium.com, June 24, 2020) ### Relating the Number of Edges to the Degrees of Vertices In the applications of graph theory to neuroscience and sociology in and YOUR TURN 12.6, there was a correlation between the degrees of vertices and the resilience of a network. Researchers in many fields have also observed a direct relationship between the number of edges in a graph and the degrees of the vertices. To begin to understand this relationship, consider a graph with five vertices and zero edges as in . Instead of being marked with a name, each vertex in is marked with its degree. In this case, all of the degrees are 0 so the sum of the degrees is also zero. Suppose that we add an edge between any two existing vertices and indicate the degrees of the vertices. This gives us a graph with five vertices and one edge like the graph in . Note that the degrees of two vertices increased, each by 1. So, the sum of the degrees is now 2. Suppose that we continue in this way, adding one edge at a time and making note of the number of edges and the sum of the degrees of the vertices as in . demonstrates a characteristic that is true of all graphs of any shape or size. When the number of edges is increased by one, the sum of the degrees increases by two. This happens because each edge has two ends and each end increases the degree of one vertex by one unit. As a result, the sum of the degrees of the vertices on any graph is always twice the number of edges. This relationship is known as the Sum of Degrees Theorem. ### Completeness Suppose that there were five strangers in a room, A, B, C, D, and E, and each one would be introduced to each of the others. How many introductions are necessary? One way to begin to answer this question is to draw a graph with each vertex representing an individual in the room and each edge representing an introduction as in . Let’s approach the problem by thinking about how many new people Person A would meet, then Person B, and so on, making sure not to repeat any introductions. The first graph in shows Person A meeting Persons B, C, D, and E, for a total of 4 introductions. The next graph shows that Person B still has to meet Persons C, D, and E, for a total of 3 more introductions. The next graph shows that Person C still has to meet Persons D and E, which is 2 more introductions. The next graph shows that Person D only remains to meet Person E, which is 1 more introduction. The final graph has edges representing 10 introductions. The last graph is an example of a complete graph because each pair of vertices is joined by an edge. Another way of saying this is that the graph is complete because each vertex is adjacent to every other vertex. shows complete graphs with three, four, five, and six vertices. Suppose we want to know the number of introductions necessary in a room with six people. This would be represented by a complete graph with six vertices, and the total number of introductions would be , the number of edges in the graph. In fact, you can always find the number of introductions in a room with people by adding all the whole numbers from 1 to . Suppose that we want to determine how many introductions are necessary in a room with 500 strangers. In other words, suppose that we want to determine the number of edges in a complete graph with 500 vertices. Adding up all the numbers from 1 to 499 could take a long time! In the next example, we use the Sum of Degrees Theorem to make the problem more manageable. Now we have a shorter way to calculate the number of introductions in a room with strangers, and the number of edges on a complete graph with vertices. Let’s update our formula. ### Subgraphs Sometimes a graph is a part of a larger graph. For example, the graph of South Florida Airports from is part of a larger graph that includes Orlando International Airport in Central Florida, which is shown in The graph in includes an additional vertex, MCO, and additional edges shown with dashed lines. The graph of direct flights between South Florida airports from is called a subgraph of the graph that also includes direct flights between Orlando and the same South Florida airports in . In general terms, if Graph B consists entirely of a set of edges and vertices from a larger Graph A, then B is called a subgraph of A. ### Identifying and Naming Cycles When you think of a cycle in everyday life, you probably think of something that begins and ends the same way. For example, the water cycle () begins with water in a lake or ocean, which evaporates into water vapor, condenses into clouds, and then returns to earth as rain or some other form of precipitation that settles into lakes or oceans. A cycle in graph theory is similar in that it begins and ends in the same way: It is a series of connected edges that begin and end at the same vertex but otherwise never repeat any vertices. In a cycle, there are always the same number of vertices as edges, and all vertices must be of degree 2. Cycles are often referred to by the number of vertices. For example, a cycle with 5 vertices can be called a 5-cycle. Cycles can also be named after polygons based on the number of edges. For example a 5-cycle is also called a pentagon. lists these names for cycles of size 3 through size 10. There are many more polygon names, including a megagon that has a million edges and a googolgon that has edges, but usually we just say -gon when the number is past 10. For example, a cycle with 11 edges could be called an 11-gon. Notice that the 10-cycle, or decagon, appears to cross over itself in . Remember, graphs can be drawn differently but represent the same connections. In , the same decagon is transformed into a graph that does not appear to overlap itself. We have done this without changing any of the connections so both diagrams represent the same relationships, and both diagrams are considered decagons. ### Cyclic Subgraphs and Cliques When cycles appear as subgraphs within a larger graph, they are called cyclic subgraphs. Cyclic subgraphs are named by listing their vertices sequentially. The vertex where you begin is not important. Graph K in has two triangle cycles (g, h, j) and (h, i, j), and one quadrilateral cycle (g, h, i, j). We have seen that sociologists use graphs to study the structures of social networks. In sociology, there is a principle known as Triadic Closure. It says that if two individuals in a social network have a friend in common, then it is more likely those two individuals will become friends too. Sociologists refer to this as an open triad becoming a closed triad. This concept can be visualized as graphs in . (Chakraborty, Dutta, Mondal, and Nath, "Application of Graph Theory in Social Media, International Journal of Computer Sciences and Engineering, 6(10):722-729) In the open triad in , person a and person b each has a friendship with person c. In the closed triad, person a and person a have also developed a friendship. Notice that the graph representing the closed triad is a three-cycle, or a triangle, in graph theory terms. Another topic of interest to sociologists, as well as computer scientists and scientists in many fields, is the concept of a clique. In a social group, a clique is a subgroup who are all friends. A triad is an example of a clique with three people, but there can be cliques of any size. In graph theory, a clique is a complete subgraph. ### Three-Cycles in Complete Graphs Just as complete graphs have a predictable number of edges, complete graphs have a predictable number of cyclic subgraphs. Let’s look at the three-cycles within complete graphs with up to six vertices, which are shown in . Let's list the names of all the triangles in each graph. . There is only one triangle in the complete graph with three vertices, (a, b, c). For the rest of the graphs, it is important that we take an organized approach. Start with the vertex that is first alphabetically, listing any triangles that include that vertex also in alphabetical order. Then, proceed to the next vertex in the alphabet, and list any triangles that include that vertex, except those that are already listed. Keep going in this way as shown in . Look at the last row in . Do you see a pattern emerge for counting triangles in a complete graph? Without drawing a complete graph with 7 vertices, we can predict that it will have triangles inside it. This pattern also appears in a famous diagram known to Western mathematicians as "Pascal’s Triangle." displays the first 11 rows of Pascal’s Triangle. Row 0 of Pascal’s Triangle only has the number 1 in it. The first and last entries in each of the other rows are also 1s. Otherwise, all the entries are formed by adding two entries from the previous row. For example, in row 6, entry 1 is 6, which was found by adding 1 and the 5 from the previous row, and entry 2 is 15, which was found by adding the 5 and the 10 from the previous row, as shown in . If we begin to list the third entries in each row of Pascal's triangle from the top down, we see 1, 4, 10, 20, 35, and so on. Notice that these values are exactly the number of triangles in a complete graph of sizes 3, 4, 5, 6, and 7, respectively. Specifically, entry 3 in row 7 is 35, the number of triangles in a complete graph of size 7. Let's practice using Pascal’s triangle to find the number of triangles in a complete graph of a particular size. STEPS TO FIND THE NUMBER OF TRIANGLES IN A COMPLETE GRAPH OF SIZE Step 1 Calculate the entries in row n of Pascal's Triangle using sums of pairs of entries in previous rows as needed. Step 2 The number of triangles is entry 3 in row n. Remember to count the entries 0, 1, 2, and 3, from left to right. ### Check Your Understanding ### Key Terms 1. complete 2. subgraph 3. cycle 4. cyclic subgraph 5. clique ### Key Concepts 1. The sum of the degrees of the vertices in a graph is twice the number of edges. 2. In a complete graph every pair of vertices is adjacent. 3. A subgraph is part of a larger graph. 4. Cycles are a sequence of connected vertices that begin and end at the same vertex but never visit any vertex twice. ### Formulas For the Sum of Degrees Theorem, or The number of edges in a complete graph with vertices is the sum of the whole numbers from 1 to , . The number of edges in a complete graph with vertices is . ### Videos 1. The Mathematical Secrets of Pascal's Triangle by Wajdi Mohamed Ratemi

# Graph Theory ## Comparing Graphs ### Learning Objectives After completing this section, you should be able to: 1. Identify the characteristics used to compare graphs. 2. Determine when two graphs represent the same relationships 3. Explore real-world examples of graph isomorphisms 4. Find the complement of a graph Maps of the same region may not always look the same. For example, a map of Earth on a flat surface looks distorted at the poles. When the same regions are mapped on a spherical globe, the countries that are closer to the polls appear smaller without the distortion. Despite these differences, the two maps still communicate the same relationships between nations such as shared boundaries, and relative position on the earth. In essence, they are the same map. This means that for every point on one map, there is a corresponding point on the other map in the same relative location. In this section, we will determine exactly what characteristics need to be the shared by two graphs in order for us to consider them “the same.” ### When Are Two Graphs Really the Same Graph? In arithmetic, when two numbers have the same value, we say they are equal, like ½ = 0.5. Although ½ and 0.5 look different, they have the same value, because they are assigned the same position on the real number line. When do we say that two graphs are equal? shows Graphs A and F, which are identical except for the labels. Graphs are visual representations of connections. As long as two graphs indicate the same pattern of connections, like Graph A and Graph F, they are considered to be equal, or in graph theory terms, isomorphic. Two graphs are isomorphic if either one of these conditions holds: 1. One graph can be transformed into the other without breaking existing connections or adding new ones. 2. There is a correspondence between their vertices in such a way that any adjacent pair in one graph corresponds to an adjacent pair in the other graph. It is important to note that if either one of the isomorphic conditions holds, then both of them do. When we need to decide if two graphs are isomorphic, we will need to make sure that one of them holds. For example, shows how Graph T can be bent and flipped to look like Graph Z, which means that Graphs T and Z satisfy condition 1 and are isomorphic. Also, notice that the vertices that were adjacent in the first graph are still adjacent in the transformed graph as shown in . For example, vertex 3 is still adjacent to vertex 4, which means they are still neighboring vertices joined by a single edge. ### When Are Two Graphs Really Different? Verifying that two graphs are isomorphic can be a challenging process, especially for larger graphs. It makes sense to check for any obvious ways in which the graphs might differ so that we don’t spend time trying to verify that graphs are isomorphic when they are not. If two graphs have any of the differences shown in , then they cannot be isomorphic. ### Recognizing Isomorphic Graphs Isomorphic graphs that represent the same pattern of connections can look very different despite having the same underlying structure. The edges can be stretched and twisted. The graph can be rotated or flipped. For example, in , each of the diagrams represents the same pattern of connections. Looking at , how can we know that these graphs are isomorphic? We will start by checking for any obvious differences. Each of the graphs in has four vertices and five edges; so, there are no differences there. Next, we will focus on the degrees of the vertices, which have been labeled in . As shown in , each graph has two vertices of degree 2 and two vertices of degree 3; so, there are no differences there. Now, let’s check for cyclic subgraphs. These are highlighted in . As shown in , each graph has two triangles and one quadrilateral; so, no differences there either. It is beginning to look likely that these graphs are isomorphic, but we will have to look further to be sure. To know with certainty that these graphs are isomorphic, we need to confirm one of the two conditions from the definition of isomorphic graphs. With smaller graphs, you may be able to visualize how to stretch and twist one graph to get the other to see if condition 1 holds. Imagine the edges are stretchy and picture how to pull and twist one graph to form the other. If you can do this without breaking or adding any connections, then the graphs are really the same. demonstrates how to change graph A4 to get A3, graph A3 to get A2, and graph A2 to get A1. Now that we have used visual analysis to see that condition 1 holds for graphs A1, A2, A3, and A4 in , we know that they are isomorphic. In , one of the edges of graph A4 crossed another edge of the graph. By transforming it into graph A3, we have “untangled” it. Graphs that can be untangled are called planar graphs. The complete graph with five vertices is an example of a nonplanar graph-that means that, no matter how hard you try, you can’t untangle it. But, when you try to figure out if two graphs are the same, it can be helpful to untangle them as much as possible to make the similarities and differences more obvious. Have you ever noticed that many popular board games may look different but are really the same game? A good example is the many variations of the board game Monopoly®, which was submitted to the U.S. Patent Office in 1935. Although the rules have been revised a bit, a very similar game board is still in use today. There have been many versions of Monopoly over the years. Many have been stylized to reflect a popular theme, such as a show or sports team, while retaining the same game board structure. If we were to represent these different versions of the game using a graph, we would find that the graphs are isomorphic. . Let's analyze some game boards using graph theory to determine if they have the same structure despite having different appearances. ### Identifying and Naming Isomorphisms When two graphs are isomorphic, meaning they have the same structure, there is a correspondence between their vertices, which can be named by listing corresponding pairs of vertices. This list of corresponding pairs of vertices in such a way that any adjacent pair in one graph corresponds to an adjacent pair in the other graph is called an isomorphism. Consider the isomorphic graphs in . In , we could replace the labels Graph F with the labels from Graph A and have an identical graph, as in . So, we can identify an isomorphism between Graph A and Graph F by listing the corresponding pairs of vertices: b-g, c-h, d-i, and e-j. Notice that b is adjacent to c and g is adjacent to h. This must be the case since b corresponds to g and c corresponds to h. The same is true for other pairs of adjacent vertices. An isomorphism between graphs is not necessarily unique. There can be more than one isomorphism between two graphs. We can see how to form a different isomorphism between Graph A and Graph F from by rotating Graph F clockwise and comparing the rotated version of F to Graph A as in YOUR TURN 12.13. Now, we can see that a second isomorphism exists, which has the correspondence: b-j, d-h, c-i, and e-g as shown in . ### Complementary Graphs Suppose that you are a camp counselor at Camp Woebegone and you are holding a camp Olympics with four events. The campers have signed up for the events. You drew a graph in to help you visualize which events have campers in common. Graph E in shows that some of the same campers will be in events a and b, as well as b and d, c and d, and a and c. What do you think the graph would look like that represented the events that do not have campers in common? It would have the same vertices, but any pair of adjacent edges in Graph E, would not be adjacent in the new graph, and vice versa. This is called a complementary graph, as shown in . Two graphs are complementary if they have the same set of vertices, but any vertices that are adjacent in one, are not adjacent in the other. In this case, we can say that one graph is the complement of the other. One way to find the complement of a graph is to draw the complete graph with the same number of vertices and remove all the edges that were in the original graph. Let’s say you wanted to find the complement of Graph E from , and you didn’t already know it was Graph F. You could start with the complete graph with four vertices and remove the edges that are in Graph E as shown in . When two graphs are really the same graph, they have the same missing edges. So, when two graphs have a lot of edges, it may actually be easier to determine if they are isomorphic by looking at which edges are missing rather than which edges are included. In other words, we can determine if two graphs are isomorphic by checking if their complements are isomorphic. ### Check Your Understanding ### Key Terms 1. isomorphic 2. isomorphism 3. planar 4. nonplanar 5. complement 6. complementary ### Key Concepts 1. Two graphs are isomorphic if they have the same structure. 2. When graphs are relatively small, we can use visual inspection to identify an isomorphism by transforming one graph into another without breaking connections or adding new ones. 3. An isomorphism between two graphs preserves adjacency. 4. If two graphs differ in number of vertices, number of edges, degrees of vertices, or types of subgraphs, they cannot be isomorphic. 5. When the complements of two graphs are isomorphic, so are the graphs themselves. ### Videos 1. Determine If Two Graphs Are Isomorphic and Identify the Isomorphism

# Graph Theory ## Navigating Graphs ### Learning Objectives After completing this section, you should be able to: 1. Describe and identify walks, trails, paths, and circuits. 2. Solve application problems using walks, trails, paths, and circuits. 3. Identify the chromatic number of a graph. 4. Describe the Four-Color Problem. 5. Solve applications using graph colorings. Now that we know the basic parts of graphs and we can distinguish one graph from another, it is time to really put our graphs to work for us. Many applications of graph theory involve navigating through a graph very much like you would navigate through a maze. Imagine that you are at the entrance to a maze. Your goal is to get from one point to another as efficiently as possible. Maybe there are treasures hidden along the way that make straying from the shortest path worthwhile, or maybe you just need to get to the end fast. Either way, you definitely want to avoid any wrong turns that would cause unnecessary backtracking. Luckily, graph theory is here to help! ### Walks Suppose is a maze you want to solve. You want to get from the start to the end. You can approach this task any way you want. The only rule is that you can’t climb over the wall. To put this in the context of graph theory, let’s imagine that at every intersection and every turn, there is a vertex. The edges that join the vertices must stay within the walls. The graph within the maze would look like . One approach to solving a maze is to just start walking. It is not the most efficient approach. You might cross through the same intersection twice. You might backtrack a bit. It’s okay. We are just out for a walk. It might look something like the black sequence of vertices and edges in . This type of sequence of adjacent vertices and edges is actually called a walk (or directed walk) in graph theory too! A walk can be identified by naming the sequence of its vertices (or by naming the sequence of its edges if those are labeled). Let’s take the graph out of the context of the maze and give each vertex a name and each edge of the walk a direction as in . The name of this walk from p to r is p → q → o → n → i → j → c → d → c → j → k → s → r. When a particular edge on our graph was traveled in both directions, it had arrows in both directions and the letters of vertices that were visited more than once had to be repeated in the name of the walk. The highlighted edges Graph Y in represent a walk between f and b. The highlighted edges in Graph X do not represent a walk between f and b, because there is a turn at a point that is not a vertex. This is like climbing over a wall when you are walking through a maze. Another way of saying this is that b → d → f is not a walk, because there is no edge between b and d. ### Paths and Trails A walk is the most basic way of navigating a graph because it has no restrictions except staying on the graph. When there are restrictions on which vertices or edges we can visit, we will call the walk by a different name. For example, if we want to find a walk that avoids travelling the same edge twice, we will say we want to find a trail (or directed trail). If we want to find a walk that avoids visiting the same vertex twice, we will say, we want to find a path (or directed path). Walks, trails, and paths are all related. 1. All paths are trails, but trails that visit the same vertex twice are not paths. 2. All trails are walks, but walks in which an edge is visited twice would not be trails. We can visualize the relationship as in . Let’s practice identifying walks, trails, and paths using the graphs in . ### Circuits In many applications of graph theory, such as creating efficient delivery routes, beginning and ending at the same location is a requirement. When a walk, path, or trail end at the same location or vertex they began, we call it closed. Otherwise, we call it open (does not begin and end at the same location or vertex). Some examples of closed walks, closed trails, and closed paths are given in in the following table. Since walks, trails, and paths are all related, closed walks, circuits, and directed cycles are related too. 1. All circuits are closed walks, but closed walks that visit the same edge twice are not circuits. 2. All directed cycles are circuits, but circuits in which a vertex is visited twice are not directed cycles. We can visualize the relationship as in . The same circuit can be named using any of its vertices as a starting point. For example, the circuit d → f → b → c → d can also be referred to in the following ways. a → b → c → d → a is the same as Let’s practice working with closed walks, circuits (closed trails), and directed cycles (closed paths). In the graph in , the vertices are major central and south Florida airports. The edges are direct flights between them. ### Graph Colorings In this section so far, we have looked at how to navigate graphs by proceeding from one vertex to another in a sequence that does not skip any vertices, but in some applications we may want to skip vertices. Remember the camp Olympics at Camp Woebegone in Comparing Graphs? You were planning a camp Olympics with four events. The campers signed up for the events. You drew a graph to help you visualize which events have campers in common. The vertices of Graph E in represent the events and adjacent vertices indicate that there are campers who are participating in both. In this case, we do NOT want events represented by two adjacent vertices to occur in the same timeslot, because that would prevent the campers who wanted to participate in both from doing so. We can use the graph in to count the timeslots we need so there are no conflicts. Let’s assign each timeslot a different color. We could categorize events that happen at 1 pm as Red; 2 pm, Purple; 3 pm, Blue; and 4 pm, Green. Then assign different colors to any pair of adjacent vertices to ensure that the events they represent do not end up in the same timeslot. shows several of the ways to do this while obeying the rule that no pair of adjacent vertices can be the same color. In , the graphs with vertices colored so that no adjacent vertices are the same color are called graph colorings. Notice that Graph 3 has the fewest colors, which means it shows us how to have the fewest number of timeslots. The events marked in red, a and d, can be held at the same time because they are not adjacent and do not have conflicts. Also, the events marked in purple, b and c, can be held at the same time. We would not need green or blue timeslots at all! A graph that uses colors is called an . The smallest number of colors needed to color a particular graph is called its chromatic number. Graph colorings can be used in many applications like the scheduling scenario at camp Woebegone. Let's look at how they work in more detail. shows two different colorings of a particular graph. Coloring A is called a four-coloring, because it uses four colors, red (R), green (G), blue (B), and purple (P). Coloring B is called a three-coloring because it uses three. The colors allow us to visually subdivide the graphs into groups. The only rule is that adjacent vertices are different colors so that they are in different groups. It turns out that a three-coloring is the best we can do with the graph in . No matter how many different patterns you try with only two colors, you will never find one in which the adjacent vertices are always different colors. In other words, the graph has a chromatic number of three. For large graphs, computer assistance is usually required to find the chromatic numbers. There is no formula for finding the chromatic number of a graph, but there are some facts that are helpful in . ### Creating Colorings to Solve Problems Let’s see how these facts can help us color the graph in . 1. Since the graph is planar, the chromatic number is no more than four. 2. The graph is not complete, but it has complete subgraphs of three vertices. In other words, it has triangles like the one shown with blue vertices in . This means that the chromatic number is at least three. We know we can color this graph in three or four colors. It is usually best to start by coloring the vertex of highest degree as shown in . In this case, we used red (R). The color is not important. We want to color as many of the vertices the same color as possible; so, we look at all the vertices that are not adjacent to the red vertex and begin to color them, red starting with the one among them of highest degree. Since the only vertices that are not adjacent are both degree 2, choose either one and color it red as shown in . Now, there is only one vertex left that is not adjacent to a red; so, color it red. Of the remaining vertices, the highest degree is four; so, color one of the vertices of degree four in a different color. These two steps are shown in . Repeat the same procedure. There are three remaining vertices that are not adjacent to a blue. Color as many blue as possible, with priority going to vertices of higher degree as shown in . All the remaining vertices are adjacent to blue. So, it is time to repeat the procedure with another color as shown in . All the vertices are now colored with a three-coloring so we know the chromatic number is at most three, but we knew the chromatic number was at least three because the graph has a triangle. So, we are now certain it is exactly three. Suppose the vertices of the graph in represented the nine events in the Camp Woebegone Olympics, and edges join any events that have campers in common, but nonadjacent vertices do not. Any vertices of the same color are nonadjacent; so, they have no conflicts. Since this graph is a three-coloring, all nine events could be scheduled in just three timeslots! ### The Four Color Problem The idea of coloring graphs to solve problems was inspired by one of the most famous problems in mathematics, the “four color problem.” The idea was that, no matter how complicated a map might be, only four colors were needed to color the map so that no two regions that shared a boundary would be the same color. For many years, everyone suspected this to be true, because no one could create a map that needed more than four colors, but they couldn’t prove it was true in general. Finally, graphs were used to solve the problem! We saw how maps can be represented as graphs in Graph Basics. from shows a map of the midwestern region of the United States. shows how this map can be associated with a graph in which each vertex represents a state and each edge indicates the states that share a common boundary. shows the final graph. Notice that the graph representing the common boundaries between midwestern states is planar, meaning that it can be drawn on a flat surface without edges crossing. As we have seen, any planar graph has a chromatic number of four or less. This very well-known fact is called the Four-Color Theorem, or Four-Color Map Theorem. ### Check Your Understanding ### Key Terms 1. walk (directed walk) 2. trail (directed trail) 3. path (directed path) 4. closed 5. open 6. closed walk 7. circuit (closed trail) 8. directed cycle (closed path) 9. coloring (graph coloring) 10. -coloring 11. chromatic number ### Key Concepts 1. Walks, trails, and paths are ways to navigate through a graph using a sequence of connected vertices and edges. 2. Closed walks, circuits, and directed cycles are ways to navigate from a vertex on a graph and return to the same vertex. 3. Colorings are a way to organize the vertices of a graph into groups so that no two members of a group are adjacent. 4. Maps can be represented with planar graphs, which can always be colored using four colors or fewer. ### Videos 1. Walks, Trails, and Paths in Graph Theory 2. Closed Walks, Closed Trails (Circuits), and Closed Paths (Directed Cycles) in Graph Theory 3. Coloring Graphs Part 1: Coloring and Identifying Chromatic Number 4. The Four Color Map Theorem – Numberphile 5. Coloring Graphs Part 2: Coloring Maps and the Four Color Problem 6. Neil deGrasse Tyson Explains the Möbius Strip

