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Or more than one car passes in a minute? The way we have it right now, we call it a success if one car passes in a minute. And if you're kind of counting, it counts as one success, even if five cars pass in that minute. And so you say, oh, OK, Sal. I know the solution there. I just have to get more granular. Instead of dividing it into minutes, why don't I divide it into seconds? | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
And so you say, oh, OK, Sal. I know the solution there. I just have to get more granular. Instead of dividing it into minutes, why don't I divide it into seconds? So the probability that I have k successes, instead of 60 intervals, I'll do 3,600 intervals. And so the probability of k successful seconds, so a second where a car is passing at that moment, out of 3,600 possible seconds, so that's 3,600, choose k, times the probability that a car passes in any given second. Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
Instead of dividing it into minutes, why don't I divide it into seconds? So the probability that I have k successes, instead of 60 intervals, I'll do 3,600 intervals. And so the probability of k successful seconds, so a second where a car is passing at that moment, out of 3,600 possible seconds, so that's 3,600, choose k, times the probability that a car passes in any given second. Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour. And we're going to have k successes, and then we're going to have, and these are the failures, the probability of failure, and you're going to have 3,600 minus k failures. And this would be even a better approximation. This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
Well, that's the probability, that's the expected number of cars in an hour divided by number of seconds in an hour. And we're going to have k successes, and then we're going to have, and these are the failures, the probability of failure, and you're going to have 3,600 minus k failures. And this would be even a better approximation. This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here. We just have to get more and more granular. We have to just make this number larger and larger and larger, and your intuition is correct. And if you do that, you'll end up getting the Poisson distribution. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
This actually would not be so bad, but still you have the situation where two cars can come within a half a second of each other, and you say, oh, OK, Sal, I see the pattern here. We just have to get more and more granular. We have to just make this number larger and larger and larger, and your intuition is correct. And if you do that, you'll end up getting the Poisson distribution. And this is really interesting, because a lot of times people give you the formula for the Poisson distribution, and you can kind of just plug in the numbers and use it, but it's neat to know that it really is just the binomial distribution, and the binomial distribution really did come from kind of the common sense of flipping coins. That's where everything is coming from. But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
And if you do that, you'll end up getting the Poisson distribution. And this is really interesting, because a lot of times people give you the formula for the Poisson distribution, and you can kind of just plug in the numbers and use it, but it's neat to know that it really is just the binomial distribution, and the binomial distribution really did come from kind of the common sense of flipping coins. That's where everything is coming from. But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt. So the first is something that you're probably reasonably familiar with by now, but I just want to make sure that the limit as x approaches infinity of 1 plus a over x to the x power is equal to e to the a x. No, sorry, is equal to e to the a. And just to prove this to you, let's make a little substitution here. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
But before we kind of prove that this, if we take the limit as, let me change colors, before we prove that as we take the limit as this number right here, the number of intervals approaches infinity, that this becomes the Poisson distribution, I'm going to make sure we have a couple of mathematical tools in our belt. So the first is something that you're probably reasonably familiar with by now, but I just want to make sure that the limit as x approaches infinity of 1 plus a over x to the x power is equal to e to the a x. No, sorry, is equal to e to the a. And just to prove this to you, let's make a little substitution here. Let's say that n is equal to, let me say 1 over n is equal to a over x, and then what would be x would be equal to n a, right, x times 1 is equal to n times a. And so the limit as x approaches infinity, what does a approach? a is, sorry, as x approaches infinity, what does n approach? | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
And just to prove this to you, let's make a little substitution here. Let's say that n is equal to, let me say 1 over n is equal to a over x, and then what would be x would be equal to n a, right, x times 1 is equal to n times a. And so the limit as x approaches infinity, what does a approach? a is, sorry, as x approaches infinity, what does n approach? Well, n is x divided by a, right? So n would also approach infinity. So this thing would be the same thing as just making our substitution. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
a is, sorry, as x approaches infinity, what does n approach? Well, n is x divided by a, right? So n would also approach infinity. So this thing would be the same thing as just making our substitution. The limit as n approaches infinity of 1 plus a over x, I mean the substitution is 1 over n, and x is, by this substitution, n times a. And this is going to be the same thing as the limit as n approaches infinity of 1 plus 1 over n to the n, all of that, to the a. And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