# Graph Theory ## Euler Circuits ### Learning Objectives After completing this section, you should be able to: 1. Determine if a graph is connected. 2. State the Chinese postman problem. 3. Describe and identify Euler Circuits. 4. Apply the Euler Circuits Theorem. 5. Evaluate Euler Circuits in real-world applications. The delivery of goods is a huge part of our daily lives. From the factory to the distribution center, to the local vendor, or to your front door, nearly every product that you buy has been shipped multiple times to get to you. If the cost and time of that delivery is too great, you will not be able to afford the product. Delivery personnel have to leave from one location, deliver the goods to various places, and then return to their original location and do all of this in an organized way without losing money. How do delivery services find the most efficient delivery route? The answer lies in graph theory. ### Connectedness Before we can talk about finding the best delivery route, we have to make sure there is a route at all. For example, suppose that you were tasked with visiting every airport on the graph in by plane. Could you accomplish that task, only taking direct flight paths between airports listed on this graph? In other words, are all the airports connected by paths? Or are some of the airports disconnected from the others? In , we can see TPA is adjacent to PBI, FLL, MIA, and EYW. Also, there is a path between TPA and MCO through FLL. This indicates there is a path between each pair of vertices. So, it is possible to travel to each of these airports only taking direct flight paths and visiting no other airports. In other words, the graph is connected because there is a path joining every pair of vertices on the graph. Notice that if one vertex is connected to each of the others in a graph, then all the vertices are connected to each other. So, one way to determine if a graph is connected is to focus on a single vertex and determine if there is a path between that vertex and each of the others. If so, the graph is connected. If not, the graph is disconnected, which means at least one pair of vertices in a graph is not joined by a path. Let’s take a closer look at graph X in . Focus on vertex a. There is a path between vertices a and b, but there is no path between vertex a and vertex c. So, Graph X is disconnected. When you are working with a planar graph, you can also determine if a graph is connected by untangling it. If you draw it so that so that none of the edges are overlapping, like we did with Graph X in , it is easier to see that the graph is disconnected. Versions 2 and 3 of Graph X in each have the same number of vertices, number of edges, degrees of the vertices, and pairs of adjacent vertices as version 1. In other words, each version retains the same structure as the original graph. Since versions 2 and 3 of Graph X, do not have overlapping edges, it is easier to identify pairs of vertices that do not have paths between them, and it is more obvious that Graph X is disconnected. In fact, there are two completely separate, disconnected subgraphs, one with the vertices in {a, b, e}, and the other with the vertices {c, d, f} These sets of vertices together with all of their edges are called components of Graph X. A component of a graph is a subgraph in which there is a path between each pair of vertices in the subgraph, but no edges between any of the vertices in the subgraph and a vertex that is not in the subgraph. Now let’s focus on Graph Y from . As in Graph X, there is a path between vertices a and b, as well as between vertices a and e, but Graph Y is different from Graph X because of vertex g. Not only is there a path between vertices a and g, but vertex g bridges the gap between a and c with the path a → b → g → c. Similarly, there is a path between vertices a and d and vertices a and f: a → b → g → d, a → b → g → d → f. Since there is a path between vertex a and every other vertex, Graph Y is connected. You can also see this a bit more clearly by untangling Graph Y as in . Even when Y is drawn so that the edges do not overlap, the graph cannot be drawn as two separate, unconnected pieces. In other words, Graph Y has only one component with the vertices {a, b, c, d, e, f}. We can give an alternate definition of connected and disconnected using the idea of components. A graph is connected if it has only one component. A graph is disconnected if it has more than one component. These alternate definitions are equivalent to the previous definitions. This means that you can confirm a graph is connected or disconnected either by checking to see if there is a path between each vertex and each other vertex, or by identifying the number of components. A graph is connected if it has only one component. ### Origin of Euler Circuits The city of Konigsberg, modern day Kaliningrad, Russia, has waterways that divide up the city. In the 1700s, the city had seven bridges over the various waterways. The map of those bridges is shown in . The question as to whether it was possible to find a route that crossed each bridge exactly once and return to the starting point was known as the Konigsberg Bridge Problem. In 1735, one of the most influential mathematicians of all time, Leonard Euler, solved the problem using an area of mathematics that he created himself, graph theory! Euler drew a multigraph in which each vertex represented a land mass, and each edge represented a bridge connecting them, as shown in . Remember from Navigating Graphs that a circuit is a trail, so it never repeats an edge, and it is closed, so it begins and ends at the same vertex. Euler pointed out that the Konigsberg Bridge Problem was the same as asking this graph theory question: Is it possible to find a circuit that crosses every edge? Since then, circuits (or closed trails) that visit every edge in a graph exactly once have come to be known as Euler circuits in honor of Leonard Euler. Euler was able to prove that, in order to have an Euler circuit, the degrees of all the vertices of a graph have to be even. He also proved that any graph with that characteristic must have an Euler circuit. So, saying that a connected graph is Eulerian is the same as saying it has vertices with all even degrees, known as the Eulerian circuit theorem. To understand why the Euler circuit theorem is true, think about a vertex of degree 3 on any graph, as shown in . First imagine the vertex of degree 3 shown in is not the starting vertex. At some point, each edge must be traveled. The first time one of the three edges is traveled, the direction will be toward the vertex, and the second time it will be away from the vertex. Then, at some point, the third edge must be traveled coming in toward the vertex again. This is a problem, because the only way to get back to the starting vertex is to then visit one of the three edges a second time. So, this vertex cannot be part of an Euler circuit. Next imagine the vertex of degree 3 shown in is the starting vertex. The first time one of the edges is traveled, the direction is away from the vertex. At some point, the second edge will be traveled coming in toward the vertex, and the third edge will be the way back out, but the starting vertex is also the ending vertex in a circuit. The only way to return to the vertex is now to travel one of the edges a second time. So, again, this vertex cannot be part of an Euler circuit. For the same reason that a vertex of degree 3 can never be part of an Euler circuit, a vertex of any odd degree cannot either. We can use this fact and the graph in to solve the Konigsberg Bridge Problem. Since the degrees of the vertices of the graph in are not even, the graph is not Eulerian and it cannot have an Euler circuit. This means it is not possible to travel through the city of Konigsberg, crossing every bridge exactly once, and returning to your starting position. ### Chinese Postman Problem At Camp Woebegone, campers travel the waterways in canoes. As part of the Camp Woebegone Olympics, you will hold a canoeing race. You have placed a checkpoint on each of the 11 different streams. The competition requires each team to travel each stream, pass through the checkpoints in any order, and return to the starting line, as shown in the . Since the teams want to go as fast as possible, they would like to find the shortest route through the course that visits each checkpoint and returns to the starting line. If possible, they would also like to avoid backtracking. Let’s visualize the course as a multigraph in which the vertices represent turns and the edges represent checkpoints as in . The teams would like to find a closed walk that repeats as few edges as possible while still visiting every edge. If they never repeat an edge, then they have found a closed trail, which is a circuit. That circuit must cover all edges; so, it would be an Euler circuit. The task of finding a shortest circuit that visits every edge of a connected graph is often referred to as the Chinese postman problem. The name Chinese postman problem was coined in honor of the Chinese mathematician named Kwan Mei-Ko in 1960 who first studied the problem. If a graph has an Euler circuit, that will always be the best solution to a Chinese postman problem. Let’s determine if the multigraph of the course has an Euler circuit by looking at the degrees of the vertices in . Since the degrees of the vertices are all even, and the graph is connected, the graph is Eulerian. It is possible for a team to complete the canoe course in such a way that they pass through each checkpoint exactly once and return to the starting line. ### Identifying Euler Circuits Solving the Chinese postman problem requires finding a shortest circuit through any graph or multigraph that visits every edge. In the case of Eulerian graphs, this means finding an Euler circuit. The method we will use is to find any circuit in the graph, then find a second circuit starting at a vertex from the first circuit that uses only edges that were not in the first circuit, then find a third circuit starting at a vertex from either of the first two circuits that uses only edges that were not in the first two circuits, and then continue this process until all edges have been used. In the end, you will be able to link all the circuits together into one large Euler circuit. Let’s find an Euler circuit in the map of the Camp Woebegone canoe race. In , we have labeled the edges of the multigraph so that the circuits can be named. In a multigraph it is necessary to name circuits using edges and vertices because there can be more than one edge between adjacent vertices. We will begin with vertex 1 because it represents the starting line in this application. In general, you can start at any vertex you want. Find any circuit beginning and ending with vertex 1. Remember, a circuit is a trail, so it doesn’t pass through any edge more than once. shows one possible circuit that we could use as the first circuit, 1 → A → 2 → B → 3 → C → 4 → G → 5 → J → 1. From the edges that remain, we can form two more circuits that each start at one of the vertices along the first circuit. Starting at vertex 3 we can use 3 → H → 5 → I → 1 → K → 3 and starting at vertex 2 we can use 2 → D → 6 → E → 7 → F → 2, as shown in . Now that each of the edges is included in one and only once circuit, we can create one large circuit by inserting the second and third circuits into the first. We will insert them at their starting vertices 2 and 3 becomes Finally, we can name the circuit using vertices, 1 → 2 → 6 → 7 → 2 → 3 → 5 → 1 → 3 → 4 → 5 → 1, or edges, A → D → E → F → B → H → I → K → C → G → J. Let's review the steps we used to find this Eulerian Circuit. Steps to Find an Euler Circuit in an Eulerian Graph Step 1 - Find a circuit beginning and ending at any point on the graph. If the circuit crosses every edges of the graph, the circuit you found is an Euler circuit. If not, move on to step 2. Step 2 - Beginning at a vertex on a circuit you already found, find a circuit that only includes edges that have not previously been crossed. If every edge has been crossed by one of the circuits you have found, move on to Step 3. Otherwise, repeat Step 2. Step 3 - Now that you have found circuits that cover all of the edges in the graph, combine them into an Euler circuit. You can do this by inserting any of the circuits into another circuit with a common vertex repeatedly until there is one long circuit. ### Eulerization The Chinese postman problem doesn’t only apply to Eulerian graphs. Recall the postal delivery person who needed to deliver mail to every block of every street in a subdivision. We used the map in to create the graph in . Since the graph of the subdivision has vertices of odd degree, there is no Euler circuit. This means that there is no route through the subdivision that visits every block of every street without repeating a block. What should our delivery person do? They need to repeat as few blocks as possible. The technique we will use to find a closed walk that repeats as few edges as possible is called eulerization. This method adds duplicate edges to a graph to create vertices of even degree so that the graph will have an Euler circuit. In , the eight vertices of odd degree in the graph of the subdivision are circled in green. We have added duplicate edges between the pairs of vertices, which changes the degrees of the vertices to even degrees so the resulting multigraph has an Euler circuit. In other words, we have eulerized the graph. The duplicate edges in the eulerized graph correspond to blocks that our delivery person would have to travel twice. By keeping these duplicate edges to a minimum, we ensure the shortest possible route. It can be challenging to determine the fewest duplicate edges needed to eulerize a graph, but you can never do better than half the number of odd vertices. In the graph in , we have found a way to fix the eight vertices of odd degree with only four duplicate edges. Since four is half of eight, we will never do better. ### Check Your Understanding ### Key Terms 1. connected 2. component 3. disconnected 4. Euler circuit 5. Eulerian graph 6. Chinese postman problem 7. Eulerization ### Key Concepts 1. A connected graph has only one component. 2. The Euler circuit theorem states that an Euler circuit exists in every connected graph in which all vertices have even degree, but not in disconnected graphs or any graph with one or more vertices of odd degree. 3. The Chinese postman problem asks how to find the shortest closed trail that visits all edges at least once. 4. If an Euler circuit exists, it is always the best solution to the Chinese postman problem. 5. Eulerization is the process of adding duplicate edges to a graph so that the new multigraph has an Euler circuit. 6. The minimum number of duplicated edges needed to eulerize a graph is half the number of odd vertices or more. ### Videos 1. Connected and Disconnected Graphs in Graph Theory 2. Recognizing Euler Trails and Euler Circuits 3. Existence of Euler Circuits in Graph Theory

# Graph Theory ## Euler Trails ### Learning Objectives After completing this section, you should be able to: 1. Describe and identify Euler trails. 2. Solve applications using Euler trails theorem. 3. Identify bridges in a graph. 4. Apply Fleury’s algorithm. 5. Evaluate Euler trails in real-world applications. We used Euler circuits to help us solve problems in which we needed a route that started and ended at the same place. In many applications, it is not necessary for the route to end where it began. In other words, we may not be looking at circuits, but trails, like the old Pony Express trail that led from Sacramento, California in the west to St. Joseph, Missouri in the east, never backtracking. ### Euler Trails If we need a trail that visits every edge in a graph, this would be called an Euler trail. Since trails are walks that do not repeat edges, an Euler trail visits every edge exactly once. ### The Five Rooms Puzzle Just as Euler determined that only graphs with vertices of even degree have Euler circuits, he also realized that the only vertices of odd degree in a graph with an Euler trail are the starting and ending vertices. For example, in , Graph H has exactly two vertices of odd degree, vertex g and vertex e. Notice the Euler trail we saw in Excercise 3 of began at vertex g and ended at vertex e. This is consistent with what we learned about vertices off odd degree when we were studying Euler circuits. We saw that a vertex of odd degree couldn't exist in an Euler circuit as depicted in . If it was a starting vertex, at some point we would leave the vertex and not be able to return without repeating an edge. If it was not a starting vertex, at some point we would return and not be able to leave without repeating an edge. Since the starting and ending vertices in an Euler trail are not the same, the start is a vertex we want to leave without returning, and the end is a vertex we want to return to and never leave. Those two vertices must have odd degree, but the others cannot. Let’s use the Euler trail theorem to solve a puzzle so you can amaze your friends! This puzzle is called the “Five Rooms Puzzle.” Suppose that you were in a house with five rooms and the exterior. There is a doorway in every shared wall between any two rooms and between any room and the exterior as shown in . Could you find a route through the house that passes through each doorway exactly once? Let’s represent the puzzle with a graph in which vertices are rooms (or the exterior) and an edge indicates a door between two rooms as shown in . To pass through each doorway exactly once means that we cross every edge in the graph exactly once. Since we have not been asked to start and end at the same position, but to visit each edge exactly once, we are looking for an Euler trail. Let’s check the degrees of the vertices. Since there are more than two vertices of odd degree as shown in , the graph of the five rooms puzzle contains no Euler path. Now you can amaze and astonish your friends! ### Bridges and Local Bridges Now that we know which graphs have Euler trails, let’s work on a method to find them. The method we will use involves identifying bridges in our graphs. A bridge is an edge which, if removed, increases the number of components in a graph. Bridges are often referred to as cut-edges. In , there are several examples of bridges. Notice that an edge that is not part of a cycle is always a bridge, and an edge that is part of a cycle is never a bridge. Edges bf, cg, and dg are “bridges” The graph in is connected, which means it has exactly one component. Each time we remove one of the bridges from the graph the number of components increases by one as shown in . If we remove all three, the resulting graph in has four components. In sociology, bridges are a key part of social network analysis. Sociologists study two kinds of bridges: local bridges and regular bridges. Regular bridges are defined the same in sociology as in graph theory, but they are unusual when studying a large social network because it is very unlikely a group of individuals in a large social network has only one link to the rest of the network. On the other hand, a local bridge occurs much more frequently. A local bridge is a friendship between two individuals who have no other friends in common. If they lose touch, there is no single individual who can pass information between them. In graph theory, a local bridge is an edge between two vertices, which, when removed, increases the length of the shortest path between its vertices to more than two edges. In , a local bridge between vertices b and e has been removed. As a result, the shortest path between b and e is b → i → j → k → e, which is four edges. On the other hand, if edge ab were removed, then there are still paths between a and b that cover only two edges, like a → i → b. The significance of a local bridge in sociology is that it is the shortest communication route between two groups of people. If the local bridge is removed, the flow of information from one group to another becomes more difficult. Let’s say that vertex b is Brielle and vertex e is Ella. Now, Brielle is less likely to hear about things like job opportunities, that Ella many know about. This is likely to impact Brielle as well as the friends of Brielle. ### Finding an Euler Trail with Fleury’s Algorithm Now that we are familiar with bridges, we can use a technique called Fleury’s algorithm, which is a series of steps, or algorithm, used to find an Euler trail in any graph that has exactly two vertices of odd degree. Here are the steps involved in applying Fleury’s algorithm. Here are the steps involved in applying Fleury’s algorithm. Step 1: Begin at either of the two vertices of odd degree. Step 2: Remove an edge between the vertex and any adjacent vertex that is NOT a bridge, unless there is no other choice, making a note of the edge you removed. Repeat this step until all edges are removed. Step 3: Write out the Euler trail using the sequence of vertices and edges that you found. For example, if you removed ab, bc, cd, de, and ef, in that order, then the Euler trail is a → b → c → d → e → f. shows the steps to find an Euler trail in a graph using Fleury’s algorithm. The Euler trail that was found in is t → v → w → u → t → w → y → x → v. In the previous section, we found Euler circuits using an algorithm that involved joining circuits together into one large circuit. You can also use Fleury’s algorithm to find Euler circuits in any graph with vertices of all even degree. In that case, you can start at any vertex that you would like to use. Step 1: Begin at any vertex. Step 2: Remove an edge between the vertex and any adjacent vertex that is NOT a bridge, unless there is no other choice, making a note of the edge you removed. Repeat this step until all edges are removed. Step 3: Write out the Euler circuit using the sequence of vertices and edges that you found. For example, if you removed ab, bc, cd, de, and ea, in that order, then the Euler circuit is a → b → c → d → e → a. ### Check Your Understanding ### Key Terms 1. algorithm 2. Fleury’s algorithm 3. Euler trail 4. bridge 5. local bridge ### Key Concepts 1. An Euler trail exists whenever a graph has exactly two vertices of odd degree. 2. When a bridge is removed from a graph, the number of components increases. 3. A bridge is never part of a circuit. 4. When a local bridge is removed from a graph, the distance between vertices increases. 5. An edge that is part of a triangle is never a local bridge. ### Videos 1. Bridges and Local Bridges in Graph Theory 2. Fluery's Algorithm to Find an Euler Circuit

# Graph Theory ## Hamilton Cycles ### Learning Objectives After completing this section, you should be able to: 1. Describe and identify Hamilton cycles. 2. Compute the number of Hamilton cycles in a complete graph. 3. Apply and evaluate weighted graphs. In Euler Circuits and Euler Trails, we looked for circuits and paths that visited each edge of a graph exactly once. In this section, we will look for circuits that visit each vertex exactly once. Like many concepts in graph theory, the idea of a circuit that visits each vertex once was inspired by games and puzzles. As early as the 9th century, Indian and Islamic intellectuals wondered whether it was possible for a knight to visit every space on a chess board of a given size, which is equivalent to visiting every vertex of a graph that represents the chess board. In 1857, a mathematician named William Rowan Hamilton invented a puzzle in which players were asked to find a route along the edges of a dodecahedron (see ), which visited every vertex exactly once. Let’s explore how graph theory provides insight into these games as well as practical applications such as the Traveling Salesperson Problem. ### Hamilton’s Puzzle Before we look at the solution to Hamilton's puzzle, let’s review some vocabulary we used in . It will be helpful to remember that directed cycle is a type of circuit that doesn’t repeat any edges or vertices. The goal of Hamilton's puzzle was to find a route along the edges of the dodecahedron, which visits each vertex exactly once. A dodecahedron is a three-dimensional space figure with faces that are all pentagons as we saw in . Since it is easier to visualize two dimensions rather than three, we will flatten out the dodecahedron and look at the edges and vertices on a flat surface. Graph A in shows a two-dimensional graph of the edges and vertices, and Graph B shows an untangled version of Graph A in which no edges are crossing. Graph B in is very similar to the design of the game board that Hamilton invented for his puzzle. We can see that this is a planar graph because it can be “untangled.” In order to solve Hamilton’s puzzle, we need to find a circuit that visits every vertex once. A solution is shown in . A circuit that doesn’t repeat any vertices, like the one in , is called a directed cycle. So, we can most accurately say that Hamilton’s puzzle asks us to find a directed cycle that visits every vertex in a graph exactly once. Because Hamilton created and solved this puzzle, these special circuits were named Hamilton cycles, or Hamilton circuits. ### Hamilton Cycles vs. Euler Circuits Let’s practice naming and identifying Hamilton cycles, as well as distinguishing them from Euler circuits. It is important to remember that Euler circuits visit all edges without repetition, while Hamilton cycles visit all vertices without repetition. Hamilton cycles are named by their vertices just like all circuits. An example is given in . Notice that the Hamilton cycle a → b → c → d for Graph Z in is NOT an Euler circuit, because it does not visit edge . Some Hamilton cycles are also Euler circuits while some are not, and some Euler circuits are Hamilton cycles while some are not. Notice that the graph is a cycle. A cycle will always be Eulerian because all vertices are degree 2. Moreover, any circuit in the graph will always be both an Euler circuit and a Hamilton cycle. It is not always as easy to determine if a graph has a Hamilton cycle as it is to see that it has an Euler circuit, but there is a large group of graphs that we know will always have Hamilton cycles, the complete graphs. Since all vertices in a complete graph are adjacent, we can always find a directed cycle that visits all the vertices. For example, look at the directed six-cycle, n → o → p → q → r → s, in the complete graph with six vertices in . That is not the only directed six-cycle in the graph though. We could find another just be reversing the direction, and we could find even more by using different edges. So, how many Hamilton cycles are in a complete graph with vertices? Before we tackle this problem, let’s look at a shorthand notation that we use in mathematics which will be helpful to us. ### Factorials In many areas of mathematics, we must calculate products like or , products that involve multiplying all the counting numbers from a particular number down to 1. Imagine that the product happened to be all the numbers from 100 down to 1. That’s a lot of writing! Instead of writing all of that out, mathematicians came up with a shorthand notation. For example, instead of , we write , which is read “7 factorial.” In other words, the product of all the counting numbers from down to 1 is called factorial and it is written A common use for factorials is counting the number of ways to arrange objects. Suppose that there were three students, Aryana, Byron, and Carlos, who wanted to line up in a row. How many arrangements are possible? There are six possibilities: ABC, ACB, BAC, BCA, CAB, or CBA. Notice that there were three students being arranged, and the number of possible arrangements is three. ### Counting Hamilton Cycles in Complete Graphs Now, let’s get back to answering the question of how many Hamilton cycles are in a complete graph. In , we have drawn all the four cycles in a complete graph with four vertices. Remember, cycles can be named starting with any vertex in the cycle, but we will name them starting with vertex . shows that there are three unique four-cycles in a complete graph with four vertices. Notice that there were two ways to name each cycle, one reading the vertices in a clockwise direction and one reading the vertices in a counterclockwise direction. This is important to us because we are interested in Hamilton cycles, which are directed cycles. Although the cycles (a, b, c, d) and (a, d, c, b) are the same cycle, the directed cycles, a → b → c → d → a and a → d → c → b → a, which travel the same route in reverse order are considered different directed cycles, as shown in . The six directed four-cycles in are the only distinct Hamilton cycles in a complete graph with four vertices. Six is also the number of ways to arrange the three letters b, c, and d. (Do you see why?) The number of ways to arrange three letters is . Similarly, the number of Hamilton cycles in a graph with five vertices is the number of ways to arrange four letters, which is . In general, to find the number of Hamilton cycles in a graph, we take one less than the number of vertices and find its factorial. ### Weighted Graphs Suppose that an officer in the U.S. Air Force who is stationed at Vandenberg Air Force base must drive to visit three other California Air Force bases before returning to Vandenberg. The officer needs to visit each base once. The vertices in the graph in represent the four U.S. Air Force bases, Vandenberg, Edwards, Los Angeles, and Beale. The edges are labeled to with the driving distance between each pair of cities. The graph in is called a weighted graph, because each edge has been assigned a value or weight. The weights can represent quantities such as time, distance, money, or any quantity associated with the adjacent vertices joined by the edges. The total weight of any walk, trail, or path is the sum of the weights of the edges it visits. Notice that the officer’s trip can be represented as a Hamilton cycle, because each of the four vertices in the graph is visited exactly once. ### Check Your Understanding ### Key Terms 1. Hamilton cycle, or Hamilton circuit 2. factorial 3. weighted graph 4. total weight ### Key Concepts 1. A Hamilton cycle is a directed cycle, or circuit, that visits each vertex exactly once. 2. Some Hamilton cycles are also Euler circuits, but some are not. 3. Hamilton cycles that follow the same undirected cycle in the same direction are considered the same cycle even if they begin at a different vertex. 4. The number of unique Hamilton cycles in a complete graph with n vertices is the same as the number of ways to arrange distinct objects. 5. Weighted graphs have a value assigned to each edge, which can represent distance, time, money and other quantities. ### Formulas The number of ways to arrange distinct objects is . The number of distinct Hamilton cycles in a complete graph with vertices is .

# Graph Theory ## Hamilton Paths ### Learning Objectives After completing this section, you should be able to: 1. Describe and identify Hamilton paths. 2. Evaluate Hamilton paths in real-world applications. 3. Distinguish between Hamilton paths and Euler trails. In the United States, school buses carry 25 million children between school and home every day. The total distance they travel is around 6 billion kilometers per year. In the city of Boston, Massachusetts, the 2016 budget for running those buses was $120 million dollars. In 2017, the city held a competition to find ways to cut costs and the Quantum Team from the MIT Operations Research Center came to the rescue, using a computer algorithm to identify the most efficient and least costly routes, which saved the city of Boston $5 million each year and even reduced daily CO2 emissions by 9,000 kilograms! (This U.S. city put an algorithm in charge of its school bus routes and saved $5 million, Sean Fleming, World Economic Forum) The problem the Quantum Team tackled involves graph theory. Imagine a graph in which vertices are the bus depot, the school, and the bus stops along a particular route. The bus must start at the depot, visit every stop exactly once, and end at the school. The route is a special kind of path that visits every vertex exactly once. Can you guess what those paths are called? ### Hamilton Paths Just as circuits that visit each vertex in a graph exactly once are called Hamilton cycles (or Hamilton circuits), paths that visit each vertex on a graph exactly once are called Hamilton paths. As we explore Hamilton paths, you might find it helpful to refresh your memory about the relationships between walks, trails, and paths by looking at . We know that paths are walks that don’t repeat any vertices or edges. So, a Hamilton path visits every vertex without repeating any vertices or edges. shows a path from vertex A to vertex E and a Hamilton path from vertex A to vertex E. ### Finding Hamilton Paths Suppose you were visiting an aquarium with some friends. The map of the aquarium is given in . The letters represent the exhibits. shows a graph of the aquarium in which each vertex represents an exhibit and each edge is a route between the pair of exhibits that doesn’t bypass another exhibit. Let’s see if we can plan a tour of the exhibits that visits each exhibit exactly once, beginning at exhibit O and ending at exhibit C. Suppose that, after exhibit O, we plan to visit exhibit Q and then exhibit M. After M, should we plan to visit N, L, or R? Take a look at . If R is not chosen next, that will cause a problem later on. Do you see what it is? If L or N is chosen next, the only way to get to R later will be to go from S to R, and then we will not be able to continue without repeating a vertex. So, we will pick R next, and then the only option is S. After S we have another choice to make. As shown in , the next choice is between B and E. Keeping in mind that the goal is to end at C, which would be the better choice? If you said vertex B, you are right! Otherwise, we will not be able to visit B later. After B, the only option is E. Then we can choose either D or G. Either will work fine. Let’s choose G as shown in . After G, you must visit H, but should you visit K or L after that? If you said to go to vertex L next, you are right! Otherwise, it will be impossible to visit N without repeating a vertex. So, next is L, then N, then K, and then at J you have another decision to make we can see in . Should you choose F, I, or P next? If you said P, you are right! If you choose either of the other two vertices, you will not be able to visit P later without passing through another vertex twice. Once P is chosen, vertex I must be next followed by F. Then you have to choose between A and D as shown in . In this case, we must go to D then to A so that we can visit C without backtracking. The complete Hamilton path is shown in . So, one Hamilton path that begins at O and ends at C is O → Q → M → R → S → B → E → G → H → L → N → K → J → P → I → F → D → A → C. There is no set sequence of steps that can be used to find a Hamilton path if it exists, but it does help to keep in mind where we are headed and avoid choices that will make returning to a particular vertex impossible without repeating vertices. Let’s practice finding Hamilton paths. ### Existence of a Hamilton Path It turns out that there is no Hamilton path between vertices A and E in Graph G in . To understand why, let’s imagine there is a red apple tree on one side of a bridge and a green apple tree on the other side of the bridge. Now suppose someone asked you to pick up all the fallen apples under each tree without crossing the bridge more than once, and making sure that the first apple you pick up and the last apple you pick up are both red. You would say, that is impossible! To have the first and last apple be red would either require leaving the green apples on the ground or crossing the bridge twice. Let’s see how this relates to finding a Hamilton path between A and E in Graph G. The edge AC is a bridge because, if it were removed, the graph would become disconnected with two components, the component {C} and the component {A, B, D, E, F}. So, we can think of the vertices A, B, D, E, and F as the red apples, vertex C as the green apple, and the edge AC is the bridge between them as in . The creation of a Hamilton path requires a visit to each vertex, just as picking up all the apples requires a visit to each apple. A and E are both red apples; so, a path from A to E would both start and end at a red apple, just as you were asked to do. And you wouldn’t be able to cross the bridge twice because that would mean visiting A twice, which is not allowed in a Hamilton path. So, it is impossible to find a Hamilton path from A to E just as it was impossible to pick up all the apples without crossing the bridge more than once. By the same reasoning, if a graph has a bridge, there will never be a Hamilton path that begins and ends on the same side of that bridge, meaning beginning and ending at vertices that would be in the same component if the bridge were removed from the graph. There is not a short way to determine if there is a Hamilton path between two vertices on a graph that works in every situation. However, there are a few common situations that can help us to quickly determine that there is no Hamilton path. Some of these are listed in . ### Hamilton Path or Euler Trail? We learned in Euler Trails that an Euler trail visits each edge exactly once, whereas a Hamilton path visits each vertex exactly once. Let’s practice distinguishing between the two. ### Check Your Understanding ### Key Terms 1. Hamilton path ### Key Concepts 1. A Hamilton path visits every vertex exactly once. 2. Some Hamilton paths are also Euler trails, but some are not.

# Graph Theory ## Traveling Salesperson Problem ### Learning Objectives After completing this section, you should be able to: 1. Distinguish between brute force algorithms and greedy algorithms. 2. List all distinct Hamilton cycles of a complete graph. 3. Apply brute force method to solve traveling salesperson applications. 4. Apply nearest neighbor method to solve traveling salesperson applications. We looked at Hamilton cycles and paths in the previous sections Hamilton Cycles and Hamilton Paths. In this section, we will analyze Hamilton cycles in complete weighted graphs to find the shortest route to visit a number of locations and return to the starting point. Besides the many routing applications in which we need the shortest distance, there are also applications in which we search for the route that is least expensive or takes the least time. Here are a few less common applications that you can read about on a website set up by the mathematics department at the University of Waterloo in Ontario, Canada: 1. Design of fiber optic networks 2. Minimizing fuel expenses for repositioning satellites 3. Development of semi-conductors for microchips 4. A technique for mapping mammalian chromosomes in genome sequencing Before we look at approaches to solving applications like these, let's discuss the two types of algorithms we will use. ### Brute Force and Greedy Algorithms An algorithm is a sequence of steps that can be used to solve a particular problem. We have solved many problems in this chapter, and the procedures that we used were different types of algorithms. In this section, we will use two common types of algorithms, a brute force algorithm and a greedy algorithm. A brute force algorithm begins by listing every possible solution and applying each one until the best solution is found. A greedy algorithm approaches a problem in stages, making the apparent best choice at each stage, then linking the choices together into an overall solution which may or may not be the best solution. To understand the difference between these two algorithms, consider the tree diagram in . Suppose we want to find the path from left to right with the largest total sum. For example, branch A in the tree diagram has a sum of . To be certain that you pick the branch with greatest sum, you could list each sum from each of the different branches: A: B: C: D: E: F: G: H: Then we know with certainty that branch E has the greatest sum. Now suppose that you wanted to find the branch with the highest value, but you only were shown the tree diagram in phases, one step at a time. After phase 1, you would have chosen the branch with 10 and 7. So far, you are following the same branch. Let’s look at the next phase. After phase 2, based on the information you have, you will choose the branch with 10, 7 and 4. Now, you are following a different branch than before, but it is the best choice based on the information you have. Let’s look at the last phase. After phase 3, you will choose branch G which has a sum of 32. The process of adding the values on each branch and selecting the highest sum is an example of a brute force algorithm because all options were explored in detail. The process of choosing the branch in phases, based on the best choice at each phase is a greedy algorithm. Although a brute force algorithm gives us the ideal solution, it can take a very long time to implement. Imagine a tree diagram with thousands or even millions of branches. It might not be possible to check all the sums. A greedy algorithm, on the other hand, can be completed in a relatively short time, and generally leads to good solutions, but not necessarily the ideal solution. ### The Traveling Salesperson Problem Now let’s focus our attention on the graph theory application known as the traveling salesperson problem (TSP) in which we must find the shortest route to visit a number of locations and return to the starting point. Recall from Hamilton Cycles, the officer in the U.S. Air Force who is stationed at Vandenberg Air Force base and must drive to visit three other California Air Force bases before returning to Vandenberg. The officer needed to visit each base once. We looked at the weighted graph in representing the four U.S. Air Force bases: Vandenberg, Edwards, Los Angeles, and Beal and the distances between them. Any route that visits each base and returns to the start would be a Hamilton cycle on the graph. If the officer wants to travel the shortest distance, this will correspond to a Hamilton cycle of lowest weight. We saw in that there are six distinct Hamilton cycles (directed cycles) in a complete graph with four vertices, but some lie on the same cycle (undirected cycle) in the graph. Since the distance between bases is the same in either direction, it does not matter if the officer travels clockwise or counterclockwise. So, there are really only three possible distances as shown in . The possible distances are: So, a Hamilton cycle of least weight is V → B → E → L → V (or the reverse direction). The officer should travel from Vandenberg to Beal to Edwards, to Los Angeles, and back to Vandenberg. ### Finding Weights of All Hamilton Cycles in Complete Graphs Notice that we listed all of the Hamilton cycles and found their weights when we solved the TSP about the officer from Vandenberg. This is a skill you will need to practice. To make sure you don't miss any, you can calculate the number of possible Hamilton cycles in a complete graph. It is also helpful to know that half of the directed cycles in a complete graph are the same cycle in reverse direction, so, you only have to calculate half the number of possible weights, and the rest are duplicates. ### The Brute Force Method The method we have been using to find a Hamilton cycle of least weight in a complete graph is a brute force algorithm, so it is called the brute force method. The steps in the brute force method are: Step 1: Calculate the number of distinct Hamilton cycles and the number of possible weights. Step 2: List all possible Hamilton cycles. Step 3: Find the weight of each cycle. Step 4: Identify the Hamilton cycle of lowest weight. Now suppose that the officer needed a cycle that visited all 5 of the Air Force bases in . There would be different arrangements of vertices and distances to compare using the brute force method. If you consider 10 Air Force bases, there would be different arrangements and distances to consider. There must be another way! ### The Nearest Neighbor Method When the brute force method is impractical for solving a traveling salesperson problem, an alternative is a greedy algorithm known as the nearest neighbor method, which always visit the closest or least costly place first. This method finds a Hamilton cycle of relatively low weight in a complete graph in which, at each phase, the next vertex is chosen by comparing the edges between the current vertex and the remaining vertices to find the lowest weight. Since the nearest neighbor method is a greedy algorithm, it usually doesn’t give the best solution, but it usually gives a solution that is "good enough." Most importantly, the number of steps will be the number of vertices. That’s right! A problem with 10 vertices requires 10 steps, not 362,880. Let’s look at an example to see how it works. Suppose that a candidate for governor wants to hold rallies around the state. They plan to leave their home in city A, visit cities B, C, D, E, and F each once, and return home. The airfare between cities is indicated in the graph in . Let’s help the candidate keep costs of travel down by applying the nearest neighbor method to find a Hamilton cycle that has a reasonably low weight. Begin by marking starting vertex as for "visited 1st." Then to compare the weights of the edges between A and vertices adjacent to A: $250, $210, $300, $200, and $100 as shown in . The lowest of these is $100, which is the edge between A and F. Mark F as for "visited 2nd" then compare the weights of the edges between F and the remaining vertices adjacent to F: $170, $330, $150 and $350 as shown in . The lowest of these is $150, which is the edge between F and D. Mark D as for "visited 3rd." Next, compare the weights of the edges between D and the remaining vertices adjacent to D: $120, $310, and $270 as shown in . The lowest of these is $120, which is the edge between D and B. So, mark B as for "visited 4th." Finally, compare the weights of the edges between B and the remaining vertices adjacent to B: $160 and $220 as shown in . The lower amount is $160, which is the edge between B and E. Now you can mark E as and mark the only remaining vertex, which is C, as . This is shown in . Make a note of the weight of the edge from E to C, which is $180, and from C back to A, which is $210. The Hamilton cycle we found is A → F → D → B → E → C → A. The weight of the circuit is . This may or may not be the route with the lowest cost, but there is a good chance it is very close since the weights are most of the lowest weights on the graph and we found it in six steps instead of finding 120 different Hamilton cycles and calculating 60 weights. Let’s summarize the procedure that we used. Step 1: Select the starting vertex and label for "visited 1st." Identify the edge of lowest weight between and the remaining vertices. Step 2: Label the vertex at the end of the edge of lowest weight that you found in previous step as where the subscript n indicates the order the vertex is visited. Identify the edge of lowest weight between and the vertices that remain to be visited. Step 3: If vertices remain that have not been visited, repeat Step 2. Otherwise, a Hamilton cycle of low weight is . ### Check Your Understanding ### Key Terms 1. brute force algorithm 2. greedy algorithm 3. traveling salesperson problem (TSP) 4. brute force method 5. nearest neighbor method ### Key Concepts 1. A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient but usually does not lead to the ideal solution. 2. A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. 3. The brute force method finds a Hamilton cycle of lowest weight in a complete graph. 4. The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a complete graph. ### Formulas 1. In a complete graph with vertices, the number of distinct Hamilton cycles is . 2. In a complete graph with vertices, there are at most different weights of Hamilton cycles.

# Graph Theory ## Trees ### Learning Objectives After completing this section, you should be able to: 1. Describe and identify trees. 2. Determine a spanning tree for a connected graph. 3. Find the minimum spanning tree for a weighted graph. 4. Solve application problems involving trees. We saved the best for last! In this last section, we will discuss arguably the most fun kinds of graphs, trees. Have you every researched your family tree? Family trees are a perfect example of the kind of trees we study in graph theory. One of the characteristics of a family tree graph is that it never loops back around, because no one is their own grandparent! ### What Is A Tree? Whether we are talking about a family tree or a tree in a forest, none of the branches ever loops back around and rejoins the trunk. This means that a tree has no cyclic subgraphs, or is acyclic. A tree also has only one component. So, a tree is a connected acyclic graph. Here are some graphs that have the same characteristic. Each of the graphs in is a tree. Let’s practice determining whether a graph is a tree. To do this, check if a graph is connected and has no cycles. ### Types of Trees Mathematicians have had a lot of fun naming graphs that are trees or that contain trees. For example, the graph in is not a tree, but it contains two components, one containing vertices a through d, and the other containing vertices e through g, each of which would be a tree on its own. This type of structure is called a forest. There are also interesting names for trees with certain characteristics. 1. A path graph or linear graph is a tree graph that has exactly two vertices of degree 1 such that the only other vertices form a single path between them, which means that it can be drawn as a straight line. 2. A star tree is a tree that has exactly one vertex of degree greater than 1 called a root, and all other vertices are adjacent to it. 3. A starlike tree is a tree that has a single root and several paths attached to it. 4. A caterpillar tree is a tree that has a central path that can have vertices of any degree, with each vertex not on the central path being adjacent to a vertex on the central path and having a degree of one. 5. A lobster tree is a tree that has a central path that can have vertices of any degree, with paths consisting of either one or two edges attached to the central path. Examples of each of these types of structures are given in . ### Characteristics of Trees As we study trees, it is helpful to be familiar with some of their characteristics. For example, if you add an edge to a tree graph between any two existing vertices, you will create a cycle, and the resulting graph is no longer a tree. Some examples are shown in . Adding edge bj to Graph T creates cycle (b, c, i, j). Adding edge rt to Graph P creates cycle (r, s, t). Adding edge tv to Graph S creates cycle (t, u, v). It is also true that removing an edge from a tree graph will increase the number of components and the graph will no longer be connected. In fact, you can see in that removing one or more edges can create a forest. Removing edge qr from Graph P creates a graph with two components, one with vertices o, p and q, and the other with vertices r, s, and t. Removing edge uw from Graph S creates two components, one with just vertex w and the other with the rest of the vertices. When two edges were removed from Graph T, edge bf and edge cd, creates a graph with three components as shown in . A very useful characteristic of tree graphs is that the number of edges is always one less than the number of vertices. In fact, any connected graph in which the number of edges is one less than the number of vertices is guaranteed to be a tree. Some examples are given in . ### Spanning Trees Suppose that you planned to set up your own computer network with four devices. One option is to use a “mesh topology” like the one in , in which each device is connected directly to every other device in the network. The mesh topology for four devices could be represented by the complete Graph A1 in where the vertices represent the devices, and the edges represent network connections. However, the devices could be networked using fewer connections. Graphs A2, A3, and A4 of show configurations in which three of the six edges have been removed. Each of the Graphs A2, A3 and A4 in is a tree because it is connected and contains no cycles. Since Graphs A2, A3 and A4 are also subgraphs of Graph A1 that include every vertex of the original graph, they are also known as spanning trees. By definition, spanning trees must span the whole graph by visiting all the vertices. Since spanning trees are subgraphs, they may only have edges between vertices that were adjacent in the original graph. Since spanning trees are trees, they are connected and they are acyclic. So, when deciding whether a graph is a spanning tree, check the following characteristics: 1. All vertices are included. 2. No vertices are adjacent that were not adjacent in the original graph. 3. The graph is connected. 4. There are no cycles. ### Constructing a Spanning Tree Using Paths Suppose that you wanted to find a spanning tree within a graph. One approach is to find paths within the graph. You can start at any vertex, go any direction, and create a path through the graph stopping only when you can’t continue without backtracking as shown in . Once you have stopped, pick a vertex along the path you drew as a starting point for another path. Make sure to visit only vertices you have not visited before as shown in . Repeat this process until all vertices have been visited as shown in . The end result is a tree that spans the entire graph as shown in . Notice that this subgraph is a tree because it is connected and acyclic. It also visits every vertex of the original graph, so it is a spanning tree. However, it is not the only spanning tree for this graph. By making different turns, we could create any number of distinct spanning trees. ### Revealing Spanning Trees Another approach to finding a spanning tree in a connected graph involves removing unwanted edges to reveal a spanning tree. Consider Graph D in . Graph D has 10 vertices. A spanning tree of Graph D must have 9 edges, because the number of edges is one less than the number of vertices in any tree. Graph D has 13 edges so 4 need to be removed. To determine which 4 edges to remove, remember that trees do not have cycles. There are four triangles in Graph D that we need to break up. We can accomplish this by removing 1 edge from each of the triangles. There are many ways this can be done. Two of these ways are shown in . ### Kruskal’s Algorithm In many applications of spanning trees, the graphs are weighted and we want to find the spanning tree of least possible weight. For example, the graph might represent a computer network, and the weights might represent the cost involved in connecting two devices. So, finding a spanning tree with the lowest possible total weight, or minimum spanning tree, means saving money! The method that we will use to find a minimum spanning tree of a weighted graph is called Kruskal’s algorithm. The steps for Kruskal’s algorithm are: Step 1: Choose any edge with the minimum weight of all edges. Step 2: Choose another edge of minimum weight from the remaining edges. The second edge does not have to be connected to the first edge. Step 3: Choose another edge of minimum weight from the remaining edges, but do not select any edge that creates a cycle in the subgraph you are creating. Step 4: Repeat step 3 until all the vertices of the original graph are included and you have a spanning tree. ### Check Your Understanding ### Key Terms 1. acyclic 2. tree 3. forest 4. path graph or linear graph 5. star tree 6. root 7. starlike tree 8. caterpillar tree 9. lobster tree 10. spanning tree 11. minimum spanning tree ### Key Concepts 1. A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient but usually does not lead to the ideal solution. 2. A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. 3. The brute force method finds a Hamilton cycle of lowest weight in a complete graph. 4. The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a complete graph. ### Videos 1. The Problem in 2. Spanning Trees in Graph Theory 3. Use Kruskal's Algorithm to Find Minimum Spanning Trees in Graph Theory ### Formulas 1. The number of edges in a tree graph with vertices is . A connected graph with n vertices and edges is a tree graph. ### Projects Everyone Gets a Turn! – Graph Colorings Let’s put your knowledge of graph colorings to work! Your task is to plan a field day following these steps. 1. Select between seven and ten activities for your field day. You can look online for ideas. 2. Create a survey asking for the participants to select the three to five events in which they would most like to participate. Survey between seven and ten people. 3. Use the results of your survey to create a graph in which each vertex represents one of the events. A pair of vertices will be adjacent if there is at least one participant who would like to participate in both events. 4. Find a minimum coloring for the graph. Explain how you found it and how you know the chromatic number of the graph. 5. Use your solution to part d to determine the minimum number of timeslots you must use to ensure that everyone has the opportunity to participate in their top three events. 6. Find the complement of the graph you created. Explain what the edges in this graph represent. A Beautiful Day in the Neighborhood – Euler Circuits Let’s apply what you have learned to the community in which you live. Using resources such as your county’s property appraiser’s website, create a detailed graph of your neighborhood in which vertices represent turns and intersections. Represent a large enough part of your community to include no fewer than 10 intersections or turns. Then use your graph to answer the following questions. 1. Label the edges of your graph. 2. Determine if your graph is Eulerian. Explain how you know. If it is not, eulerize it. 3. Find an Euler circuit for your graph. Give the sequence of vertices that you found. 4. What does the Euler circuit you found in part c represent for your community? 5. Describe an application for which this Euler circuit might be used. Dream Vacation – Hamilton Cycles and Paths Where in the world would you like to travel most: the Eiffel Tower in Paris, a Broadway musical in New York city, a bike tour of Amsterdam, the Tenerife whale and dolphin cruises in the Canary Islands, the Giza Pyramid in Cairo, or maybe the Jokhang Temple in Tibet? Let's plan your dream vacation! 1. Which four destinations are at the top of your bucket list? 2. Draw a complete weighted graph with five vertices representing the four destinations and your home city, and the weights representing the cost of travel between cities. 3. Use a website (such as Travelocity) to find the best airfare between each pair of cities. List the airlines and flight numbers along with the prices. Include cost for ground transportation from the nearest airport if there is no airport at the destination you want to visit. 4. Use the nearest neighbor algorithm to find a Hamilton cycle of low weight beginning and ending in your hometown. What is the weight of this circuit and what does it represent? 5. Use the brute force method to find a Hamilton cycle of lowest weight beginning and ending in your hometown. What is the weight of this circuit? Is it the same or different from the weight of the Hamilton cycle you found in Exercise 4? 6. Suppose that instead of returning home, you planned to move to your favorite location on the list, but you wanted to stop at the other three destinations once along the way. Where would you move? List all Hamilton paths between your hometown and your favorite location. 7. Find the weights of all the Hamilton paths you found in Exercise 6. ### Chapter Review ### Graph Basics ### Graph Structures ### Comparing Graphs ### Navigating Graphs ### Euler Circuits ### Euler Trails ### Hamilton Cycles ### Hamilton Paths ### Traveling Salesperson Problem ### Trees ### Chapter Test

# Math and... ## Introduction Where do we find math around us? Math can be found in areas that are expected and sometimes in areas that are surprising. There are many ways that mathematical concepts, such as those in this text, are infused in the world around us. In this chapter, we will explore a sampling of five distinct areas from everyday life where math’s impact plays a meaningful role. Math's impact on art can be found in numerical relationships that are known to create or enhance beauty. The Fibonacci numbers are one mathematical example that can be found in nature such as in petal count of a rose. On a different note, a mathematical exploration can aid in making a convincing argument on how we can positively impact our environment. Whether looking at the choices of a single individual or the larger impact offered from a collaborative effort, there are measurable responses to positively address climate change. Turning to medicine, which has been a topic of global importance in recent years, we will explore how math is used to determine drug dosage rates and test the validity of a vaccine. Switching back to items of aesthetic nature, we will examine some foundational components of music which, like art, brings beauty and joy to our lives. Finally, we will explore some ways that math is used in sports to predict future performance and analyze tournaments styles.