So this thing would be the same thing as just making our substitution. The limit as n approaches infinity of 1 plus a over x, I mean the substitution is 1 over n, and x is, by this substitution, n times a. And this is going to be the same thing as the limit as n approaches infinity of 1 plus 1 over n to the n, all of that, to the a. And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a. And this is our definition, or one of the ways to get to e, if you watch the videos on compound interests and all of that. This is how we got to e. And if you try it on your calculator, just try larger and larger n's here, and you'll get e. So this is equal to, this inner part is equal to e. So, and we raised it to the a power, so it's equal to e to the a. So hopefully you're pretty satisfied that this limit is equal to e to the a. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
And since there's no n out here, we could just take the limit of this and then take that to the a power, so that's going to be equal to the limit as n approaches infinity of 1 plus 1 over n to the nth power, all of that, to the a. And this is our definition, or one of the ways to get to e, if you watch the videos on compound interests and all of that. This is how we got to e. And if you try it on your calculator, just try larger and larger n's here, and you'll get e. So this is equal to, this inner part is equal to e. So, and we raised it to the a power, so it's equal to e to the a. So hopefully you're pretty satisfied that this limit is equal to e to the a. And then one other toolkit I want on our belt, and I'll probably actually do the proof in the next video, the other toolkit is to recognize that x factorial, let me do this, x factorial over x minus k factorial is equal to x times x minus 1 times x minus 2, all the way down to times x minus k plus 1. And we've done this a lot of times, but this is the most abstract we've ever written it. I can give you a couple of examples. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
So hopefully you're pretty satisfied that this limit is equal to e to the a. And then one other toolkit I want on our belt, and I'll probably actually do the proof in the next video, the other toolkit is to recognize that x factorial, let me do this, x factorial over x minus k factorial is equal to x times x minus 1 times x minus 2, all the way down to times x minus k plus 1. And we've done this a lot of times, but this is the most abstract we've ever written it. I can give you a couple of examples. And they'll be exactly, and just so you know, they'll be exactly k terms here. 1, 2, 3, this is the first term, second term, third term, all the way, and this is the kth term. And this is important to our derivation of the Poisson distribution. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
I can give you a couple of examples. And they'll be exactly, and just so you know, they'll be exactly k terms here. 1, 2, 3, this is the first term, second term, third term, all the way, and this is the kth term. And this is important to our derivation of the Poisson distribution. But just to make this in real numbers, if I had 7 factorial over 7 minus 2 factorial, that's equal to 7 times 6 times 5 times 4 times 3 times 2 times 1 over 2 times, no sorry, 7 minus 2, this is 5. So it's over 5 times 4 times 3 times 2 times 1. These cancel out, and you just have 7 times 6. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
And this is important to our derivation of the Poisson distribution. But just to make this in real numbers, if I had 7 factorial over 7 minus 2 factorial, that's equal to 7 times 6 times 5 times 4 times 3 times 2 times 1 over 2 times, no sorry, 7 minus 2, this is 5. So it's over 5 times 4 times 3 times 2 times 1. These cancel out, and you just have 7 times 6. And so it's 7, and then the last term is 7 minus 2 plus 1, which is 6. 7 minus 2 plus 1. And you had, in this example, k was 2, and you had exactly 2 terms. | Poisson process 1 Probability and Statistics Khan Academy.mp3 |
We will now begin our journey into the world of statistics, which is really a way to understand or get our head around data. So statistics is all about data. And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently. Anyway, I'll leave you there. | Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3 |
And maybe, well, I mean, if you go into almost any scientific field, you might even argue it's the single most important concept. And I've actually told people that it's kind of sad that they don't cover this in the core curriculum. Everyone should know about this, because it touches on every single aspect of our lives. And that's the normal distribution, or the Gaussian distribution, or the bell curve. And just to kind of give you a preview of what it is, and my preview will actually make it seem pretty strange, but as we go through this video, hopefully you'll get a little bit more intuition of what it's all about. But the Gaussian distribution, or the normal distribution, they're just two words for the same thing. It was actually Gauss who came up with it. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And that's the normal distribution, or the Gaussian distribution, or the bell curve. And just to kind of give you a preview of what it is, and my preview will actually make it seem pretty strange, but as we go through this video, hopefully you'll get a little bit more intuition of what it's all about. But the Gaussian distribution, or the normal distribution, they're just two words for the same thing. It was actually Gauss who came up with it. I think he was studying astronomical phenomenon when he did. But it's a probability density function, just like we studied the Poisson distribution. It's just like that. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