# Math and... ## Math and Art ### Learning Objectives After completing this section, you should be able to: 1. Identify and describe the golden ratio. 2. Identify and describe the Fibonacci sequence and its application to nature. 3. Apply the golden ratio and the Fibonacci sequence relationship. 4. Identify and compute golden rectangles. Art is the expression or application of human creative skill and imagination, typically in a visual form such as painting or sculpture, producing works to be appreciated primarily for their beauty or emotional power. Oxford Dictionary Art, like other disciplines, is an area that combines talent and experience with education. While not everyone considers themself skilled at creating art, there are mathematical relationships commonly found in artistic masterpieces that drive what is considered attractive to the eye. Nature is full of examples of these mathematical relationships. Enroll in a cake decorating class and, when you learn how to create flowers out of icing, you will likely be directed as to the number of petals to use. Depending on the desired size of a rose flower, the recommendation for the number of petals to use is commonly 5, 8, or 13 petals. If learning to draw portraits, you may be surprised to learn that eyes are approximately halfway between the top of a person’s head and their chin. Studying architecture, we find examples of buildings that contain golden rectangles and ratios that add to the beautifying of the design. The Parthenon (), which was built around 400 BC, as well as modern-day structures such the Washington Monument are two examples containing these relationships. These seemingly unrelated examples and many more highlight mathematical relationships that we associate with beauty in artistic form. ### Golden Ratio The golden ratio, also known as the golden proportion, is a ratio aspect that can be found in beauty from nature to human anatomy as well as in golden rectangles that are commonly found in building structures. The golden ratio is expressed in nature from plants to creatures such as the starfish, honeybees, seashells, and more. It is commonly noted by the Greek letter ϕ (pronounced “fee”). , which has a decimal value approximately equal to 1.618. Consider : Note how the building is balanced in dimension and has a natural shape. The overall structure does not appear as if it is too wide or too tall in comparison to the other dimensions. The golden ratio has been used by artists through the years and can be found in art dating back to 3000 BC. Leonardo da Vinci is considered one of the artists who mastered the mathematics of the golden ratio, which is prevalent in his artwork such as Virtuvian Man (). This famous masterpiece highlights the golden ratio in the proportions of an ideal body shape. The golden ratio is approximated in several physical measurements of the human body and parts exhibiting the golden ratio are simply called golden. The ratio of a person’s height to the length from their belly button to the floor is ϕ or approximately 1.618. The bones in our fingers (excluding the thumb), are golden as they form a ratio that approximates ϕ. The human face also includes several ratios and those faces that are considered attractive commonly exhibit golden ratios. ### Fibonacci Sequence and Application to Nature The Fibonacci sequence can be found occurring naturally in a wide array of elements in our environment from the number of petals on a rose flower to the spirals on a pine cone to the spines on a head of lettuce and more. The Fibonacci sequence can be found in artistic renderings of nature to develop aesthetically pleasing and realistic artistic creations such as in sculptures, paintings, landscape, building design, and more. It is the sequence of numbers beginning with 1, 1, and each subsequent term is the sum of the previous two terms in the sequence (1, 1, 2, 3, 5, 8, 13, …). The petal counts on some flowers are represented in the Fibonacci sequence. A daisy is sometimes associated with plucking petals to answer the question “They love me, they love me not.” Interestingly, a daisy found growing wild typically contains 13, 21, or 34 petals and it is noted that these numbers are part of the Fibonacci sequence. The number of petals aligns with the spirals in the flower family. ### Golden Ratio and the Fibonacci Sequence Relationship Mathematicians for years have explored patterns and applications to the world around us and continue to do so today. One such pattern can be found in ratios of two adjacent terms of the Fibonacci sequence. Recall that the Fibonacci sequence = 1, 1, 3, 5, 8, 13,… with 5 and 8 being one example of adjacent terms. When computing the ratio of the larger number to the preceding number such as 8/5 or 13/8, it is fascinating to find the golden ratio emerge. As larger numbers from the Fibonacci sequence are utilized in the ratio, the value more closely approaches ϕ, the golden ratio. ### Golden Rectangles Turning our attention to man-made elements, the golden ratio can be found in architecture and artwork dating back to the ancient pyramids in Egypt () to modern-day buildings such as the UN headquarters. The ancient Greeks used golden rectangles—any rectangles where the ratio of the length to the width is the golden ratio—to create aesthetically pleasing as well as solid structures, with examples of the golden rectangle often being used multiple times in the same building such as the Parthenon, which is shown in . Golden rectangles can be found in twentieth-century buildings as well, such as the Washington Monument. Looking at another man-made element, artists paintings often contain golden rectangles. Well-known paintings such as Leonardo da Vinci’s The Last Supper and the Vitruvian Man contain multiple golden rectangles as do many of da Vinci’s masterpieces. Whether framing a painting or designing a building, the golden rectangle has been widely utilized by artists and are considered to be the most visually pleasing rectangles. ### Check Your Understanding ### Key Terms 1. golden ratio 2. ϕ 3. Fibonacci sequence 4. golden rectangle ### Key Concepts 1. The golden ratio, ϕ, can be found in nature, and the relationship is often associated with beauty and balance. 2. The Fibonacci sequence reflects a pattern of numbers that can be found in various places in nature. The sequence can be used to predict other values that follow the Fibonacci pattern. 3. State some naturally occurring applications of the Fibonacci sequence. 4. State some naturally occurring applications of the golden ratio. 5. Determine if a rectangle is golden. 6. State some artistic applications of the golden rectangle.

# Math and... ## Math and the Environment ### Learning Objectives After completing this section, you should be able to: 1. Compute how conserving water can positively impact climate change. 2. Discuss the history of solar energy. 3. Compute power needs for common devices in a home. 4. Explore advantages of solar power as it applies to home use. Climate change and emissions management have been debated topics in recent years. However, more and more people are recognizing the impacts that have resulted in temperature changes and are seeking timely and effective action. The World Meteorological Organization shared in a June 2021 publication that “2021 is a make-or-break year for climate action, with the window to prevent the worst impacts of climate change—which include ever more frequent more intense droughts, floods and storms—closing rapidly.” The problem no longer belongs to a few countries or regions but rather is a worldwide concern measured with increasing temperatures leading to decreased glacier coverage and resulting rise in sea levels. The good news is, there are small steps that each of us can do that collectively can positively impact climate change. ### Making a Positive Impact on Climate Change—Water Usage Our use of water is one element that impacts climate change. Having access to clean, potable water is critical for not only our health but also for the health of our ecosystem. About 1 out of 10 people on our planet do not have easy access to clean water to drink. As each of us conserves water, we prolong the life span of fresh water from our lakes and rivers and also reduce the impact on sewer systems and drainage in our communities. Additionally, as we conserve water, we also conserve electricity that is used to bring water to and in our homes. So, what can we do to help conserve water? ### History of Solar Energy In the mid-1800s, Willoughby Smith discovered photoconductive responsiveness in selenium. Shortly thereafter, William Grylls Adams and Richard Evans Day discovery that selenium can produce electricity if exposed to the sun was a major breakthrough. Less than 10 years later, Charles Fritts invented the first solar cells using selenium. Jumping a mere 100 years later, Bell Labs in the United States produced the first practical photovoltaic cells in the mid-1950s and developed versions used to power satellites in the same decade. Solar panel use has exploded in recent decades and is now used by residences, organizations, businesses, and government buildings such as the White House, space to power satellites, and various methods of transportation. One reason for the expansion is a continuing drop in cost combined with an increase in performance and durability. In the mid-1950s, the cost of a solar panel was around $300 per watt capability. Twenty years later, the cost was a third of the 1950s’ cost. Currently, solar panel cost has dropped to less than $1 per watt while decreasing in size as well as increasing in longevity. The dropping price and improved performance has moved solar to a modest investment that can pay for itself in less than half the time of systems from 15 years ago. ### Compute Power Needs for Common Home Devices A kilowatt (kW) is 1,000 watts (W). A kilowatt-hour (kWh) is a measurement of energy use, which is the amount of energy used by a 1,000-watt device to run for an hour. Using the definition of a kilowatt-hour, to calculate how long it would take to consume 1 kWh of power, we divide 1,000 by the watts use of a device. For example, a 75 W bulb would take to use 1 kW of power. ### Solar Advantages There are multiple advantages that solar power can offer us today including reducing greenhouse gas and CO2 emissions, powering vehicles, reducing water pollution, reducing strain on limited supply of other power options such as fossil fuels. We will look further at reducing greenhouse gas and CO2 emissions. Any gas that prevents infrared radiation from escaping Earth's atmosphere is a greenhouse gas. There are 24 currently identified greenhouse gases of which carbon dioxide is one. When measuring the impact of any of the greenhouse gases, the measurements are given in units of carbon dioxide emissions. For this reason, greenhouse gas and carbon dioxide have become interchangeable in discussions. ### Check Your Understanding ### Key Terms 1. greenhouse gas 2. CO2 emissions 3. watt 4. kilowatt (kW) ### Key Concepts 1. Recognize how water conservation by one person, family, community, or nation can positively impact the world’s freshwater supply. 2. Recall components from the history of solar use by mankind. 3. Calculate electrical demand given watts. 4. Recognize advantages of residential solar power. ### Formulas

# Math and... ## Math and Medicine ### Learning Objectives After completing this section, you should be able to: 1. Compute the mathematical factors utilized in concentrations/dosages of drugs. 2. Describe the history of validating effectiveness of a new drug. 3. Describe how mathematical modeling is used to track the spread of a virus. The pandemic that rocked the world starting in 2020 turned attention to finding a cure for the Covid-19 strain into a world race and dominated conversations from major news channels to households around the globe. News reports decreeing the number of new cases and deaths locally as well as around the world were part of the daily news for over a year and progress on vaccines soon followed. How was a vaccine able to be found so quickly? Is the vaccine safe? Is the vaccine effective? These and other questions have been raised through communities near and far and some remain debatable. However, we can educate ourselves on the foundations of these discussions and be more equipped to analyze new information related to these questions as it becomes available. ### Concentrations and Dosages of Drugs Consider any drug and the recommended dosage varies based on several factors such as age, weight, and degree of illness of a person. Hospitals and medical dispensaries do not stock every possible needed concentration of medicines. Drugs that are delivered in liquid form for intravenous (IV) methods in particular can be easily adjusted to meet the needs of a patient. Whether administering anesthesia prior to an operation or administering a vaccine, calculation of the concentration of a drug is needed to ensure the desired amount of medicine is delivered. The formula to determine the volume needed of a drug in liquid form is a relatively simple formula. The volume needed is calculated based on the required dosage of the drug with respect to the concentration of the drug. For drugs in liquid form, the concentration is noted as the amount of the drug per the volume of the solution that the drug is suspended in which is commonly measured in g/mL or mg/mL. Suppose a doctor writes a prescription for 6 mg of a drug, which a nurse calculates when retrieving the needed prescription from their secure pharmaceutical storage space. On the shelves, the drug is available in liquid form as 2 mg per mL. This means that 1 mg of the drug is found in 0.5 mL of the solution. Multiplying 6 mg by 0.5 mL yields 3 mL, which is the volume of the prescription per single dose. A common calculation for the weight of a liquid drug is measured in grams of a drug per 100 mL of solution and is also called the percentage weight by volume measurement and labeled as % w/v or simply w/v. Suppose you visit your doctor with symptoms of an upset stomach and unrelenting heartburn. One possible recourse is sodium bicarbonate, which aids in reducing stomach acid. ### Validating Effectiveness of a New Vaccine The process to develop a new vaccine and be able to offer it to the public typically takes 10 to 15 years. In the United States, the system typically involves both public and private participation in a process. During the 1900s, several vaccines were successfully developed, including the following: polio vaccine in the 1950s and chickenpox vaccine in the 1990s. Both of these vaccines took years to be developed, tested, and available to the public. Knowing the typical timeline for a vaccine to move from development to administration, it is not surprising that some people wondered how a vaccine for Covid-19 was released in less than a year’s time. Lesser known is that research on coronavirus vaccines has been in process for approximately 10 years. Back in 2012, concern over the Middle Eastern respiratory syndrome (MERS) broke out and scientists from all over the world began working on researching coronaviruses and how to combat them. It was discovered that the foundation for the virus is a spike protein, which, when delivered as part of a vaccine, causes the human body to generate antibodies and is the platform for coronavirus vaccines. When the Covid-19 pandemic broke out, Operation Warp Speed, fueled by the U.S. federal government and private sector, poured unprecedented human resources into applying the previous 10 years of research and development into targeting a specific vaccine for the Covid-19 strain. ### Mathematical Modeling to Track the Spread of a Vaccine With a large number of people receiving a Covid-19 vaccine, the concern at this time is how to create an affordable vaccine to reach people all over the world. If a world solution is not found, those without access to a vaccine will serve as incubators to variants that might be resistant to the existing vaccines. As we work to vaccinate the world, attention continues with tracking the spread of the Covid-19 and its multiple variants. Mathematical modeling is the process of creating a representation of the behavior of a system using mathematical language. Digital mathematical modeling plays a key role in analyzing the vast amounts of data reported from a variety of sources such as hospitals and apps on cell phones. When attempting to represent an observed quantitative data set, mathematical models can aid in finding patterns and concentrations as well as aid in predicting growth or decline of the system. Mathematical models can also be useful to determine strengths and vulnerabilities of a system, which can be helpful in arresting the spread of a virus. The chapter on Graph Theory explores one such method of mathematical modeling using paths and circuits. Cell phones have been helpful in tracking the spread of the Covid-19 virus using apps regulated by regional government public health authorities to collect data on the network of people exposed to an individual who tests positive for the Covid-19 virus. Consider the following graph (): At the center of the graph, we find Alyssa, whom we will consider positive for a virus. Utilizing the technology of phone apps voluntarily installed on each phone of the individuals in the graph, tracking of the spread of the virus among the 6 individuals that Alyssa had direct contact with can be implemented, namely Suad, Rocio, Braeden, Soren, and Sandra. Let’s look at José’s exposure risk as it relates to Alyssa. There are multiple paths connecting José with Alyssa. One path includes the following individuals: José to Mikaela to Nate to Sandra to Alyssa. This path contains a length of 4 units, or people, in the contact tracing line. There are 2 more paths connecting José to Alyssa. A second path of the same length consists of José to Lucia to Rocio to Braeden to Alyssa. Path 3 is the shortest and consists of José to Lucia to Rocio to Alyssa. Tracking the spread of positive cases in the line between Alyssa and José aids in monitoring the spread of the infection. Now consider the complexity of tracking a pandemic across the nation. Graphs such as the one above are not practical to be drawn on paper but can be managed by computer programs capable of computing large volumes of data. In fact, a computer-generated mathematical model of contact tracing would look more like a sphere with paths on the exterior as well as on the interior. Mathematical modeling of contact tracing is complex and feasible through the use of technology. ### Check Your Understanding ### Key Terms 1. concentrations/dosages of drugs 2. mathematical modeling ### Key Concepts 1. Compute volumes of prescription drugs in liquid and pill form. 2. Validate the effectiveness of a new drug. 3. Mathematical modeling can be used to describe and track the spread of a virus. ### Formula .

# Math and... ## Math and Music ### Learning Objectives After completing this section, you should be able to: 1. Describe the basics of frequency related to sound. 2. Describe the basics of pitch as it relates to music. 3. Describe and evaluate musical notes, half-steps, whole steps, and octaves. 4. Describe and find frequencies of octaves. “The world’s most famous and popular language is music.” Psy, South Korean singer, rapper, songwriter, and record producer Imagine a world without music and many of us would struggle to fill the void. Music uplifts, inspires, heals, and generally adds dimension to virtually every aspect of our lives. But what is music? For some it is a song; for others it may be the sounds of birds or the rhythmic sound of drumming or a myriad of other sounds. Whatever you consider music, it is all around us and is an integral part of our lives. “Music can raise someone’s mood, get them excited, or make them calm and relaxed. Music also—and this is important—allows us to feel nearly or possibly all emotions that we experience in our lives. The possibilities are endless” (Galindo, 2009). What music you listen to can impact your mood and emotions. In similar fashion, the music we choose can often tell those around us something about our current moods and emotions. Consider the music you may have been listening to as you today or even as you are reading this text. What cues to your mood do your music selections share? Albert Einstein is quoted as saying, “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.” What clues do your recent music choices say about your mood or how your day is going? ### Basics of Frequency as It Relates to Sound Every sound is created by an object vibrating and these vibrations travel in waves that are captured by our ears. Some vibrations we may be able to see, such as a plucked guitar string moving, whereas other vibrations we may not be able to see, such as the sound created when we hold our breath when accidentally dropping our cell phone on a hard floor. We don’t see the vibrations of our cell phone hitting the floor; however, any audible sound created in the fall is the result of vibrations in the form of sound waves, which can be pictured similarly to waves moving through the ocean. The waves of sounds each have a frequency, or rate of vibration of sound waves, that measures the number of waves completed in a single second and are measured in hertz (Hz; one Hz is one cycle per second). Louder sounds have stronger vibrations or are created closer to our ear. The further an ear is from the source of the sound, the quieter the sound will appear. Sounds range in frequency from 16 Hz to ultrasonic values, with humans able to hear sounds in a frequency range of about 20 Hz to 20,000 Hz. Adults lose the ability to hear the upper end of the range and typically top out in the ability to hear in a frequency of 15,000–17,000 Hz. Sounds with a frequency above 17,000 Hz are less likely to be heard by adults while still being audible to children. While frequency plays a key role in audible sounds, so too does the sound level, which can be measured in decibels (dB), which are the units of measure for the intensity of a sound or the degree of loudness. As a sound level increases, the decibel level increases. A person with average hearing can hear sounds down to 0 dB. Those with exceptionally good hearing can hear even quieter sounds, down to approximately –5 dB. The following table includes sample sounds with their related decibel values. ### Basics of Pitch When considering the various sound levels the human ear can hear, the ear perceives sound both from the frequency level and the pitch of a sound. The quality of the sound is referred to as pitch, the tonal quality of a sound and how high or low the tone. Sounds with a high frequency have a high pitch, such as 900 Hz, and sounds with a low pitch have a low frequency, such as 50 Hz. Let’s take a look at frequency and pitch using a string instrument such as a guitar or piano. When a string is plucked on a guitar or a key is played on a piano, the related string vibrates at a frequency that is related to the length and thickness of the string. The frequency is measurable and has a singular value. The pitch of the note played is open for interpretation, as the pitch is a function of personal opinion. ### Note Values, Half-Steps, Whole Steps, and Octaves “There are not more than five musical notes, yet the combinations of these five give rise to more melodies than can ever be heard.” Sun Tzu, Chinese strategist Moving our exploration to note values, the frequency of all notes is well defined by a specific and unique frequency for each note that is measurable. We will explore keys on a keyboard to discuss notes that have the same relationships with any instrument or musical piece. Let’s look at . The white keys are labeled with the letters A–G and the photo begins with middle C, which can be found in the middle of a keyboard. This labeling of the keys repeats across an entire keyboard and keys to the right have a higher pitch and frequency than keys to the left. Each of the keys correlates to a musical note. Movement up or down between any two consecutive keys (black and white) or notes constitutes a half-step. Movement of one half-step sometimes involves a sharp (#) or a flat (♭) symbol. For example, D# is one half-step above D and D♭ is one half-step below D. Note that this is not always true as one half-step above B is C, and one half-step below F is E. In similar fashion, a whole step is movement up or down between any two half-steps on a keyboard. You may have noticed that there are eight letters of the alphabet used to label notes. Selecting any one note and counting up 12 half-steps you will find that the numbering for notes begins at the same value as you started from. This collection of 12 consecutive half-notes is called an octave and is a basic foundational component in music theory. ### Frequencies of Octaves Notes that are one octave apart have the same name and are related in frequency values. Given the frequency of any note, the frequency of same note one octave higher is doubled and this pattern continues as you move up and down the notes on a keyboard or any other musical instrument. Song writers and singers use this knowledge to change the pitch of a note up or down to align with a person’s vocal range. Regardless of which C is played or sung, the pitch is the same and the frequency is related by a power or two. Labeled keys on a keyboard are numbered for ease in identification. For example, middle C is labeled as C4 on a full keyboard as it is the fourth C from the left in a set of eight notes. The frequency of C4 is 262 Hz, rounded to the nearest whole number. We have explored some basics components of frequency, pitch, note relationships, and octaves, which are building blocks of music. It may be exciting to learn that the mathematical relationships found in music are vast and grow in complexity beyond the math commonly studied in high school. ### Check Your Understanding ### Key Terms 1. frequency 2. pitch 3. Hertz 4. decibel 5. half-step 6. whole step 7. flat 8. sharp 9. octave ### Key Concepts 1. As frequency of a sound increases, the pitch of the sound increases. 2. Hertz is a unit of measurement for frequency. 3. Decibel is a unit of measurement for the intensity of sound. 4. Half-steps and whole-steps describe one type of movement between two notes on a keyboard. 5. Octaves are a collection of any 12 consecutive notes, which is a foundation in music theory.

# Math and... ## Math and Sports ### Learning Objectives After completing this section, you should be able to: 1. Describe why data analytics (statistics) is crucial to advance a team’s success. 2. Describe single round-robin method of tournaments. 3. Describe single-elimination method of tournaments. 4. Explore math in baseball, fantasy football, hockey, and soccer (projects at the end of the section). Sports are big business and entertainment around the world. In the United States alone, the revenue from professional sports is projected to bring in over $77 billion, which includes admission ticket costs, merchandise, media coverage access rights, and advertising. So, whether or not you enjoy watching professional sports, you probably know someone who does. Some celebrities compete to be part of half-time shows and large companies vie for commercial spots that are costly but reach a staggering number of viewers, some who only watch the half-time shows and advertisements. ### Data Analytics (Statistics) Is Crucial to Advance a Team’s Success Analyzing the vast data that today’s world has amassed to find patterns and to make predictions for future results has created a degree field for data analytics at many colleges, which is in high demand in places that might surprise you. One such place is in sports, where being able to analyze the available data on your team’s players, potential recruits, opposing team strategies, and opposing players can be paramount to your team’s success. Hollywood turned the notion of using data analytics into a major motion picture back in 2011 with the release of Moneyball, starring Brad Pitt, which grossed over $110 million. The critically acclaimed movie, based on a true story as shared in a book by Michael Lewis, follows the story of a general manager for the Oakland Athletics who used data analytics to take a team comprised of relatively unheard of players to ultimately win the American League West title in a year’s time. The win caught the eye of other team managers and owners, which started an avalanche of other teams digging into the data of players and teams. In today’s world of sports, a team has multiple positions utilizing data analytics from road scouts who evaluate a potential recruit’s skills and potential to the ultimate position of general manager who is typically the highest-paid (non-player) employee with the exception of the coaches. Being able to understand and evaluate the available data is big business and is a highly sought after skill set. In college and professional sports, it is no longer sufficient to have a strong playbook and great players. The science to winning is in understanding the math of the data and using it to propel your team to excelling. ### Single Round-Robin Tournaments A common tournament style is single round-robin tournaments (), where each team or opponent plays every other team or opponent, and the champion is determined by the team that wins the most games. Ties are possible and are resolved based on league rules. An advantage of the round-robin tournament style is that no one team has the advantage of seeding, which eliminates some teams from playing against each other based on rank of their prior performance. Rather, each team plays every other team, providing equal opportunity to triumph over each team. In this sense, round-robin tournaments are deemed the fairest tournament style. One hindrance to employing a round-robin-style tournament is the potential for the number of games involved in tournament play to determine a winner. Determining the number of games can be found easily using a formula which, as we will see, can quickly grow in the number of games required for a single round-robin tournament. As the examples show, single round-robin tournament play can quickly grow in the number of games required to determine a champion. As such, some tournaments elect to employ variations of single round-robin tournament play as well as other tournament styles such as elimination tournaments. ### Single-Elimination Tournaments When desiring a more efficient tournament style to determine a champion, one option is single-elimination tournaments (), where teams are paired up and the winner advances to the next round of play. The losing team is defeated from tournament play and does not advance in the tournament, although some leagues offer consolation matches. A single-elimination tournament offers an advantage over single round-robin tournament style of play in the number of games needed to complete the tournament. As you can see, in comparing the number of games in a single round-robin tournament in with the number of games in single-elimination tournament as shown in , the number of games required for single round-robin can quickly become unmanageable to schedule. There are modifications to both the round-robin and elimination tournament styles such as double round-robin and double-elimination tournaments. Next time you observe a college or professional sporting event, see if you can determine the tournament style of play. ### Check Your Understanding ### Key Terms 1. data analytics 2. seeding ### Key Concepts 1. Describe how a round-robin tournament is organized. 2. Compute the number of games played in a round-robin tournament. 3. Describe how a single-elimination tournament is organized. 4. Compute the number of games played in a single-elimination tournament. ### Formulas Number of games in a single round-robin tournament with teams is . Number of games in a single-elimination tournament with teams is . ### project ### Lucas Sequence and Fibonacci Sequence The Lucas numbers bear some similarity to the Fibonacci numbers and exhibit a stronger link to the golden ratio. Edouard Lucas is credited with naming the Fibonacci numbers and the Lucas numbers were so named in his honor. The Lucas numbers play a role in finding prime numbers that are utilized in encrypting data for actions such as using your debit card to obtain money at a cash machine or when making a credit card purchase for point of sale as well as when shopping online. Complete the following questions to explore numbers in the Lucas sequence as well as their relationships to the numbers in the Fibonacci sequence. ### Solar Array for a Residence One of the first steps in adding solar to a residence is determining the size of a system to achieve the desired output. In this project, we will explore the solar needs of a residence and estimate needs of a solar array to supply electrical output to meet various percentages of electrical need. Step 1: Obtain an electric bill from your apartment/home. Find the average monthly or yearly usage if listed or call the electric company to inquire. If an electric bill is not available, use the Internet to find an average monthly or yearly electric usage for your area. Step 2: Determine a daily and hourly usage. Divide the average monthly usage by 30 or the yearly average by 365. Divide again by 24 to calculate an average hourly electric usage, which will yield the average kilowatt-hours for how much electrical power your is being utilized in an hour. Step 3: Multiply your average hourly use (kilowatts) by 1,000 to convert to watts. Step 4: Use the Internet to determine the average daily peak hours of sunlight where you live. Step 5: Divide your average hourly watts (Step 3) by the average daily peak hours (Step 4) to calculate the average energy needed for a solar array to produce every hour. Step 6: Determine the average energy needed in a solar array per hour to meet each of the following: Step 7: Using the values computed in Step 6, compute the residential savings based on an average cost of 12 cents per watt. ### Vaccine Validation Validation of vaccines is a topic that exploded in the news when the Covid-19 pandemic spread across the world. As governments and organizations looked for a vaccine to curb the spread and minimize the severity of infection, concern was expressed by some for what appeared to be a quick discovery for a Covid-19 vaccine. Conduct an Internet search to explore the following questions. Pay special attention to the sources you select to ensure that they are credible sources. ### Frequency and Ultrasonic Sounds Ultrasonic sounds have been utilized for a variety of reasons, from purportedly repelling rodents and other animals as well as a variety of other applications. Using an Internet search, complete the following questions to explore some of these applications and examine the validity of various claims. ### Repelling Insects, Rodents, and Small Animals Some radio stations purport to play a high pitch sound dually with their music to aid in deterring insects and other annoying bugs to aid in providing a bug-reduced listening environment. To deter small rodents, some products claim to emit ultrasonic sounds that drive away mice and other similar pests. ### Disbursing Teenagers Some business owners and communities have turned to products such as the “mosquito” sonic deterrent device to discourage groups of teenagers from loitering around storefronts and community landmarks, citing a public nuisance issue and public safety concerns. ### Jewelry Cleaner Use of pastes and liquid chemicals to clean jewelry can be harsh on stones as well as metals. So how can we safely obtain the sparkling clean look at home that jewelry stores provide? Some would say the answer is to use an ultrasonic jewelry cleaner, but do these really work? ### Specialized Ringtones As the use of cell phones has become commonplace and families grow towards each member having their own cell phone, specialized ringtones have become popular and can aid in identifying who is calling just by the ringtone. Ever hear of ringtones that can be heard by teens but often not their teachers? The banning of cell phone use by K–12 students during class time as been implemented across a wide array of schools and some students have purportedly found ways to get around teachers hearing a cell phone ring through the use of ultrasonic ringtones. ### Streaming Services and Math With the ability to stream music virtually anywhere you are, it is not surprising that Google Play Music, Apple Music, and a slew of other companies such as Spotify, Amazon Music, YouTube Music, Sound Cloud, Pandora, Deezer Music, Tidal, Napster, and Bandcamp have invested heavily to bring streaming service to users worldwide. Streaming services have expanded to offer virtually every genre of music with vast libraries to meet diverse user requests. Considering all of the choices available for streaming music, there is a wide array of options for subscribing. Conduct an Internet research to review your current streaming choices, if any, and evaluate competitors’ products. ### Math and Baseball Baseball is known to have one of the largest pools of statistics related to the game and its players. Managers, coaches, and pitchers study the statistics of the players on opposing teams to give their team an edge by knowing what pitches to throw for the best probability to be missed by a batter. In similar fashion, batters study pitchers’ statistics to learn a pitcher’s strength and how to predict what a pitcher will throw and how to best hit against a pitcher. The three primary baseball statistics are batting average, home runs, and runs batted in (RBIs), which are the components of the title of Triple Crown winner that is awarded to players who dominate in these three areas. However, there is a wealth of other statistics to evaluate when studying the performance of a player. Conduct an Internet search to research statistics and how they are calculated in the following categories: ### Batting Statistics 1. There are about 30 batting statistics. Select a minimum of 10 batting statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: AB/HR represents at bats per home run and is calculated by the number of times a player is at bat divided by home runs. ### Pitching Statistics 1. There are about 40 pitching statistics. Select a minimum of 10 pitching statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: K/9 represents strikeouts per nine innings and is calculated by the number of strikeouts times nine divided by the number of innings pitched. ### Fielding Statistics 1. There are around 10 fielding statistics. Select a minimum of five fielding statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: FP represents fielding percentage is calculated by the number of total plays divided by the number of total chances. ### Overall 1. Select a player from recent years to evaluate. The player can be one that you have followed, one from a favorite team, or any current player. Find the statistics shared in Questions 1–3 to use in evaluating the potential strengths and weaknesses of the player. Write a short paragraph analyzing your selected player, supported by the statistics from the answers to Questions 1–3. ### Math and Fantasy Football Fantasy football offers spectators an added dimension to football season with a competitive math-based game where the active components are real-life players in the current season. For clarity in this exercise, the actual fantasy football players will be denoted as FFP and actual professional team members will be denoted as players. While some fantasy football leagues have slightly different setups or scoring systems, most share some common elements. Often using a lottery system to determine who picks first, second, and so on, FFPs select 15 current players to comprise their personal fantasy football team. The players selected can be from any professional teams and a FFP can utilize any team recognized by their league. FFPs can elect to keep the same players on their team for the whole league play or trade for any player not selected by another FFP in their league. At the start of each week during football season, each fantasy football player selects their roster of actual players to comprise their roster of starting players. Typically, a starting roster consists of the following players: As actual professional games are played, points are tallied based on your league’s scoring system. The points the team members on your starting roster make during the week are computed and whichever FFP has the highest score for the week wins that week. The FFPs with the best records of wins versus losses enters fantasy football playoffs to determine the ultimate league champion and collects the pot. The above overview of fantasy football describes the basic game play. The fun comes in understanding and analyzing the math behind the scoring. ### Math and Hockey Hockey is full of math from obvious components such as scoring and statistics to the shape of the rink and the angles involved in puck movement. Collegiate and professional hockey games are 60 minutes long and are divided into three periods of 20 (60/3) minutes each. At any one time, there are five players and one goalie on the ice for each team. If a player is called on a penalty and is placed in the penalty box, that team now has four players, which is 20% less players on the ice competing against five opponents. In some instances, a team may have two players in the penalty box at one time, resulting in three players, or 40% less players on the ice compared to a full team. Being one player down for a 2-minute penalty or potentially 5 minutes for a major penalty leads to an imbalance on the ice and calls for a quick change of offense and defense strategy. ### Rink Composition—North American An ice rink is comprised of various geometrical shapes, each with precise dimensions. ### Statistics Using an Internet search, select two top hockey players from the same league to answer the following questions: 1. Write a paragraph sharing a minimum of five statistics for each player you have selected. Describe how each of the statistics are calculated and what each statistic means. 2. Write a paragraph comparing the two players and determine who you believe is the better player. Support your choice. ### Scoring The basics of scoring in ice hockey is simple, the team with the most goals is the winner. But, how to score the most goals involves much math! ### Math and Soccer As the world’s most popular sport, you’ll be excited to confirm that soccer is full of mathematics ranging from scoring and statistics to footwork, angles, and field shape. Soccer requires understanding of mathematical concepts and equations as well as skill, fitness, and game knowledge. One such example is angles, which you all will remember from your geometry class. While players are not carrying protractors and measuring angles during play, mental calculation of angles is a constant in any successful player’s thinking. A goalie is not physically able to cover the entire open net region and a player must calculate an angle to kick the ball consistent with the net opening while predicting the ability of the goalie to stop the ball from entering the net. Using an Internet search, select two top soccer players from the same league to answer the following questions: ### Chapter Review ### Section 13.1 Math and Art ### Section 13.2 Math and the Environment ### Section 13.3 Math and Medicine ### Section 13.4 Math and Music ### Section 13.5 Math and Sports ### Chapter Test

# Introduction ## What is MATLAB? MATLAB stands for MATrix LABoratory (see wikipedia) and is a commercial software application written by The MathWorks, Inc. When you first use MATLAB, you can think of it as a glorified calculator allowing you to perform engineering calculations and plot data. However, MATLAB is more than an advanced scientific calculator, for example MATLAB's sophisticated numerical computation environment also allows us to analyze data, simulate engineering systems, document and share our code with others. ### Why Use MATLAB? MATLAB has become a defacto standard in many fields of engineering and science. Even a casual exploration of MATLAB should unveil its computational power however a closer look at MATLAB's graphics and data analysis tools as well as interaction with other applications and programing languages prove why MATLAB is a very strong application for technical computing. The standard MATLAB installation includes graphics features to visualize engineering and scientific data in 2-D and 3-D plots. We can interactivity build graphs and generate MATLAB command output that can be saved for use in the future. The saved-instructions can be called again with different data set to build new plots. The plots created with MATLAB can be exported in various file formats (e.g. .jpg, .png) to embed in Microsoft Word documents or PowerPoint slideshows. MATLAB also contains interactive tools to explore and analyze data. For example, we can visualize data with one of the many plotting routines, zoom in to plots to take measurements, perform statistical calculations, fit curves to data and evaluate the obtained expression for a desired value. MATLAB interacts with other applications (e.g. Microsoft Excel) and can be called from C code, C++ or Fortran programming language. ### Running MATLAB To use MATLAB, it must be installed on your computer and you can start it just like you start any application on your system or you must have access to a network where it is available. BCIT holds a Total Student Headcount (TSH) license for Mathworks software and this allows students to install MathWorks software on their personally-owned computers. Matlab download and installation instructions for BCIT students For your install, choose the latest version available and install MATLAB, Curve Fitting Toolbox and Symbolic Math Toolbox. In addition, MATLAB Online provides access to MATLAB from your web browser. Just log in to use MATLAB: Access MATLAB From Your Web Browser ### The MATLAB Desktop When you start the MATLAB program, it displays the MATLAB desktop. The desktop is a set of tools (graphical user interfaces or GUIs) for managing files, variables, and applications associated with MATLAB. The first time you start MATLAB, the desktop appears with the default layout, as shown in the following illustration. ### Command Window The Command Window is where we execute MATLAB commands. We enter statements at the Command Window prompt. The prompt can be any one of the following: 1. Trial>> indicates that the Command Window is in normal mode and the MATLAB license will expire after the trial period ends. 2. EDU>> indicates that the Command Window is in normal mode, in MATLAB Student Version. 3. >> indicates that the Command Window is in normal mode. ### Command History The Command History is a log of the commands we have executed in the command window. ### Workspace The workspace consists of a set of variables stored in memory during a MATLAB session. To open the Workspace browser, select Desktop > Workspace in the MATLAB desktop, or type at the Command Window prompt. ### Current Folder The Current Folder is like the Finder in Mac OS X or Windows Explorer in Windows operating systems and allows us to browse through the files and folders. The Current Folder also displays details about files in your current directory and within the hierarchy of the folders it contains. ### Tool Strip The tool strip contains global tabs, Home, Plots and Apps. Contextual tabs become available when you need them. The plots tab allows us to plot various types of graphs quickly and easily. The apps tab gives quick access to interactive applications within MATLAB environment. Layout button allows us to change the desktop layout or go back to the default configuration. ### Toolbar The MATLAB toolbar provides on-screen buttons to access frequently used features such as, copy, paste, undo and redo. ### Keyboard shortcuts MATLAB provides keyboard shortcuts for viewing a history of commands and listing contextual help. ### The Up Arrow Key Suppose we want to enter the following equation: But we mistakenly entered MATLAB returns the following prompt: Instead of retyping the equation, press the up arrow key, the mistakenly entered line is displayed. Using the left arrow key, move the cursor to the misspelled letter. Make the correction and press Return or Enter to execute the command. Pressing the up arrow key repeatedly recalls the previously entered commands. Likewise, typing the first characters of previously entered line and pressing the up arrow key displays the full command line. To execute that line, simply press the Return or Enter key. ### The Tab Key Suppose you forgot how to enter the square root command. Begin typing y=sq in the command prompt: Then press the tab key and scroll down to sqrt. Select it and press Return or Enter key. ### The Semicolon Symbol The semicolon symbol at the end of a line suppresses the screen output. This is useful when you want to keep your command window clean. Type the following entry and press the Return key: The following output is displayed: Now, press the up arrow key to recall our initial entry And insert a semicolon as follows: No numerical result is displayed however MATLAB stores the value of y in the memory. We can recall the value y by simply typing y and pressing Return. ### MATLAB Help MATLAB comes with three forms of online help: help, doc and demos. ### Help Typing help in the Command Window lists all primary help topics. You can display a topic by clicking on the link. Or if you know the command or function you need help with, you can type help followed by the command or function. For example to learn about clc command, type help clc at the command prompt: Also try the following command: >> help clear To learn about sine function, type help sin at the command prompt: ### Doc Obviously, to use help effectively, you need to know what you are looking for. Often times, especially when you first start learning an application, it is usually difficult to ask the right questions. In the case of MATLAB, doc command is generally better than help. If you type doc in the command prompt, MATLAB opens a browser from where you can obtain help easier: Like using help sin, try typing doc sin in the command prompt: ### Demos You can learn more about MATLAB through demos by typing demo in the command prompt, a list of links to demos will open in Help Browser. Demos and online seminars are available at product demos and online seminars. ### Useful Commands and Functions For a detailed explanation and examples for each of the following type ‘help function’ (without quotes) at the MATLAB prompt. ### Summary of Key Points 1. MATLAB is a popular technical computing application and MathWorks offers a trial version of MATLAB on their website, 2. The MATLAB Desktop consists of Command Window, Command History, Workspace, Current Folder and Start Button, 3. The up/down arrow keys, the tab key and the semicolon are convenient tools to use the Command Window, 4. MATLAB features an online help, doc and demo, 5. Various commands and functions make MATLAB experience easier, for example, clc, clear and exit.

# Getting Started ## MATLAB Essentials Learning a new skill, especially a computer program in this case, can be overwhelming. However, if we build on what we already know, the process can be handled rather effectively. In the preceding chapter we learned about MATLAB Graphical User Interface (GUI) and how to get help. Knowing the GUI, we will use basic math skills in MATLAB to solve linear equations and find roots of polynomials in this chapter. ### Basic Computation ### Mathematical Operators The evaluation of expressions is accomplished with arithmetic operators as we use them in scientific calculators. Note the addtional operators shown in the table below: ### Operator Precedence MATLAB allows us to build mathematical expressions with any combination of arithmetic operators. The order of operations are set by precedence levels in which MATLAB evaluates an expression from left to right. The precedence rules for MATLAB operators are shown in the list below from the highest precedence level to the lowest. 1. Parentheses () 2. Power (^) 3. Multiplication (*), right division (/), left division (\) 4. Addition (+), subtraction (-) ### Mathematical Functions MATLAB has all of the usual mathematical functions found on a scientific calculator including square root, logarithm, and sine. Practice the following examples to familiarize yourself with the common mathematical functions. Be sure to read the relevant help and doc pages for functions that are not self explanatory. ### The format Function The format function is used to control how the numeric values are displayed in the Command Window. The short format is set by default and the numerical results are displayed with 4 digits after the decimal point (see the examples above). The long format produces 15 digits after the decimal point. ### Variables In MATLAB, a named value is called a variable. MATLAB comes with several predefined variables. For example, the name pi refers to the mathematical quantity π, which is approximately pi ans = 3.1416 ### Declaring Variables Variables in MATLAB are generally represented as matrix quantities. Scalars and vectors are special cases of matrices having size 1x1 (scalar), 1xn (row vector) or nx1 (column vector). ### Declaration of a Scalar The term scalar as used in linear algebra refers to a real number. Assignment of scalars in MATLAB is easy, type in the variable name followed by = symbol and a number: ### Declaration of a Row Vector Elements of a row vector are separated with blanks or commas. ### Declaration of a Column Vector Elements of a column vector is ended by a semicolon: ### Declaration of a Matrix Matrices are typed in rows first and separated by semicolons to create columns. Consider the examples below: ### Linear Equations Systems of linear equations are very important in engineering studies. In the course of solving a problem, we often reduce the problem to simultaneous equations from which the results are obtained. As you learned earlier, MATLAB stands for Matrix Laboratory and has features to handle matrices. Using the coefficients of simultaneous linear equations, a matrix can be formed to solve a set of simultaneous equations. ### Polynomials In the preceding section, we briefly learned about how to use MATLAB to solve linear equations. Equally important in engineering problem solving is the application of polynomials. Polynomials are functions that are built by simply adding together (or subtracting) some power functions. (see Wikipedia). The coeffcients of a polynominal are entered as a row vector beginning with the highest power and including the ones that are equal to 0. ### The polyval Function We can evaluate a polynomial p for a given value of x using the syntax polyval(p,x) where p contains the coefficients of polynomial and x is the given number. ### The roots Function Consider the following equation: Probably you have solved this type of equations numerous times. In MATLAB, we can use the roots function to find the roots very easily. ### Splitting a Statement You will soon find out that typing long statements in the Command Window or in the the Text Editor makes it very hard to read and maintain your code. To split a long statement over multiple lines simply enter three periods "..." at the end of the line and carry on with your statement on the next line. ### Comments Comments are used to make scripts more "readable". The percent symbol % separates the comments from the code. Examine the following examples: ### Basic Operations ### Special Characters ### Summary of Key Points 1. MATLAB has the common functions found on a scientific calculator and can be operated in a similar way, 2. MATLAB can store values in variables. Variables are case sensitive and some variables are reserved by MATLAB (e.g. pi stores 3.1416), 3. Variable Editor can be used to enter or manipulate matrices, 4. The coefficients of simultaneous linear equations and polynomials are used to form a row vector. MATLAB then can be used to solve the equations, 5. The format function is used to control the number of digits displayed, 6. Three periods "..." at the end of the line is used to split a long statement over multiple lines, 7. The percent symbol % separates the comments from the code, anything following % symbol is ignored by MATLAB.

# Graphics ## Graphing with MATLAB A picture is worth a thousand words, particularly visual representation of data in engineering is very useful. MATLAB has powerful graphics tools and there is a very helpful section devoted to graphics in MATLAB Help: Graphics. Students are encouraged to study that section; what follows is a brief summary of the main plotting features. ### Two-Dimensional Plots ### The plot Statement Probably the most common method for creating a plot is by issuing plot(x, y) statement where function y is plotted against x. Having variables assigned in the Workspace, x and y=sin(x) in our case, we can also select x and y, and right click on the selected variables. This opens a menu from which we choose plot(x,y). See the figure below. ### Annotating Plots Graphs without labels are incomplete and labeling elements such as plot title, labels for x and y axes, and legend should be included. Using up arrow, recall the statement above and add the annotation commands as shown below. Run the file and compare your result with the first one. ### Superimposed Plots If you want to merge data from two graphs, rather than create a new graph from scratch, you can superimpose the two using a simple trick: ### Multiple Plots in a Figure Multiple plots in a single figure can be generated with subplot in the Command Window. However, this time we will use the built-in Plot Tools. Before we initialize that tool set, let us create the necessary variables using the following script: Note that the above script clears the command window and variable workspace. It also closes any open Figures. After running the script, we will have X1, Y1, Y2, Y3 and Y4 loaded in the workspace. Next, select File > New > Figure, a new Figure window will open. Click "Show Plot Tools and Dock Figure" on the tool bar. Under New Subplots > 2D Axes, select four vertical boxes that will create four subplots in one figure. Also notice, the five variables we created earlier are listed under Variables. After the subplots have been created, select the first supblot and click on "Add Data". In the dialog box, set X Data Source to X1 and Y Data Source to Y1. Repeat this step for the remaining subplots paying attention to Y Data Source (Y2, Y3 and Y4 need to be selected in the subsequent steps while X1 is always the X Data Source). Next, select the first item in "Plot Browser" and activate the "Property Editor". Fill out the fields as shown in the figure below. Repeat this step for all subplots. Save the figure as sinxcosx.fig in the current directory. ### Three-Dimensional Plots 3D plots can be generated from the Command Window as well as by GUI alternatives. This time, we will go back to the Command Window. ### The plot3 Statement With the X1,Y1,Y2 and Y2 variables still in the workspace, type in plot3(X1,Y1,Y2) at the MATLAB prompt. A figure will be generated, click "Show Plot Tools and Dock Figure". Use the property editor to make the following changes. The final result should look like this: Use help or doc commands to learn more about 3D plots, for example, image(x), surf(x) and mesh(x). ### Quiver or Velocity Plots To plot vectors, it is useful to draw arrows so that the direction of the arrow points the direction of the vector and the length of the arrow is vector’s magnitude. However the standard plot function is not suitable for this purpose. Fortunately, MATLAB has quiver function appropriately named to plot arrows. quiver(x,y,u,v) plots vectors as arrows at the coordinates (x,y) with components (u,v). The matrices x, y, u, and v must all be the same size and contain corresponding position and velocity components. ### Summary of Key Points 1. plot(x, y) and plot3(X1,Y1,Y2) statements create 2- and 3-D graphs respectively, 2. Plots at minimum should contain the following elements: title, xlabel, ylabel and legend, 3. Annotated plots can be easily generated with GUI Plot Tools, 4. quiver and quiver3 plots are useful for making vector diagrams.