It was actually Gauss who came up with it. I think he was studying astronomical phenomenon when he did. But it's a probability density function, just like we studied the Poisson distribution. It's just like that. And just to give you the preview, it looks like this. The probability of getting any x, and it's a class of probability distribution functions. Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
It's just like that. And just to give you the preview, it looks like this. The probability of getting any x, and it's a class of probability distribution functions. Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters. But it's equal to, and this is how you would traditionally see it written in a lot of textbooks. And if we have time, I'd like to rearrange the algebra, just so you get a little bit more intuition of how it all works out. Or maybe we could get some insights on where it all came from. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Just like the binomial distribution is and the Poisson distribution is, based on a bunch of parameters. But it's equal to, and this is how you would traditionally see it written in a lot of textbooks. And if we have time, I'd like to rearrange the algebra, just so you get a little bit more intuition of how it all works out. Or maybe we could get some insights on where it all came from. I'm not going to prove it in this video. That's a little bit beyond our scope. Although I do want to do it. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Or maybe we could get some insights on where it all came from. I'm not going to prove it in this video. That's a little bit beyond our scope. Although I do want to do it. And there's actually some really neat mathematics that might show up in, if you're a math lead, there's something called Stirling's formula, which you might want to do a Wikipedia search on, which is really fascinating. It approximates factorials with essentially a continuous function, but I won't go into that right now. But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Although I do want to do it. And there's actually some really neat mathematics that might show up in, if you're a math lead, there's something called Stirling's formula, which you might want to do a Wikipedia search on, which is really fascinating. It approximates factorials with essentially a continuous function, but I won't go into that right now. But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half. Well, I like to write it this way. It's easier to remember. Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But the normal distribution is 1 over, this is how it's normally written, the standard deviation times the square root of 2 pi times e to the minus 1 half. Well, I like to write it this way. It's easier to remember. Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared. And so if you think about it, actually, this is a good thing to just notice right now. This is how far I'm from the mean, and we're dividing that by the standard deviation of whatever our distribution is. And this is a preview of actually a normal distribution that I've plotted. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution squared. And so if you think about it, actually, this is a good thing to just notice right now. This is how far I'm from the mean, and we're dividing that by the standard deviation of whatever our distribution is. And this is a preview of actually a normal distribution that I've plotted. The purple line right here is the normal distribution. And just so you know, the whole exercise here, I know I jump around a little bit, is to show you that the normal distribution is a good approximation for the binomial distribution and vice versa. If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And this is a preview of actually a normal distribution that I've plotted. The purple line right here is the normal distribution. And just so you know, the whole exercise here, I know I jump around a little bit, is to show you that the normal distribution is a good approximation for the binomial distribution and vice versa. If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting. Because we're saying how far are we away from the mean? We're dividing by the standard deviation. We're saying, so this whole term right here is how many standard deviations we are away from the mean. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If the binomial distribution, if you take enough trials in your binomial distribution, we'll touch on that in a second, but the intuition of this term right here I think is interesting. Because we're saying how far are we away from the mean? We're dividing by the standard deviation. We're saying, so this whole term right here is how many standard deviations we are away from the mean. And this is actually called a standard z-score. One thing I've found in statistics is there's a lot of words and a lot of definitions and they all sound very fancy. The standard z-score. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
We're saying, so this whole term right here is how many standard deviations we are away from the mean. And this is actually called a standard z-score. One thing I've found in statistics is there's a lot of words and a lot of definitions and they all sound very fancy. The standard z-score. But the underlying concept is pretty straightforward. Let's say I had a probability distribution and I get some x value that's out here and it's 3 and 1 half standard deviations away from the mean. Then it's standard z-score is 3 and 1 half. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