# Introductory Programming ## Introductory Programming with MATLAB MATLAB provides scripting and automation tools that can simplify repetitive computational tasks. For example, a series of commands executed in a MATLAB session to solve a problem can be saved in a script file called an m-file. An m-file can be executed from the command line by typing the name of the file or by pressing the run button in the built-in text editor tool bar. ### Script Files A script is a file containing a sequence of MATLAB statements. Script files have a filename extension of .m. By typing the filename at the command prompt, we can run the script and obtain results in the command window. A sample m-file named ThermalConductivity.m is displayed in Text Editor below. Note the triangle (in green) run button in the tool bar, pressing this button executes the script in the command window. Now let us see how an m-file is created and executed. ### The input Function Notice that the script we have created above is not interactive and computes the total volume only for the variables defined in the m-file. To make this script interactive we will make some changes to the existing AcetyleneBottle.m by adding input function and save it as AcetyleneBottleInteractive.m. The syntax for input is as follows: ### The disp Function As you might have noticed, the output of our script is not displayed in a well-formatted fashion. Using disp, we can control how text or arrays are displayed in the command window. For example, to display a text string on the screen, type in disp('Hello world!'). This command will return our friendly greeting as follows: Hello world! disp(variable) can be used to display only the value of a variable. To demonstrate this, issue the following command in the command window: We have created a row vector with 5 elements. The following is displayed in the command window: Now if we type in disp(b) and press enter, the variable name will not be displayed but its value will be printed on the screen: The following example demonstrates the usage of disp function. ### The num2str Function The num2str function allows us to convert a number to a text string. Basic syntax is str = num2str(A) where variable A is converted to a text and stored in str. Let's see how it works in AcetyleneBottleInteractiveDisp.m. Remember to save the file with a different name before editing it, for example, AcetyleneBottleInteractiveDisp1.m. ### The fopen and fclose Functions The first command is used to open or create a file. The basic syntax for fopen is as follows: For example, fo = fopen('output.txt', 'w'); opens or creates a new file named output.txt and sets the permission for writing. If the file already exists, it discards the existing contents. fclose command is used to close a file. For example, if we type in fclose(fo);, we close the file that was created above. ### The fprintf Function fprintf function writes formatted data to the computer monitor or a file. This command can be used to save the results of a calculation to a file. To do this, first we create or open an output file with fopen, second we issue the fprintf command and then we close the output file with fclose. The simplified syntax for fprintf is as follows: Upon running the file, the output.txt file will display the following: ### Loops In programming, a loop executes a set of code a specified number of times or until a condition is met. ### For Loop This loop iterates an index variable from an initial value using a specified increment to a final value and runs a set of code. The for loop syntax is the following: for loop_index=vector_statement code ... code end ### While Loop Like the for loop, a while loop executes blocks of code over and over again however it runs as long as the test condition remains true. The syntax of a while loop is while test_condition code ... code end ### The diary Function Instead of writing a script from scratch, we sometimes solve problems in the Command Window as if we are using a scientific calculator. The steps we perform in this fashion can be used to create an m-file. For example, the diary function allows us to record a MATLAB session in a file and retrieve it for review. Reviewing the file and by copying relevant parts of it and pasting them in to an m-file, a script can be written easily. Typing diary at the MATLAB prompt toggles the diary mode on and off. As soon as the diary mode is turned on, a file called diary is created in the current directory. If you like to save that file with a specific name, say for example problem16, type >> diary problem16.txt. A file named problem16.txt will be created. The following is the content of a diary file called problem16.txt. Notice that in that session, the user is executing the four files we created earlier. The user's keyboard input and the resulting display output is recorded in the file. The session is ended by typing diary which is printed in the last line. This might be useful to create a record of your work to hand in with a lab or to create the beginnings of an m-file. ### Style Guidelines Try to apply the following guidelines when writing your scripts: 1. Share your code or programs with others, consider adopting one of Creative Commons or GNU General Public License schemes 2. Include your name and contact info in the opening lines 3. Use comments liberally 4. Group your code and use proper indentation 5. Use white space liberally 6. Use descriptive names for your variables 7. Use descriptive names for your m-files ### Summary of Key Points 1. A script is a file containing a sequence of MATLAB statements. Script files have a filename extension of .m. 2. Functions such as input, disp and num2str can be used to make scripts interactive, 3. fopen, fprintf and fclose functions are used to create output files, 4. A for loop is used to repeat a specific block of code a definite number of times. 5. A while loop is used to repeat a specific block of code an indefinite number of times, until a condition is met. 6. The diary function is useful to record a MATLAB command window session from which an m-file can be easily created, 7. Various style guidelines covered here help improve our code.

# Interpolation ## Interpolation with MATLAB Linear interpolation is one of the most common techniques for estimating values between two given data points. For example, when using steam tables, we often have to carry out interpolations. With this technique, we assume that the function between the two points is linear. MATLAB has a built-in interpolation function. ### The interp1 Function Give an x-y table, y_new = interp1(x,y,x_new) interpolates to find y_new. Consider the following examples: ### Summary of Key Points 1. Linear interpolation is a technique for estimating values between two given data points, 2. Problems involving steam tables often require interpolated data, 3. MATLAB has a built-in interpolation function.

# Numerical Integration ## Numerical Integration with MATLAB This chapter essentially deals with the problem of computing the area under a curve. First, we will employ a basic approach and form trapezoids under a curve. From these trapezoids, we can calculate the total area under a given curve. This method can be tedious and is prone to errors, so in the second half of the chapter, we will utilize a built-in MATLAB function to carry out numerical integration. ### A Basic Approach There are various methods to calculating the area under a curve, for example, Rectangle Method, Trapezoidal Rule and Simpson's Rule. The following procedure is a simplified method. Consider the curve below: Each segment under the curve can be calculated as follows: Therefore, if we take the sum of the area of each trapezoid, given the limits, we calculate the total area under a curve. Consider the following example. ### The Trapezoidal Rule Sometimes it is rather convenient to use a numerical approach to solve a definite integral. The trapezoid rule allows us to approximate a definite integral using trapezoids. ### The trapz Command Z = trapz(Y) computes an approximation of the integral of Y using the trapezoidal method. Now, let us see a typical problem. ### The integral Function As we have seen earlier, trapz gives a good approximation for definite integrals. The integral function streamlines numerical integration even further. Before we learn about integral function, first we will look at anonymous functions. ### Anonymous Functions An anonymous function is a function that can be defined in the command window (i.e. it does not need to be stored in a program file). Anonymous functions can accept inputs and return outputs, just as standard functions do such as sqrt(X) or log(X). To define an anonymous function, first we create a handle with @(x) and type in the function: myfunction=@(x) x^2+1. If you want to evaluate myfunction at 1, just type in a=myfunction(1) at the command window and you get the result of 2. Syntax for To evaluate an integral from a minimum to a maximum value, we specify a function and its minimum and maximum Z = integral(fun,xmin,xmax). ### Summary of Key Points 1. In its simplest form, numerical integration involves calculating the areas of segments that make up the area under a curve, 2. MATLAB has built-in functions to perform numerical integration, 3. Z = trapz(Y) computes an approximation of the integral of Y using the trapezoidal method. 4. Anonymous functions are inline statements that we can define with @(x), 5. Z = integral(fun,xmin,xmax) numerically integrates function fun from xmin to xmax.

# Regression Analysis ## Regression Analysis with MATLAB ### What is Regression Analysis? Suppose we calculate some variable of interest, y, as a function of some other variable x. We call y the dependent variable and x the independent variable. For example, consider the data set below, taken from a simple experiment involving a vehicle, its velocity versus time is tabulated. In this case, velocity is a function of time, thus velocity is the dependent variable and the time is the independent variable. In its simplest form regression analysis involves fitting the best straight line relationship to explain how the variation in a dependent variable, y, depends on the variation in an independent variable, x. In our example above, once the relationship (in this case a linear relationship) has been estimated we can produce a linear equation in the following form: And once an analytic equation such as the one above has been determined, dependent variables at intermediate independent values can be computed. ### Performing Linear Regression Regression analysis with MATLAB is easy. The MATLAB Basic Fitting GUI allows us to interactively to do "curve fitting" which is a method to arrive at the best "straight line" fit for linear equations or the best curve fit for a polynomial up to the tenth degree. The procedure to perform a curve fitting with MATLAB is as follows: 1. Input the variables, 2. Plot the data, 3. Initialize the Basic Fitting GUI, 4. Select the desired regression analysis parameters. Now let us do another curve fitting and obtain an equation for the function. Using that equation, we can evaluate the function at a desired value with polyval. ### Summary of Key Points 1. Linear regression involves fitting the best straight line relationship to explain how the variation in a dependent variable, y, depends on the variation in an independent variable, x, 2. Basic Fitting GUI allows us to interactively perform curve fitting, 3. Some of the plot fits available are linear, quadratic and cubic functions, 4. Basic Fitting GUI can evaluate functions at given points.

# Publishing with MATLAB ## Publishing with MATLAB MATLAB includes an automatic report generator called publisher. The publisher publishes a script in several formats, including HTML, XML, MS Word and PowerPoint. The published file can contain the following: 1. Commentary on the code, 2. MATLAB code, 3. Results of the executed code, including the Command Window output and figures created by the code. ### The publish Function The most basic syntax is publish('file','format') where the m-file is called and executed line by line then saved to a file in specified format. All published files are placed in the html directory although the published output might be a doc file. ### Publishing with Editor The publisher is easily accessible from the Tool Strip: ### The Double Percentage %% Sign The scripts sometimes can be very long and their readability might be reduced. To improve the publishing result, sections are introduced by adding descriptive lines to the script preceded by %%. Consider the following example. ### Summary of Key Points 1. MATLAB can generate reports containing commentary on the code, MATLAB code and the results of the executed code, 2. The publisher generates a script in several formats, including HTML, XML, MS Word and PowerPoint. 3. The Double Percentage %% can be used to creates hyper-linked sections.

# Sampling and Data ## Introduction You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or fact. Statistical methods can help you make the best educated guess. Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics. Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what good data can be distinguished from bad.

# Sampling and Data ## Definitions of Statistics, Probability, and Key Terms ### The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives. In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers, for example, finding an average. After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from good data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct. Effective interpretation of data, or inference, is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life. ### Statistical Models Statistics, like all other branches of mathematics, uses mathematical models to describe phenomena that occur in the real world. Some mathematical models are deterministic. These models can be used when one value is precisely determined from another value. Examples of deterministic models are the quadratic equations that describe the acceleration of a car from rest or the differential equations that describe the transfer of heat from a stove to a pot. These models are quite accurate and can be used to answer questions and make predictions with a high degree of precision. Space agencies, for example, use deterministic models to predict the exact amount of thrust that a rocket needs to break away from Earth’s gravity and achieve orbit. However, life is not always precise. While scientists can predict to the minute the time that the sun will rise, they cannot say precisely where a hurricane will make landfall. Statistical models can be used to predict life’s more uncertain situations. These special forms of mathematical models or functions are based on the idea that one value affects another value. Some statistical models are mathematical functions that are more precise—one set of values can predict or determine another set of values. Or some statistical models are mathematical functions in which a set of values do not precisely determine other values. Statistical models are very useful because they can describe the probability or likelihood of an event occurring and provide alternative outcomes if the event does not occur. For example, weather forecasts are examples of statistical models. Meteorologists cannot predict tomorrow’s weather with certainty. However, they often use statistical models to tell you how likely it is to rain at any given time, and you can prepare yourself based on this probability. ### Probability Probability is a mathematical tool used to study randomness. It deals with the chance of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is or .5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction is equal to .498 which is very close to .5, the expected probability. The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. ### Key Terms In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion, or subset, of the larger population and study that portion—the sample—to gain information about the population. Data are the result of sampling from a population. Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16-ounce can contains 16 ounces of carbonated drink. From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class as a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. Since we do not have the data for all math classes, that statistic is our best estimate of the average for the entire population of math classes. If we happen to have data for all math classes, we can find the population parameter. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter. One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. In order to have an accurate sample, it must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter. A variable, usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population. Variables may describe values like weight in pounds or favorite subject in school. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X—calculate the average number of points earned, for example—but it makes no sense to do math with values of Y—calculating an average party affiliation makes no sense. Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value. Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three. Your mean score would be 84.3 to one decimal place. If, in your math class, there are 40 students and 22 are males and 18 females, then the proportion of men students is and the proportion of women students is . Mean and proportion are discussed in more detail in later chapters. ### References The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html. ### Chapter Review The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. ### Practice Use the following information to answer the next five exercises. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once they start the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected. Researcher A Researcher B Determine what the key terms refer to in the example for Researcher A. ### HOMEWORK For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.

# Sampling and Data ## Data, Sampling, and Variation in Data and Sampling Data may come from a population or from a sample. Lowercase letters like or generally are used to represent data values. Most data can be put into the following categories: 1. Qualitative 2. Quantitative Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O–, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type. Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous. All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three. Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data. Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data. ### Qualitative Data Discussion Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts, frequencies, and percentages or proportions, relative frequencies. For instance, to calculate the percentage of part time students at De Anza College, divide 9,200/22,496 to get .4089. Round to the nearest thousandth—third decimal place and then multiply by 100 to get the percentage, which is 40.9 percent. So, the percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College. Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. Two graphs that are used to display qualitative data are pie charts and bar graphs. In a pie chart, categories of data are shown by wedges in a circle that represent the percent of individuals/items in each category. We use pie charts when we want to show parts of a whole. In a bar graph, the length of the bar for each category represents the number or percent of individuals in each category. Bars may be vertical or horizontal. We use bar graphs when we want to compare categories or show changes over time. A Pareto chart consists of bars that are sorted into order by category size (largest to smallest). Look at and and determine which graph (pie or bar) you think displays the comparisons better. It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the best graph depending on the data and the context. Our choice also depends on what we are using the data for. ### Percentages That Add to More (or Less) Than 100 Percent Sometimes percentages add up to be more than 100 percent (or less than 100 percent). In the graph, the percentages add to more than 100 percent because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100 percent. ### Omitting Categories/Missing Data The table displays Ethnicity of Students but is missing the Other/Unknown category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart. The following graph is the same as the previous graph but the Other/Unknown percent (9.6 percent) has been included. The Other/Unknown category is large compared to some of the other categories (Native American, .6 percent, Pacific Islander 1.0 percent). This is important to know when we think about what the data are telling us. This particular bar graph in can be difficult to understand visually. The graph in is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret. ### Pie Charts: No Missing Data The following pie charts have the Other/Unknown category included since the percentages must add to 100 percent. The chart in b is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in a. ### Marginal Distributions in Two-Way Tables Below is a two-way table, also called a contingency table, showing the favorite sports for 50 adults: 20 women and 30 men. This is a two-way table because it displays information about two categorical variables, in this case, gender and sports. Data of this type (two variable data) are referred to as bivariate data. Because the data represent a count, or tally, of choices, it is a two-way frequency table. The entries in the total row and the total column represent marginal frequencies or marginal distributions. Note—The term marginal distributions gets its name from the fact that the distributions are found in the margins of frequency distribution tables. Marginal distributions may be given as a fraction or decimal: For example, the total for men could be given as .6 or 3/5 since Marginal distributions require bivariate data and only focus on one of the variables represented in the table. In other words, the reason 20 is a marginal frequency in this two-way table is because it represents the margin or portion of the total population that is women (20/50). The reason 25 is a marginal frequency is because it represents the portion of those sampled who favor football (25/50). Note: The values that make up the body of the table (e.g., 20, 8, 2) are called joint frequencies. ### Conditional Distributions in Two-Way Tables The distinction between a marginal distribution and a conditional distribution is that the focus is on only a particular subset of the population (not the entire population). For example, in the table, if we focused only on the subpopulation of women who prefer football, then we could calculate the conditional distributions as shown in the two-way table below. To find the first sub-population of women who prefer football, read the value at the intersection of the Women row and Football column which is 5. Then, divide this by the total population of football players which is 25. So, the subpopulation of football players who are women is 5/25 which is .2. Similarly, to find the subpopulation of women who play football, use the value of 5 which is the number of women who play football. Then, divide this by the total population of women which is 20. So, the subpopulation of women who play football is 5/20 which is .25. ### Presenting Data After deciding which graph best represents your data, you may need to present your statistical data to a class or other group in an oral report or multimedia presentation. When giving an oral presentation, you must be prepared to explain exactly how you collected or calculated the data, as well as why you chose the categories, scales, and types of graphs that you are showing. Although you may have made numerous graphs of your data, be sure to use only those that actually demonstrate the stated intentions of your statistical study. While preparing your presentation, be sure that all colors, text, and scales are visible to the entire audience. Finally, make sure to allow time for your audience to ask questions and be prepared to answer them. ### Sampling Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. In a simple random sample, each group has the same chance of being selected. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in . Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. The most common random number generators are five digit numbers where each digit is a unique number from 0 to 9. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows: .94360, .99832, .14669, .51470, .40581, .73381, .04399. Lisa reads two-digit groups until she has chosen three class members (That is, she reads .94360 as the groups 94, 43, 36, 60.) Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers. The table below shows how Lisa reads two-digit numbers form each random number. Each two-digit number in the table would represent each student in the roster above in . The random numbers .94360 and .99832 do not contain appropriate two digit numbers. However the third random number, .14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Lundquist, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Lundquist, Cuningham, and Cuarismo. Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample. To choose a stratified sample, divide the population into groups called strata and then the sample is selected by picking the same number of values from each strata until the desired sample size is reached. For example, you could stratify (group) your high school student population by year (freshmen, sophomore, juniors, and seniors) and then choose a proportionate simple random sample from each stratum (each year) to get a stratified random sample. To choose a simple random sample from each year, number each student of the first year, number each student of the second year, and do the same for the remaining years. Then use simple random sampling to choose proportionate numbers of students from the first year and do the same for each of the remaining years. Those numbers picked from the first year, picked from the second year, and so on represent the students who make up the stratified sample. To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four homeroom classes from your student population, the four classes make up the cluster sample. Each class is a cluster. Number each cluster, and then choose four different numbers using random sampling. All the students of the four classes with those numbers are the cluster sample. So, unlike a stratified example, a cluster sample may not contain an equal number of randomly chosen students from each class. To choose a systematic sample, establish and follow a rule. The most common way to select a systematic sample is to list the members of the population and choose every entry from a random starting point. For example, suppose you have 100,000 individuals in your population and you want to choose a sample of 1,000. Use a random number generator to select your starting point. Now, 100,000/1,000 = 100, so to ensure coverage throughout the list, choose every entry in the list. When you reach the end of the list, continue the count from the beginning until you have selected the complete sample. A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others. Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased. They may favor a certain group. It is better for the person conducting the survey to select the sample respondents. When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error. In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error. In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others. Remember, each member of the population should have an equally likely chance of being chosen. When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied. For instance, if a survey of all students is conducted only during noon lunchtime hours is biased. This is because the students who do not have a noon lunchtime would not be included. ### Critical Evaluation We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include the following: 1. Problems with samples: —A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not reliable. Reliability in statistical measures must also be considered when analyzing data. Reliability refers to the consistency of a measure. A measure is reliable when the same results are produced given the same circumstances. 2. Self-selected samples—Responses only by people who choose to respond, such as internet surveys, are often unreliable. 3. Sample size issues—: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples include crash testing cars or medical testing for rare conditions. 4. Undue influence—: collecting data or asking questions in a way that influences the response. 5. Non-response or refusal of subject to participate: —The collected responses may no longer be representative of the population.  Often, people with strong positive or negative opinions may answer surveys, which can affect the results. 6. Causality: —A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable. 7. Self-funded or self-interest studies—: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done. 8. Misleading use of data—: These can be improperly displayed graphs, incomplete data, or lack of context. If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural. ### Variation in Data Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage: 15.8, 16.1, 15.2, 14.8, 15.8, 15.9, 16.0, 15.5. Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range. Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy. ### Variation in Samples It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their high school sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however. Think about what contributes to making Doreen’s and Jung’s samples different. If Doreen and Jung took larger samples, that is, the number of data values is increased, their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This is called sampling variability. In other words, it refers to how much a statistic varies from sample to sample within a population. The larger the sample size, the smaller the variability between samples will be. So, the large sample size makes for a better, more reliable statistic. ### Size of a Sample The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200–1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals. Be aware that many large samples are biased. For example, internet surveys are invariably biased, because people choose to respond or not. ### References Gallup. Retrieved from http://www.well-beingindex.com/. Gallup. Retrieved from http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4. Gallup. Retrieved from http://www.gallup.com/175196/gallup-healthways-index-methodology.aspx. Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President. LBCC Distance Learning (DL) Program. Retrieved from http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus. Lusinchi, D. (2012). “President” Landon and the 1936 Literary Digest poll: Were automobile and telephone owners to blame? Social Science History 36(1), 23-54. Retrieved from https://muse.jhu.edu/article/471582/pdf. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html. The Mercury News. Retrieved from http://www.mercurynews.com/. Virtual Laboratories in Probability and Statistics. Retrieved from http://www.math.uah.edu/stat/data/LiteraryDigest.html. The Mercury News. Retrieved from http://www.mercurynews.com/. ### Chapter Review Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete. Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data. Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted. ### Practice Use the following information to answer the next four exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Antonio, Texas. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed. For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). Use the following information to answer the next seven exercises: Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected: Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data: Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning, and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in . The second study collected the data in . ### HOMEWORK For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data. Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed. ### Bringing It Together

# Sampling and Data ## Frequency, Frequency Tables, and Levels of Measurement Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible. ### Answers and Rounding Off A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. Expect that some of your answers will vary from the text due to rounding errors. It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it is more helpful to leave an answer as an unreduced fraction. ### Levels of Measurement The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are as follows (from lowest to highest level): 1. Nominal scale level 2. Ordinal scale level 3. Interval scale level 4. Ratio scale level Data that is measured using a nominal scale is qualitative (categorical). Categories, colors, names, labels, and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful. Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, but there is no agreed upon order of these brands, even though people may have personal preferences. Nominal scale data cannot be used in calculations. Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data. Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are excellent, good, satisfactory, and unsatisfactory. These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations. Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point. Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like –10 °F and –15 °C exist and are colder than 0. Interval level data can be used in calculations, but one type of comparison cannot be done. 80 °C is not four times as hot as 20 °C (nor is 80 °F four times as hot as 20 °F). There is no meaning to the ratio of 80 to 20 (or four to one). Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded. The data can be put in order from lowest to highest 20, 68, 80, 92. The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20. ### Frequency Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3. lists the different data values in ascending order and their frequencies. A frequency is the number of times a value of the data occurs. According to , there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample. A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample, in this case, 20. Relative frequencies can be written as fractions, percents, or decimals. The sum of the values in the relative frequency column of is , or 1. Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in . In the first row, the cumulative frequency is simply .15 because it is the only one. In the second row, the relative frequency was .25, so adding that to .15, we get a relative frequency of .40. Continue adding the relative frequencies in each row to get the rest of the column. The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated. The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured. 60, 60.5, 61, 61, 61.5, 63.5, 63.5, 63.5, 64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5, 70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71, 72, 72, 72, 72.5, 72.5, 73, 73.5, 74 summarizes the heights in this sample. Since heights are expressed in tenths, the frequency table will use labels measured in hundredths. This ensures that no data value will coincide with the upper or lower limit of an interval. The data in this table have been grouped into the following intervals: 1. 59.95–61.95 inches 2. 61.95–63.95 inches 3. 63.95–65.95 inches 4. 65.95–67.95 inches 5. 67.95–69.95 inches 6. 69.95–71.95 inches 7. 71.95–73.95 inches 8. 73.95–75.95 inches In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints. ### References Levels of Measurement. Retrieved from http://cnx.org/content/m10809/latest/. National Hurricane Center. Retrieved from http://www.nhc.noaa.gov/gifs/table5.gif. ThoughtCo. Retrieved from https://www.thoughtco.com/levels-of-measurement-in-statistics-3126349. U.S. Census Bureau. Retrieved from https://www.census.gov/quickfacts/table/PST045216/00. Levels of measurement. Retrieved from https://www.cos.edu/Faculty/georgew/Tutorial/Data_Levels.htm. ### Chapter Review Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth. Expect that some of your answers will vary from the text due to rounding errors. In addition to rounding your answers, you can measure your data using the following four levels of measurement: 1. Nominal scale level data that cannot be ordered nor can it be used in calculations 2. Ordinal scale level data that can be ordered; the differences cannot be measured 3. Interval scale level data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio 4. Ratio scale level data with a starting point that can be ordered; the differences have meaning and ratios can be calculated When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these. ### HOMEWORK Use the following information to answer the next two exercises: contains data on hurricanes that have made direct hits on the United States. Between 1851-2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

# Sampling and Data ## Experimental Design and Ethics Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as speeding? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data. The purpose of an experiment is to investigate the relationship between two variables. In an experiment, there is the explanatory variable which affects the response variable. In a randomized experiment, the researcher manipulates the explanatory variable and then observes the response variable. Each value of the explanatory variable used in an experiment is called a treatment. You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention. Additional variables that can cloud a study are called lurking variables. In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments randomly, all of the potential lurking variables are spread equally among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables. Confounding occurs when the effects of multiple factors on a response cannot be separated, for instance, if a student guesses on the even-numbered questions on an exam and sits in a favorite spot on exam day. Why does the student get a high test scores on the exam? It could be the increased study time or sitting in the favorite spot or both. Confounding makes it difficult to draw valid conclusions about the effect of each factor on the outcome. The way around this is to test several outcomes with one method (treatment). This way, we know which treatment really works. The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing substances, researchers noted the following: Results showed that believing one had taken the substance resulted in [performance] times almost as fast as those associated with consuming the substance itself. In contrast, taking the substance without knowledge yielded no significant performance increment.McClung, M. and Collins, D. (2007 June). "Because I know it will!" Placebo effects of an ergogenic aid on athletic performance. When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group. This group is given a placebo treatment, a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill that contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment designed to reduce bias by hiding information. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded. Sometimes, it is neither possible nor ethical for researchers to conduct experimental studies. For example, if you want to investigate whether malnutrition affects elementary school performance in children, it would not be appropriate to assign an experimental group to be malnourished. In these cases, observational studies or surveys may be used. In an observational study, the researcher does not directly manipulate the independent variable. Instead, he or she takes recordings and measurements of naturally occurring phenomena. By sorting these data into control and experimental conditions, the relationship between the dependent and independent variables can be drawn. In a survey, a researcher’s measurements consist of questionnaires that are answered by the research participants. ### Ethics The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do. A recent investigation of famous social psychologist, Diederik Stapel, has led to the retraction of his articles from some of the world’s top journals including, Journal of Experimental Social Psychology, Social Psychology, Basic and Applied Social Psychology, British Journal of Social Psychology, and the magazine Science. Diederik Stapel is a former professor at Tilburg University in the Netherlands. Over the past two years, an extensive investigation involving three universities where Stapel has worked concluded that the psychologist is guilty of fraud on a colossal scale. Falsified data taints over 55 papers he authored and 10 Ph.D. dissertations that he supervised. Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct results that journals found attractive. “It was a quest for aesthetics, for beauty—instead of the truth,” he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud. Bhattacharjee, Y. (2013, April 26). The mind of a con man. The committee investigating Stapel concluded that he is guilty of several practices including 1. creating datasets, which largely confirmed the prior expectations, 2. altering data in existing datasets, 3. changing measuring instruments without reporting the change, and 4. misrepresenting the number of experimental subjects. Clearly, it is never acceptable to falsify data the way this researcher did. Sometimes, however, violations of ethics are not as easy to spot. Researchers have a responsibility to verify that proper methods are being followed. The report describing the investigation of Stapel’s fraud states that, “statistical flaws frequently revealed a lack of familiarity with elementary statistics.”Tillburg University. (2012, Nov. 28). Flawed science: the fraudulent research practices of social psychologist Diederik Stapel. Retrieved from https://www.tilburguniversity.edu/upload/3ff904d7-547b-40ae-85fe-bea38e05a34a_Final%20report%20Flawed%20Science.pdf. Many of Stapel’s co-authors should have spotted irregularities in his data. Unfortunately, they did not know very much about statistical analysis, and they simply trusted that he was collecting and reporting data properly. Many types of statistical fraud are difficult to spot. Some researchers simply stop collecting data once they have just enough to prove what they had hoped to prove. They don’t want to take the chance that a more extensive study would complicate their lives by producing data contradicting their hypothesis. Professional organizations, like the American Statistical Association, clearly define expectations for researchers. There are even laws in the federal code about the use of research data. When a statistical study uses human participants, as in medical studies, both ethics and the law dictate that researchers should be mindful of the safety of their research subjects. The U.S. Department of Health and Human Services oversees federal regulations of research studies with the aim of protecting participants. When a university or other research institution engages in research, it must ensure the safety of all human subjects. For this reason, research institutions establish oversight committees known as Institutional Review Boards (IRB). All planned studies must be approved in advance by the IRB. Key protections that are mandated by law include the following: 1. Risks to participants must be minimized and reasonable with respect to projected benefits. 2. Participants must give informed consent. This means that the risks of participation must be clearly explained to the subjects of the study. Subjects must consent in writing, and researchers are required to keep documentation of their consent. 3. Data collected from individuals must be guarded carefully to protect their privacy. These ideas may seem fundamental, but they can be very difficult to verify in practice. Is removing a participant’s name from the data record sufficient to protect privacy? Perhaps the person’s identity could be discovered from the data that remains. What happens if the study does not proceed as planned and risks arise that were not anticipated? When is informed consent really necessary? Suppose your doctor wants a blood sample to check your cholesterol level. Once the sample has been tested, you expect the lab to dispose of the remaining blood. At that point the blood becomes biological waste. Does a researcher have the right to take it for use in a study? It is important that students of statistics take time to consider the ethical questions that arise in statistical studies. How prevalent is fraud in statistical studies? You might be surprised—and disappointed. There is a website dedicated to cataloging retractions of study articles that have been proven fraudulent. A quick glance will show that the misuse of statistics is a bigger problem than most people realize. Vigilance against fraud requires knowledge. Learning the basic theory of statistics will empower you to analyze statistical studies critically. ### References Econoclass.com. Retrieved from http://www.econoclass.com/misleadingstats.html. Bloomberg Businessweek. Retrieved from www.businessweek.com. Ethics in statistics. Retrieved from http://cnx.org/content/m15555/latest/. Forbes. Retrieved from www.forbes.com. Forbes. http://www.forbes.com/best-small-companies/list/. Harvard School of Public Health. Retrieved from https://www.hsph.harvard.edu/nutritionsource/vitamin-e/. Jacskon, M.L., et al. (2013). Cognitive components of simulated driving performance: Sleep loss effect and predictors. Accident Analysis and Prevention Journal 50, 438-44. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/22721550. International Business Times. Retrieved from http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacks-study-300443. National Highway Traffic Safety Administration. Retrieved from http://www-fars.nhtsa.dot.gov/Main/index.aspx. Athleteinme.com. Retrieved from http://www.athleteinme.com/ArticleView.aspx?id=1053. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html. U.S. Department of Health and Human Services. Retrieved from https://www.hhs.gov/ohrp/regulations-and-policy/regulations/45-cfr-46/index.html. U.S. Department of Transportation. Retrieved from http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report. U.S. Geological Survey. Retrieved from http://earthquake.usgs.gov/earthquakes/eqarchives/year/. ### Chapter Review A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. Participants in the control group receive a placebo treatment that looks exactly like the active treatments but cannot influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due to the influence of the explanatory variable. “An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.”Gelman, A. (2013, May 1). Open data and open methods. Ethical violations in statistics are not always easy to spot. Professional associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis. ### HOMEWORK

# Descriptive Statistics ## Introduction Once you have a data collection, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data. In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called descriptive statistics. You will learn how to calculate and, even more important, how to interpret these measurements and graphs. A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data values cluster and where there are only a few data values. Newspapers and the internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then more formal tools may be applied. Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon—a type of broken line graph—the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs as well as frequency polygons, time series graphs, and dot plots. Our emphasis will be on histograms and box plots.

# Descriptive Statistics ## Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The stem consists of the leading digit(s), while the leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem. Make sure the leaves show a space between values, so that the exact data values may be easily determined. The frequency of data values for each stem provides information about the shape of the distribution. The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes, for example, writing 50 instead of 500, while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later. Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in , the (horizontal axis) consists of data values and the (vertical axis) consists of frequency points. The frequency points are connected using line segments. Bar graphs consist of bars that are separated from each other. The bars can be rectangles, or they can be rectangular boxes, used in three-dimensional plots, and they can be vertical or horizontal. The bar graph shown in has age-groups represented on the and proportions on the . ### References Burbary, K. (2011, March 7). Facebook demographics revisited – 2001 statistics. Social Media Today. Retrieved from http://www.kenburbary.com/2011/03/facebook-demographics-revisited-2011-statistics-2/ Centers for Disease Control and Prevention. (n.d.). Overweight and obesity: Adult obesity facts. Available online http://www.cdc.gov/obesity/data/adult.html CollegeBoard. (2013). The 9th annual AP report to the nation. Retrieved from http://apreport.collegeboard.org/goals-andfindings/promoting-equity ### Chapter Review A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends, that is, finding a general pattern in data sets, including temperature, sales, employment, company profit, or cost, over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Bar graphs are especially useful when categorical data are being used. ### For each of the following data sets, create a stemplot and identify any outliers. For the next three exercises, use the data to construct a line graph. ### Homework

# Descriptive Statistics ## Histograms, Frequency Polygons, and Time Series Graphs For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is more or less a number line, labeled with what the data represents, for example, distance from your home to school. The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The shape of the data refers to the shape of the distribution, whether normal, approximately normal, or skewed in some direction, whereas the center is thought of as the middle of a data set, and the spread indicates how far the values are dispersed about the center. In a skewed distribution, the mean is pulled toward the tail of the distribution. The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. Remember, frequency is defined as the number of times an answer occurs. If 1. f = frequency, 2. n = total number of data values (or the sum of the individual frequencies), and 3. RF = relative frequency, then For example, if three students in Mr. Ahab's English class of 40 students received from ninety to 100 percent, then f = 3, n = 40, and RF = = = 0.075. Thus, 7.5 percent of the students received 90 to 100 percent. Ninety to 100 percent is a quantitative measures. To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The width of each bar is also referred to as the bin size, which may be calculated by dividing the range of the data values by the desired number of bins (or bars). There is not a set procedure for determining the number of bars or bar width/bin size; however, consistency is key when determining which data values to place inside each interval. ### Frequency Polygons Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals and resulting interval size, for both the x-axis and y-axis. The x-axis will show the lower and upper bound for each interval, containing the data values, whereas the y-axis will represent the frequencies of the values. Each data point represents the frequency for each interval. For example, if an interval has three data values in it, the frequency polygon will show a 3 at the upper endpoint of that interval. After choosing the appropriate intervals, begin plotting the data points. After all the points are plotted, draw line segments to connect them. Frequency polygons are useful for comparing distributions. This comparison is achieved by overlaying the frequency polygons drawn for different data sets. Suppose that we want to study the temperature range of a region for an entire month. Every day at noon, we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected. One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph. ### Constructing a Time Series Graph To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By using the axes in that way, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur. ### Uses of a Time Series Graph Time series graphs are important tools in various applications of statistics. When a researcher records values of the same variable over an extended period of time, it is sometimes difficult for him or her to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot. ### References Bureau of Labor Statistics, U.S. Department of Labor. (n.d.). Consumer price index. Retrieved from https://www.bls.gov/cpi/ CIA World Factbook. (n.d.). Demographics: Children under the age of 5 years underweight. Available at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en Centers for Disease Control and Prevention. (n.d.). Overweight and obesity: Adult obesity facts. Available online http://www.cdc.gov/obesity/data/adult.html Food and Agriculture Organization of the United Nations. (n.d.). Food security statistics. Retrieved from http://www.fao.org/economic/ess/ess-fs/en/ General Register Office for Scotland. Births time series data. (2013). Retrieved from http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html Gunst, R., and Mason, R. (1980). Regression analysis and its application: A data-oriented approach. Boca Raton, FL: CRC Press. Sandbox Networks. (2007). Presidents. Available online at http://www.factmonster.com/ipka/A0194030.html Scholastic. (2013). Timeline: Guide to the U.S. presidents. Retrieved from http://www.scholastic.com/teachers/article/timeline-guide-us-presidents World Bank Group. (2013). DataBank: CO2 emissions (kt). Retrieved from http://databank.worldbank.org/data/home.aspx ### Chapter Review A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually go on the y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time. ### Homework Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each.

# Descriptive Statistics ## Measures of the Location of the Data The common measures of location are quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1, is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile. To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the test scores are the same as or greater than your test score. Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75th percentile. That translates into a score of at least 1220. Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant. The median is a number that measures the center of the data. You can think of the median as the middle value, but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data: 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1 Ordered from smallest to largest: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order. Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two. The median is seven. Half of the values are smaller than seven and half of the values are larger than seven. Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as and equals 7. So, the median, or second quartile ( ), is 7. The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( ) will be the middle value, or 2. Using the same procedure, we can see that the median of the upper half, or the third quartile ( ) will be the middle value of the upper half, or 9. The quartiles are illustrated below: The interquartile range is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (Q3) and the first quartile (Q1) IQR = Q3 – Q1. The IQR for this data set is calculated as 9 minus 2, or 7. The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than 1.5 × . Potential outliers always require further investigation. In the example above, you just saw the calculation of the median, first quartile, and third quartile. These three values are part of the five number summary. The other two values are the minimum value (or min) and the maximum value (or max). The five number summary is used to create a box plot. ### A Formula for Finding the kth Percentile If you were to do a little research, you would find several formulas for calculating the kth percentile. Here is one of them. k = the kth percentile. It may or may not be part of the data. i = the index (ranking or position of a data value) n = the total number of data 1. Order the data from smallest to largest. 2. Calculate 3. If i is an integer, then the kth percentile is the data value in the ith position in the ordered set of data. 4. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example. ### A Formula for Finding the Percentile of a Value in a Data Set 1. Order the data from smallest to largest. 2. x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile. 3. y = the number of data values equal to the data value for which you want to find the percentile. 4. n = the total number of data. 5. Calculate (100). Then round to the nearest integer. ### Interpreting Percentiles, Quartiles, and Median A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15 percent of data values are less than or equal to the 15th percentile. 1. Low percentiles always correspond to lower data values. 2. High percentiles always correspond to higher data values. A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person's height doesn't carry any value judgment. Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text. ### References Cauchon, D., and Overberg, P. (2012). Census data shows minorities now a majority of U.S. births. USA Today. Retrieved from http://usatoday30.usatoday.com/news/nation/story/2012-05-17/minority-birthscensus/55029100/1 The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/ Time. (n.d.). Survey by Yankelovich Partners, Inc. U.S. Census Bureau. (1990). 1990 census. Retrieved from http://www.census.gov/main/www/cen1990.html U.S. Census Bureau. (n.d.). Data. Retrieved from http://www.census.gov/ ### Chapter Review The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50th percentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quartile (Q1) is the 25th percentile, the second quartile (Q2 or median) is the 50th percentile, and the third quartile (Q3) is the 75th percentile. The interquartile range, or IQR, is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q1 from Q3 and can help determine outliers by using the following two expressions. 1. Q3 + IQR(1.5) 2. Q1 – IQR(1.5) ### Formula Review where i = the ranking or position of a data value, k = the kth percentile, n = total number of data. Expression for finding the percentile of a data value (100) where x = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile, y = the number of data values equal to the data value for which you want to find the percentile, n = total number of data. Use to calculate the following values. ### Homework

# Descriptive Statistics ## Box Plots Box plots, also called box-and-whisker plots or box-whisker plots, give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. As mentioned previously, a box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value. We use these values to compare how close other data values are to them. To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box, and the third quartile marks the other end of the box. Approximately the middle 50 percent of the data fall inside the box. The whiskers extend from the ends of the box to the smallest and largest data values. A box plot easily shows the range of a data set, which is the difference between the largest and smallest data values (or the difference between the maximum and minimum). Unless the median, first quartile, and third quartile are the same value, the median will lie inside the box or between the first and third quartiles. The box plot gives a good, quick picture of the data. Consider, again, this data set: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 The first quartile is two, the median is seven, and the third quartile is nine. The smallest value is one, and the largest value is 11.5. The following image shows the constructed box plot. The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line. For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same. For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like the following: In this case, at least 25 percent of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25 percent of the values are equal to five. The top 25 percent of the values fall between five and seven, inclusive. ### References West Magazine. (n.d.). Retrieved from https://westmagazine.net/ ### Chapter Review Box plots are a type of graph that can help visually organize data. Before a box plot can be graphed, the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. ### Homework ### Bringing It Together

# Descriptive Statistics ## Measures of the Center of the Data The center of a data set is also a way of describing location. The two most widely used measures of the center of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “x bar”): . The sample mean is a statistic. The Greek letter μ (pronounced "mew") represents the population mean. The population mean is a parameter. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. To see that both ways of calculating the mean are the same, consider the following sample: 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4 In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4). You can quickly find the location of the median by using the expression . The letter n is the total number of data values in the sample. As discussed earlier, if n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by two after the data have been ordered. For example, if the total number of data values is 97, then = = 49. The median is the 49th value in the ordered data. If the total number of data values is 100, then = = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The uppercase letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median. Another measure of the center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. ### The Law of Large Numbers and the Mean The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean of the sample is very likely to get closer and closer to µ. This law is discussed in more detail later in the text. ### Sampling Distributions and Statistic of a Sampling Distribution You can think of a sampling distribution as a relative frequency distribution with a great many samples. See Chapter 1: Sampling and Data for a review of relative frequency. Suppose 30 randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below. A relative frequency distribution includes the relative frequencies of a number of samples. Recall that a statistic is a number calculated from a sample. Statistic examples include the mean, the median, and the mode as well as others. The sample mean is an example of a statistic that estimates the population mean μ. ### Calculating the Mean of Grouped Frequency Tables When only grouped data is available, you do not know the individual data values (we know only intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table, we can apply the basic definition of mean: mean = We simply need to modify the definition to fit within the restrictions of a frequency table. Since we do not know the individual data values, we can instead find the midpoint of each interval. The midpoint is . We can now modify the mean definition to be where f = the frequency of the interval, m = the midpoint of the interval, and sigma (∑) is read as "sigma" and means to sum up. So this formula says that we will sum the products of each midpoint and the corresponding frequency and divide by the sum of all of the frequencies. ### References CIA World Factbook. (n.d.). Obesity – adult prevalence rate. Available at http://www.indexmundi.com/g/r.aspx?t=50&v=2228&l=en World Bank Group. (n.d.). Retrieved from http://www.worldbank.org ### Chapter Review The mean and the median can be calculated to help you find the center of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your data set. The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges that lack specific values, the mean may seem impossible to calculate. However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval. Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set. ### Formula Review where f = interval frequencies and m = interval midpoints. Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40 Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Calculate the following. ### Homework ### Bringing It Together Use the following information to answer the next three exercises: We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section.