The standard z-score. But the underlying concept is pretty straightforward. Let's say I had a probability distribution and I get some x value that's out here and it's 3 and 1 half standard deviations away from the mean. Then it's standard z-score is 3 and 1 half. But anyway, let's focus on the purpose of this video. So that's what the normal distribution, I guess the probability density function for the normal distribution looks like. But how did it get there? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Then it's standard z-score is 3 and 1 half. But anyway, let's focus on the purpose of this video. So that's what the normal distribution, I guess the probability density function for the normal distribution looks like. But how did it get there? I guess even more importantly, by the end of this video you should at least feel comfortable that this is a good approximation for the binomial distribution if you take enough trials. And that's what's fascinating about the normal distribution is that if you have the sum, and I'll do a whole other video on the central limit theorem, but if you have the sum of many, many independent trials approaching infinity, that the distribution of those, even though the distribution of each of those trials might have been non-normal, but the distribution of the sum of all of those trials approaches the normal distribution. And I'll talk more about that later. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But how did it get there? I guess even more importantly, by the end of this video you should at least feel comfortable that this is a good approximation for the binomial distribution if you take enough trials. And that's what's fascinating about the normal distribution is that if you have the sum, and I'll do a whole other video on the central limit theorem, but if you have the sum of many, many independent trials approaching infinity, that the distribution of those, even though the distribution of each of those trials might have been non-normal, but the distribution of the sum of all of those trials approaches the normal distribution. And I'll talk more about that later. But that's why it's such a good distribution to kind of assume for a lot of underlying phenomenon if you're kind of modeling weather patterns or drug interactions. And we'll talk about where it might work well and where it might not work so well. Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And I'll talk more about that later. But that's why it's such a good distribution to kind of assume for a lot of underlying phenomenon if you're kind of modeling weather patterns or drug interactions. And we'll talk about where it might work well and where it might not work so well. Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up. But anyway, let's get back to this. And this is a spreadsheet right here. I just made a black background. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Like sometimes people might assume things like a normal distribution in finance and we see in the financial crisis that's led to a lot of things blowing up. But anyway, let's get back to this. And this is a spreadsheet right here. I just made a black background. And you can download it at, let me write it right here, at khanacademy.org slash downloads. And actually if you just do that, you'll see all of the downloads, I haven't put it there yet, I'm going to do it right after I record the video. This downloads slash normal distribution. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I just made a black background. And you can download it at, let me write it right here, at khanacademy.org slash downloads. And actually if you just do that, you'll see all of the downloads, I haven't put it there yet, I'm going to do it right after I record the video. This downloads slash normal distribution. Normal distribution. That's distribution.xls. If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
This downloads slash normal distribution. Normal distribution. That's distribution.xls. If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there. And you'll see that this spreadsheet. And I encourage you to play with it. And maybe do other spreadsheets where you experiment with it. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If you just go up to khanacademy.org slash downloads slash, you'll see all of the things there. And you'll see that this spreadsheet. And I encourage you to play with it. And maybe do other spreadsheets where you experiment with it. So this spreadsheet, what we do is we're doing a game where let's say I'm sitting, I'm on a street, and I flip a coin. I flip a completely fair coin. If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And maybe do other spreadsheets where you experiment with it. So this spreadsheet, what we do is we're doing a game where let's say I'm sitting, I'm on a street, and I flip a coin. I flip a completely fair coin. If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left. And if I get a tails, I take a step to the right. So in general, I always have a 50, this is a completely fair coin. I have a 50% chance of taking a step to the left. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left. And if I get a tails, I take a step to the right. So in general, I always have a 50, this is a completely fair coin. I have a 50% chance of taking a step to the left. And I have a 50% chance of taking a step to the right. So your intuition there is, if I told you I took 1,000 flips of the coin, you're going to keep going left and right. If by chance you get a bunch of heads, you might end up really kind of moving over to the left. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I have a 50% chance of taking a step to the left. And I have a 50% chance of taking a step to the right. So your intuition there is, if I told you I took 1,000 flips of the coin, you're going to keep going left and right. If by chance you get a bunch of heads, you might end up really kind of moving over to the left. If you get a bunch of tails, you might move over to the right. And we learned already that the odds of getting a bunch of tails, or many, many, many, many more tails than heads, is a lot lower than things kind of being equal or close to equal. And right here, what I've done, this is the mean number of left steps. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If by chance you get a bunch of heads, you might end up really kind of moving over to the left. If you get a bunch of tails, you might move over to the right. And we learned already that the odds of getting a bunch of tails, or many, many, many, many more tails than heads, is a lot lower than things kind of being equal or close to equal. And right here, what I've done, this is the mean number of left steps. And all I did is I got the probability, and we figured out the mean of the binomial distribution. The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials. And so that's equal to 5. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And right here, what I've done, this is the mean number of left steps. And all I did is I got the probability, and we figured out the mean of the binomial distribution. The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials. And so that's equal to 5. That's where that number comes from. And then the variance, and I'm not sure if I went over this. I need to prove this to you if I have. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And so that's equal to 5. That's where that number comes from. And then the variance, and I'm not sure if I went over this. I need to prove this to you if I have. And I'll make a whole other video on the variance of the binomial distribution. But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial. That's what I'm defining left as a successful trial, but could be right as well. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I need to prove this to you if I have. And I'll make a whole other video on the variance of the binomial distribution. But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial. That's what I'm defining left as a successful trial, but could be right as well. Times the probability of 1 minus a successful trial, or non-successful trial. In this case, they're equally probable, and that's where I got the 2.5 from. And that's all in the spreadsheet. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
That's what I'm defining left as a successful trial, but could be right as well. Times the probability of 1 minus a successful trial, or non-successful trial. In this case, they're equally probable, and that's where I got the 2.5 from. And that's all in the spreadsheet. If you actually click on the cell and look at the actual formula, I did that. Although sometimes when you see it in Excel, it's a little bit confusing. And this is just the square root of that number, right? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And that's all in the spreadsheet. If you actually click on the cell and look at the actual formula, I did that. Although sometimes when you see it in Excel, it's a little bit confusing. And this is just the square root of that number, right? The standard deviation is just the square root of the variance. So that's just the square root of 2.5. And so if you look here, this says, OK, what is the probability that I take 0 steps? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And this is just the square root of that number, right? The standard deviation is just the square root of the variance. So that's just the square root of 2.5. And so if you look here, this says, OK, what is the probability that I take 0 steps? So I take a total of 10 steps, just to understand the spreadsheet. So if I take a total of 10 steps, what is the probability that I take 0 left steps? Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And so if you look here, this says, OK, what is the probability that I take 0 steps? So I take a total of 10 steps, just to understand the spreadsheet. So if I take a total of 10 steps, what is the probability that I take 0 left steps? Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps. And I calculate this probability, and I should have drawn maybe a line here. I calculate this using the binomial distribution. And how do I do that? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
Well, and just to clarify, if I take 0 left steps, that means I must have taken 10 right steps. And I calculate this probability, and I should have drawn maybe a line here. I calculate this using the binomial distribution. And how do I do that? I say, what is the probability that I take a total of 10 steps, so let me actually switch colors just to make things interesting. Let's see, do they have a purple here? I'll do a blue. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And how do I do that? I say, what is the probability that I take a total of 10 steps, so let me actually switch colors just to make things interesting. Let's see, do they have a purple here? I'll do a blue. So blue for binomial. So what I have here is, how many total steps? So there's a total of 10 steps, so 10 factorial. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I'll do a blue. So blue for binomial. So what I have here is, how many total steps? So there's a total of 10 steps, so 10 factorial. That's kind of the number of trials I have. Of that, I'm choosing 0 to go left. So 0 factorial divided by 10 minus 0 factorial, right? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So there's a total of 10 steps, so 10 factorial. That's kind of the number of trials I have. Of that, I'm choosing 0 to go left. So 0 factorial divided by 10 minus 0 factorial, right? This is 10 choose 0. I'm choosing 0 left steps of the total 10 steps I'm taking times the probability of 0 left steps. So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So 0 factorial divided by 10 minus 0 factorial, right? This is 10 choose 0. I'm choosing 0 left steps of the total 10 steps I'm taking times the probability of 0 left steps. So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those. So that's where this number came from, this 0.001. That's what the binomial distribution tells us. And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So it's the probability of a left step, I'm only taking 0 of them, times the probability of a right step, and I'm taking 10 of those. So that's where this number came from, this 0.001. That's what the binomial distribution tells us. And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one. And once again, if you click on the actual cell, you'll see that explained. So we've done this multiple times, it's just the binomial calculation. And then right here, after this line right here, you can almost ignore it. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And then this one, similarly, is 10 factorial over 1 factorial over 10 minus 1 factorial, and that's how I get that one. And once again, if you click on the actual cell, you'll see that explained. So we've done this multiple times, it's just the binomial calculation. And then right here, after this line right here, you can almost ignore it. And I did that so that I can do a bunch of different scenarios. So for example, if I were to go to my spreadsheet, and instead of doing 10 steps, I wanted to do 20 steps, then everything changes. And that's why down here, after you get to a certain point, the whole thing just repeats. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And then right here, after this line right here, you can almost ignore it. And I did that so that I can do a bunch of different scenarios. So for example, if I were to go to my spreadsheet, and instead of doing 10 steps, I wanted to do 20 steps, then everything changes. And that's why down here, after you get to a certain point, the whole thing just repeats. And I'll let you think about why I do that. Maybe I should have made a cleaner spreadsheet. But it doesn't affect the scatter plot chart that I did. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And that's why down here, after you get to a certain point, the whole thing just repeats. And I'll let you think about why I do that. Maybe I should have made a cleaner spreadsheet. But it doesn't affect the scatter plot chart that I did. And so this plot in blue, and you can't see it because the purple is almost right over it. Actually, let me make it smaller so that you can see. So if I only took 6 steps, well, it's still hard to see the difference between the two. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
But it doesn't affect the scatter plot chart that I did. And so this plot in blue, and you can't see it because the purple is almost right over it. Actually, let me make it smaller so that you can see. So if I only took 6 steps, well, it's still hard to see the difference between the two. And then once again, the whole point of this is to see that the normal distribution is a good approximation, but they're so close that you can't even see the difference on mine. If you only took 4 steps, OK, I think you can see here, the blue here is definitely, let me get my screen drawer on. So let me draw this. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So if I only took 6 steps, well, it's still hard to see the difference between the two. And then once again, the whole point of this is to see that the normal distribution is a good approximation, but they're so close that you can't even see the difference on mine. If you only took 4 steps, OK, I think you can see here, the blue here is definitely, let me get my screen drawer on. So let me draw this. The blue curve is right around there. So this is the binomial. There's only a few points here. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So let me draw this. The blue curve is right around there. So this is the binomial. There's only a few points here. The points only go up to here. This is if I take 0 steps left, 1 step left, 2 steps left, 3 steps left, 4 steps left, and then I plot it. And then I say, what's the probability using the binomial distribution? | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
There's only a few points here. The points only go up to here. This is if I take 0 steps left, 1 step left, 2 steps left, 3 steps left, 4 steps left, and then I plot it. And then I say, what's the probability using the binomial distribution? And then this is my final position. If I take 0 steps to the left, then I take 4 steps to the right, so my final position is at 4. So that's this scenario right here. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
And then I say, what's the probability using the binomial distribution? And then this is my final position. If I take 0 steps to the left, then I take 4 steps to the right, so my final position is at 4. So that's this scenario right here. Let me switch my color back to yellow. It's easier to see. If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
So that's this scenario right here. Let me switch my color back to yellow. It's easier to see. If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4. It's going to be here. If I take an equal amount of both, so that's this scenario, then I'm neutral. I'm just stuck in the middle right here. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
If I take 4 steps to the left, I take 0 steps to the right, and so my final position is going to be at minus 4. It's going to be here. If I take an equal amount of both, so that's this scenario, then I'm neutral. I'm just stuck in the middle right here. I take 2 steps to the right, and then I take 2 steps to the left, or vice versa. I take 2 steps to the left, and then I take 2 steps right, and I end up right there. Hopefully that makes a little sense of how this is going to seem. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
I'm just stuck in the middle right here. I take 2 steps to the right, and then I take 2 steps to the left, or vice versa. I take 2 steps to the left, and then I take 2 steps right, and I end up right there. Hopefully that makes a little sense of how this is going to seem. My phone is ringing. No, I'll ignore that. Because the normal distribution is so important. | Normal distribution excel exercise Probability and Statistics Khan Academy.mp3 |
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