# Descriptive Statistics ## Skewness and the Mean, Median, and Mode Consider the following data set: 4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10 This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the middle of an interval. The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems chopped off compared to the left-hand side. A distribution of this type is called skewed to the left because it is pulled out to the left. A skewed left distribution has more high values. The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution. The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is skewed to the right. A skewed right distribution has more low values. The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most. To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean. Skewness and symmetry become important when we discuss probability distributions in later chapters. ### Chapter Review Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like . A left (or negative) skewed distribution has a shape like . A symmetrical distribution looks like . Use the following information to answer the next three exercises. State whether the data are symmetrical, skewed to the left, or skewed to the right. ### Homework

# Descriptive Statistics ## Measures of the Spread of the Data An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean. ### The standard deviation 1. provides a numerical measure of the overall amount of variation in a data set and 2. can be used to determine whether a particular data value is close to or far from the mean. ### The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation. Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes. Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average. ### The standard deviation can be used to determine whether a data value is close to or far from the mean. Suppose that both Rosa and Binh shop at Supermarket A. Rosa waits at the checkout counter for seven minutes, and Binh waits for one minute. At Supermarket A, the mean waiting time is five minutes, and the standard deviation is two minutes. The standard deviation can be used to determine whether a data value is close to or far from the mean. A z-score is a standardized score that lets us compare data sets. It tells us how many standard deviations a data value is from the mean and is calculated as the ratio of the difference in a particular score and the population mean to the population standard deviation. We can use the given information to create the table below. Since Rosa and Binh only shop at Supermarket A, we can ignore the row for Supermarket B. We need the values from the first row to determine the number of standard deviations above or below the mean each individual wait time is; we can do so by calculating two different z-scores. Rosa waited for seven minutes, so the z-score representing this deviation from the population mean may be calculated as The z-score of one tells us that Rosa’s wait time is one standard deviation above the mean wait time of five minutes. Binh waited for one minute, so the z-score representing this deviation from the population mean may be calculated as The z-score of −2 tells us that Binh’s wait time is two standard deviations below the mean wait time of five minutes. A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if they are more than two standard deviations away is more of an approximate rule of thumb than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters. The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is one standard deviation to the right of five because 5 + (1)(2) = 7. If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (–2)(2) = 1. 1. In general, a value = mean + (#ofSTDEV)(standard deviation) 2. where #ofSTDEVs = the number of standard deviations 3. #ofSTDEV does not need to be an integer 4. One is two standard deviations less than the mean of five because 1 = 5 + (–2)(2). The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population as follows: The lowercase letter s represents the sample standard deviation and the Greek letter σ (lower case) represents the population standard deviation. The symbol is the sample mean, and the Greek symbol is the population mean. ### Calculating the Standard Deviation If x is a number, then the difference x – mean is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is x – μ. For sample data, in symbols, a deviation is x – . The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar but not identical. Therefore, the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lowercase letter s represents the sample standard deviation and the Greek letter σ (lowercase sigma) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ. To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations (the x – values for a sample or the x – μ values for a population). The symbol σ2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s2 represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations. If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N, the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by , one less than the number of items in the sample. ### Formulas for the Sample Standard Deviation 1. or 2. For the sample standard deviation, the denominator is ; that is, the sample size minus 1. ### Formulas for the Population Standard Deviation 1. or 2. For the population standard deviation, the denominator is N, the number of items in the population. In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three. ### Types of Variability in Samples When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. Variability is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following: 1. Observational or measurement variability 2. Natural variability 3. Induced variability 4. Sample variability Here are some examples to describe each type of variability: Example 1: Measurement variability Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability. Example 2: Natural variability Natural variability arises from the differences that naturally occur because members of a population differ from each other. For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability. Example 3: Induced variability Induced variability is the counterpart to natural variability. This occurs because we have artificially induced an element of variation that, by definition, was not present naturally. For example, we assign people to two different groups to study memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may be affected by induced variability. Example 4: Sample variability Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability. ### Sampling Variability of a Statistic The statistic of a sampling distribution was discussed in Descriptive Statistics: Measures of the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is , where σ is the standard deviation of the population and n is the size of the sample. ### Explanation of the standard deviation calculation shown in the table The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11, which is indicated by the deviations .97 and .47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero. We can sum the products of the frequencies and deviations to show that the sum of the deviations is always zero. For , there are n = 20 deviations. So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation. The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data. Notice that instead of dividing by n = 20, the calculation divided by n – 1 = 20 – 1 = 19 because the data is a sample. For the sample variance, we divide by the sample size minus one (n – 1). Why not divide by n? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by (n – 1) gives a better estimate of the population variance. The standard deviation, s or σ, is either zero or larger than zero. Describing the data with reference to the spread is called variability. The variability in data depends on the method by which the outcomes are obtained, for example, by measuring or by random sampling. When the standard deviation is zero, there is no spread; that is, all the data values are equal to each other. The standard deviation is small when all the data are concentrated close to the mean and larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make s or σ very large. The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better feel for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful, but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data. Display your data in a histogram or a box plot. ### Standard deviation of Grouped Frequency Tables Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula where interval frequencies and m = interval midpoints. Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how unusual individual data are when compared to the mean. ### Comparing Values from Different Data Sets As explained before, a z-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading. 1. For each data value, calculate how many standard deviations away from its mean the value is. 2. In symbols, the formulas for calculating z-scores become the following. As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used. The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data. 2. At least 75 percent of the data is within two standard deviations of the mean. 3. At least 89 percent of the data is within three standard deviations of the mean. 4. At least 95 percent of the data is within 4.5 standard deviations of the mean. 5. This is known as Chebyshev's Rule. A bell-shaped distribution is one that is normal and symmetric, meaning the curve can be folded along a line of symmetry drawn through the median, and the left and right sides of the curve would fold on each other symmetrically.. With a bell-shaped distribution, the mean, median, and mode are all located at the same place. 2. Approximately 68 percent of the data is within one standard deviation of the mean. 3. Approximately 95 percent of the data is within two standard deviations of the mean. 4. More than 99 percent of the data is within three standard deviations of the mean. 5. This is known as the Empirical Rule. 6. It is important to note that this rule applies only when the shape of the distribution of the data is bell-shaped and symmetric; we will learn more about this when studying the Normal or Gaussian probability distribution in later chapters. ### References King, B. (2005, Dec.). Graphically Speaking. Retrieved from http://www.ltcc.edu/web/about/institutional-research Microsoft Bookshelf. (n.d.). ### Chapter Review The standard deviation can help you calculate the spread of data. There are different equations to use if you are calculating the standard deviation of a sample or of a population. 1. The standard deviation allows us to compare individual data or classes to the data set mean numerically. 2. s = or s = is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ, and the formula σ = or σ = . ### Formula Review where and ### For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability. Use the following information to answer the next two exercises. The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles. 29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150 ### Homework Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005. 1. μ = 1,000 FTES 2. median = 1,014 FTES 3. σ = 474 FTES 4. first quartile = 528.5 FTES 5. third quartile = 1,447.5 FTES 6. n = 29 years ### Bringing It Together Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility.

# Probability Topics ## Introduction It is often necessary to guess about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs. You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach.

# Probability Topics ## Terminology Probability is a measure that is associated with how certain we are of results, or outcomes, of a particular activity. When the activity is a planned operation carried out under controlled conditions, it is called an experiment. If the result is not predetermined, then the experiment is said to be a chance experiment. Each time the experiment is attempted is called a trial. Examples of chance experiments include the following: 1. flipping a fair coin, 2. spinning a spinner, 3. drawing a marble at random from a bag, and 4. rolling a pair of dice. A result of an experiment is called an outcome. The sample space of an experiment is the set, or collection, of all possible outcomes. There are four main ways to represent a sample space: *We will investigate tree diagrams and Venn diagrams in Section 3.5. Note—when represented as a set, the sample space is denoted with an uppercase S. An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like A and B are commonly used to represent events. For example, if the experiment is to flip three fair coins, event A might be getting at most one head. The probability of an event A is written P(A), and means the event A can never happen. P(A) = 1 means the event A always happens. means the event A is equally likely to occur or not to occur. If two outcomes or events are equally likely, then they have equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer. To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. This is known as the theoretical probability of A. Theoretical Probability of Event A For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. Let A represent the outcome getting one head. There are two outcomes that meet this condition {HT, TH}, so Theoretical probability is not sufficient in all situations, however. Suppose we want to calculate the probability that a randomly selected car will run a red light at a given intersection. In this case, we need to look at events that have occurred, not theoretical possibilities. We could install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a period of time. These data will allow us to calculate the experimental, or empirical, probability that a car runs the red light. Experimental Probability of Event A While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two possible outcomes. So, the theoretical probability of heads is . Probability does not predict short-term results, however. If an experiment involves flipping a coin 10 times, you should not expect exactly five heads and five tails. The probability of any outcome measures the long-term relative frequency of that outcome. If you continue to flip the coin (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches .5 (the probability of heads).This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed, or empirical, relative frequency will approach the theoretical probability. Suppose you roll one fair, six-sided die with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five. There are two outcomes {5, 6}. If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, of the rolls would result in an outcome of at least five. You would not expect exactly , but the long-term relative frequency of obtaining this result would approach the theoretical probability of as the number of repetitions grows larger and larger. It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one-euro coin and discovered that in 250 trials, a head was obtained 56 percent of the time and a tail was obtained 44 percent of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely. OR EventAn outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are not listed twice. AND EventAn outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}. The complement of event A is denoted A′ (read "A prime"). A′ consists of all outcomes that are not in A. Notice that P(A) + P(A′) = 1. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then, A′ = {5, 6}. P(A) = , P(A′) = , and P(A) + P(A′) = = 1. The conditional probability of A given B is written P(A|B), read "the probability of A, given B." P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional probability reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) is P(A|B) = where P(B) is greater than zero. For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = {2, 3} and B = {2, 4, 6}. P(A|B) represents the probability that a randomly selected outcome is in A given that it is in B. We know that the outcome must lie in B, so there are three possible outcomes. There is only one outcome in B that also lies in A, so P(A|B) = . We get the same result by using the formula. Remember that S has six outcomes. P(A|B) = Understanding Terminology and SymbolsIt is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any. ### References Worldatlas. (2013). Countries list by continent. Retrieved from http://www.worldatlas.com/cntycont.htm ### Chapter Review In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive. ### Formula Review A and B are events P(S) = 1 where S is the sample space 0 ≤ P(A) ≤ 1 P(A|B) = Use the following information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, 10 finger traps, and five bags of confetti. Let H = the event of getting a hat. Let N = the event of getting a noisemaker. Let F = the event of getting a finger trap. Let C = the event of getting a bag of confetti. Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Let B = the event of getting a blue jelly bean Let G = the event of getting a green jelly bean. Let O = the event of getting an orange jelly bean. Let P = the event of getting a purple jelly bean. Let R = the event of getting a red jelly bean. Let Y = the event of getting a yellow jelly bean. Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 countries in Oceania (Pacific Ocean region). Let A = the event that a country is in Asia. Let E = the event that a country is in Europe. Let F = the event that a country is in Africa. Let N = the event that a country is in North America. Let O = the event that a country is in Oceania. Let S = the event that a country is in South America. Use the following information to answer the next two exercises. You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area. Let B = the event of landing on blue. Let R = the event of landing on red. Let G = the event of landing on green. Let Y = the event of landing on yellow. Use the following information to answer the next 10 exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters. Let I = the event that a player in an infielder. Let O = the event that a player is an outfielder. Let H = the event that a player is a great hitter. Let N = the event that a player is not a great hitter. Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let E = the event that it lands on an even number. Let M = the event that it lands on a multiple of three. ### Homework

# Probability Topics ## Independent and Mutually Exclusive Events Independent and mutually exclusive do not mean the same thing. ### Independent Events Two events are independent if the following are true: 1. P(A|B) = P(A) 2. P(B|A) = P(B) 3. P(A AND B) = P(A)P(B) Two events A and B are independent events if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are not independent, then we say that they are dependent events. Sampling may be done with replacement or without replacement. 1. With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick. A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is . When James draws a marble from the bag a second time, the probability of drawing blue is still . James replaced the marble after the first draw, so there are still four blue and three white marbles. 1. Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent. The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is . Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now three blue and three white marbles. So, the probability of drawing blue is now . Removing the first marble without replacing it influences the probabilities on the second draw. If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise. ### Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0. For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}. A AND B = {4, 5}. P(A AND B) = and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so P(A AND C) = 0. Therefore, A and C are mutually exclusive. If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms. ### References Gallup. (n.d.). Retrieved from www.gallup.com/ Lopez, S., and Sidhu, P. (2013, March 28). U.S. teachers love their lives, but struggle in the workplace. Gallup Wellbeing. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx ### Chapter Review Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other. ### Formula Review If A and B are independent, P(A AND B) = P(A)P(B), P(A|B) = P(A), and P(B|A) = P(B). If A and B are mutually exclusive, P(A OR B) = P(A) + P(B) and P(A AND B) = 0. ### Homework Use the following information to answer the next 12 exercises. The graph shown is based on more than 170,000 interviews that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Health Index Scores are the sample space. We randomly sample one type of Health Index Score, the emotional well-being score. ### Bringing It Together

# Probability Topics ## Two Basic Rules of Probability In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not. ### The Multiplication Rule If A and B are two events defined on a sample space, then P(A AND B) = P(B)P(A|B). This equation can be rewritten as P(A AND B) = P(B)P(A|B), the multiplication rule. If A and B are independent, then P(A|B) = P(A). In this special case, P(A AND B) = P(A|B)P(B) becomes P(A AND B) = P(A)P(B). A bag contains four green marbles, three red marbles, and two yellow marbles. Mark draws two marbles from the bag without replacement. The probability that he draws a yellow marble and then a green marble is Notice that . After the yellow marble is drawn, there are four green marbles in the bag and eight marbles in all. ### The Addition Rule If A and B are defined on a sample space, then P(A OR B) = P(A) + P(B) − P(A AND B). Draw one card from a standard deck of playing cards. Let H = the card is a heart, and let J = the card is a jack. These events are not mutually exclusive because a card can be both a heart and a jack. If A and B are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) − P(A AND B) becomes P(A OR B) = P(A) + P(B). Draw one card from a standard deck of playing cards. Let H = the card is a heart and S = the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade is ### References Baseball Almanac. (2013). Retrieved from www.baseball-almanac.com DiCamillo, Mark, and Field, M. The file poll. Field Research Corporation. Retrieved from http://www.field.com/fieldpollonline/subscribers/Rls2443.pdf Field Research Corporation. (n.d.). Retrieved from www.field.com/fieldpollonline Forum Research. (n.d.). Mayor’s approval down. Retrieved from http://www.forumresearch.com/forms/News Archives/News Releases/74209_TO_Issues_-_Mayoral_Approval_%28Forum_Research%29%2820130320%29.pdf Rider, D. (2011, Sept. 14). Ford support plummeting, poll suggests. The Star. Retrieved from http://www.thestar.com/news/gta/2011/09/14/ford_support_plummeting_poll_suggests.html Shin, H. B., and Kominski, R. A. (2010 April). Language use in the United States: 2007 (American Community Survey Reports, ACS-12). Washington, DC: United States Census Bureau. Retrieved from http://www.census.gov/hhes/socdemo/language/data/acs/ACS-12.pdf The Roper Center for Public Opinion Research. (n.d.). Archives. Retrieved from http://www.ropercenter.uconn.edu/ The Wall Street Journal. (n.d.). Retrieved from https://www.wsj.com/ U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/ Wikipedia. (n.d.). Roulette. Retrieved from http://en.wikipedia.org/wiki/Roulette ### Chapter Review The multiplication rule and the addition rule are used for computing the probability of A and B, as well as the probability of A or B for two given events A, B defined on the sample space. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common. ### Formula Review The multiplication rule—P(A AND B) = P(A|B)P(B) The addition rule—P(A OR B) = P(A) + P(B) − P(A AND B) Use the following information to answer the next 10 exercises. Forty-eight percent of all voters of a certain state prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Among Latino registered voters in this state, 55 percent prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Of all citizens in this state, 37.6 percent are Latino. In this problem, let Suppose that one citizen is randomly selected. ### Homework . A local restaurant sells pork chops and chicken breasts. The given values below are the weights (in ounces) of pork chops and chicken breasts listed on the menu. Your server will randomly select one piece of meat (pork chop or chicken breast) that you will be served.

# Probability Topics ## Contingency Tables A two-way table provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a contingency table. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. We used two-way tables in Chapters 1 and 2 to calculate marginal and conditional distributions. These tables organize data in a way that supports the calculation of relative frequency and, therefore, experimental (empirical) probability. Later on, we will use contingency tables again, but in another manner. ### References American Red Cross. (2013). Blood Types. Retrieved from http://www.redcrossblood.org/learn-about-blood/bloodtypes Centers for Disease Control and Prevention/National Center for Health Statistics, United States Department of Health and Human Services. (n.d.). Retrieved from https://www.cdc.gov/nchs/ Haiman, C. A., et al. (2006, Jan. 26). Ethnic and racial differences in the smoking-related risk of lung cancer. The New England Journal of Medicine. Retrieved from http://www.nejm.org/doi/full/10.1056/NEJMoa033250 Samuels, T. M. (2013). Strange facts about RH negative blood. eHow Health. Retrieved from http://www.ehow.com/facts_5552003_strange-rh-negative-blood.html The Disaster Center Crime Pages. (n.d.). United States: Uniform crime report – state statistics from 1960–2011. Retrieved from http://www.disastercenter.com/crime/ United Blood Services. (2011). Human blood types. Retrieved from http://www.unitedbloodservices.org/learnMore.aspx United States Senate. (n.d.). Retrieved from www.senate.gov ### Chapter Review There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables, also known as two-way tables, help display data and are particularly useful when calculating probabilites that have multiple dependent variables. ### Use the following information to answer the next four exercises. shows a random sample of musicians and how they learned to play their instruments. ### Bringing It Together Use the following information to answer the next seven exercises. An article in the , reported about a study of people who use a product in California and Hawaii. In one part of the report, the self-reported ethnicity and using the product levels per day were given. Of the people using the product at most 10 times a day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 whites. Of the people using the product 11 to 20 times per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people using the product 21 to 30 times per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people using the product at least 31 times per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites. ### Homework Use the information in the The table shows the political party affiliation of each of 67 members of the U.S. Senate in June 2012, and when they would next be up for reelection. Use the following information to answer the next two exercises. The table of data obtained from shows hit information for four well-known baseball players. Suppose that one hit from the table is randomly selected.

# Probability Topics ## Tree and Venn Diagrams Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities. ### Tree Diagrams A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of branches that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram: ### Venn Diagram A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. ### References American Cancer Society. (n.d.). Retrieved from https://www.cancer.org/ Clara County Public Health Department. (n.d.). Retrieved from https://www.sccgov.org/sites/sccphd/en-us/pages/phd.aspx Federal Highway Administration, U.S. Department of Transportation. (n.d.). Retrieved from https://www.fhwa.dot.gov/ The Data and Story Library. (1996). Retrieved from http://lib.stat.cmu.edu/DASL/ The Roper Center for Public Opinion Research. (2013). Search for datasets. Retrieved from http://www.ropercenter.uconn.edu/data_access/data/search_for_datasets.html USA Today. (n.d.). Retrieved from https://www.usatoday.com/ U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/ World Bank Group. (2013). Environment. Available online at http://data.worldbank.org/topic/environment ### Chapter Review A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize. A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities. ### ### Homework Use the following information to answer the next two exercises. This tree diagram shows the tossing of an unfair coin followed by drawing one bead from a cup containing three red (R), four yellow (Y), and five blue (B) beads. For the coin, P(H) = and P(T) = where H is heads and T is tails. ### Bringing It Together Use the following information to answer the next two exercises. Suppose that you have eight cards. Five are green and three are yellow. The cards are well shuffled. Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) who are female is 48.60. Of the females, 5.03 percent are age 19 and under; 81.36 percent are age 20–64; 13.61 percent are age 65 or over. Of the licensed U.S. male drivers, 5.04 percent are age 19 and under; 81.43 percent are age 20–64; 13.53 percent are age 65 or over.

# Discrete Random Variables ## Introduction A student takes a 10-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a 70 percent? Small companies might be interested in the number of long-distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long-distance phone calls during the peak time? These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable is a variable whose values are numerical outcome of a probability experiment. We always describe a random variable in words and its values in numbers. The values of a random variable can vary with each repetition of an experiment. ### Random Variable Notation Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If The following are examples of random variables: Example 1: Suppose a jar contains three marbles, one blue, one red, and one white. Randomly draw one marble from the jar. Let X = the possible number of red marbles to be drawn. The sample space for the drawing is red, white, and blue. Then, x = 0,1. If the marble we draw is red, then x = 1; otherwise, x = 0. Example 2: Let X = the number of female children in a randomly selected family with only two kids. Here we are only interested in families with two kids, not families with one kid or more than two kids. The sample space for the genders of two-kid families is MM, MF, FM, FF. Here the first letter represents the gender of the older child and the second letter represents the gender of the younger child. F represents a female child and M represents a male child. For example, FM represents that the older child is a girl and the younger child is a boy, while MF represents that the older child is a boy and the younger child is a girl. Then, x = 0,1,2. A family has 0 female children if it has two boys (MM), a family has one female child if it has one boy and one girl (MF or FM), and a family has two female children if both kids are girls (FF). Example 3: Let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. Here the first letter represents the result of the first toss, the second letter represents the result of the second toss, and the third letter represents the result of the third toss. T represents a tail and H represents a head. For example, THH means we get a tail in the first toss but a head in the second and third toss, while HHT means we get a head in the first and second toss but a tail in the third toss. Then, x = 0, 1, 2, 3. There are 0 heads if the result is TTT, one head if the result is THT, TTH, or HTT, two heads if the result is THH, HTH, or HHT, and three heads if the result is HHH.

# Discrete Random Variables ## Probability Distribution Function (PDF) for a Discrete Random Variable There are two types of random variables, discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. For example, let X = temperature of a randomly selected day in June in a city. The value of X can be 68°, 71.5°, 80.6°, or 90.32°. These values are obtained by measuring by a thermometer. Another example of a continuous random variable is the height of a randomly selected high school student. The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler. A discrete probability distribution function has two characteristics: 1. Each probability is between zero and one, inclusive. 2. The sum of the probabilities is one. ### Chapter Review The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows: 1. Each probability is between zero and one, inclusive (inclusive means to include zero and one) 2. The sum of the probabilities is one ### Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution: Let X = the number of years a new hire will stay with the company. Let P(x) = the probability that a new hire will stay with the company x years. Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random. Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time. ### Homework

# Discrete Random Variables ## Mean or Expected Value and Standard Deviation The expected value of a discrete random variable X, symbolized as E(X), is often referred to as the long-term average or (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average. As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below. Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce rounding errors. For some probability distributions, there are shortcut formulas for calculating μ and σ. Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions. A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions. ### References Florida State University. (n.d.). Class catalogue at the Florida State University. Retrieved from https://apps.oti.fsu.edu/RegistrarCourseLookup/SearchFormLegacy World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction ### Chapter Review The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes. ### Formula Review Mean or Expected Value: Standard Deviation: Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution: Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution: 1. Let X = the number of years a student will study ballet with the teacher. 2. Let P(x) = the probability that a student will study ballet x years. ### HOMEWORK

# Discrete Random Variables ## Binomial Distribution (Optional) There are three characteristics of a binomial experiment: The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a binomial probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book. Here n is the number of trials, p is the probability of a success, and q is the probability of a failure. ### Notation for the Binomial: B = Binomial Probability Distribution Function X ~ B(n, p) Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial. The following is the interpretation of the mean and standard deviation : If you randomly select 20 adult workers, and do that over and over, you expect around eight adult workers out of 20 to have a high school diploma but do not pursue any further education on average. And you expect that to vary by about two workers on average. ### References American Cancer Society. (2013). What are the key statistics about pancreatic cancer? Retrieved from http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html ESPN NBA. (2013). NBA statistics – 2013. Retrieved from http://espn.go.com/nba/statistics/_/seasontype/2 Newport, F. (2013). Americans still enjoy saving rather than spending: Few demographic differences seen in these views other than by income. GALLUP Economy. Retrieved from http://www.gallup.com/poll/162368/americansenjoy-saving-rather-spending.aspx Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles, CA: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf Wikipedia. (n.d.). Distance education. Retrieved from http://en.wikipedia.org/wiki/Distance_education World Bank Group. (2013). Access to electricity (% of population). Retrieved from http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc ### Chapter Review A statistical experiment can be classified as a binomial experiment if the following conditions are met: 1. There are a fixed number of trials, n 2. There are only two possible outcomes, called success and failure, for each trial; the letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial 3. The n trials are independent and are repeated using identical conditions The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np, and the standard deviation is given by the formula σ = . ### Formula Review X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p. X = the number of successes in n independent trials n = the number of independent trials X takes on the values x = 0, 1, 2, 3, . . . , n p = the probability of a success for any trial q = the probability of a failure for any trial p + q = 1 q = 1 – p The mean of X is μ = np. The standard deviation of X is σ = . Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law. ### HOMEWORK Use the following information to answer the next four exercises: Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. Use the following information to answer the next three exercises: The probability that a local hockey team will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. Let X = the number of games won in that upcoming month.

# Discrete Random Variables ## Geometric Distribution (Optional) There are three main characteristics of a geometric experiment: 1. Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials n = 1. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a success so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success, stop. 2. In theory, the number of trials could go on forever. There must be at least one trial. 3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is . This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is q = , the probability of a failure. The probability of getting a three on the fifth roll is = .0804. X = the number of independent trials until the first success. p = the probability of a success, q = 1 – p = the probability of a failure. There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book. ### Notation for the Geometric: G = Geometric Probability Distribution Function X ~ G(p) Read this as X is a random variable with a . The parameter is p; p = the probability of a success for each trial. ### References Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html Pew Research Center. (n.d.). Millennials: A portrait of generation next. Retrieved from http://www.pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf Pew Research. (2013). Millennials: confident. Executive Summary: Pew Research Social & Demographic Trends. Retrieved from http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-tochange/ Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/ TheAmericanFreshman2011.pdf The European Union and ICON-Institute. (2007/8). Summary of the national risk and vulnerability assessment 2007/8: A profile of Afghanistan. Retrieved from http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf The World Bank. (2013). Prevalence of HIV, total (% of populations ages 15-49). Retrieved from http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc UNICEF Television. (n.d.). UNICEF reports on female literacy centers in Afghanistan established to teach women and girls basic reading and writing skills. (Video). Retrieved from http://www.unicefusa.org/assets/video/afghan-femaleliteracy-centers.html ### Chapter Review There are three characteristics of a geometric experiment: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success 2. In theory, the number of trials could go on forever; there must be at least one trial 3. The probability, p, of a success and the probability, q, of a failure are the same for each trial In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success. We say that X has a geometric distribution and write X ~ G(p) where p is the probability of success in a single trial. The mean of the geometric distribution X ~ G(p) is μ = = . ### Formula Review X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p. X = the number of independent trials until the first success X takes on the values x = 1, 2, 3, . . . p = the probability of a success for any trial q = the probability of a failure for any trial p + q = 1 : q = 1 – p The mean is μ = . The standard deviation is σ = = . Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask. ### HOMEWORK