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So let me write that down. This is equal to 178 centimeters minus 170 centimeters, which is going to be equal to, I'll do it in this color, this is going to be equal to eight centimeters. So this is eight right over here. Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64. And then we have six squared. This right over here is six. Six squared is going to be 36. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So eight squared is going to be 64. And then we have six squared. This right over here is six. Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution? | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man? | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1⁄10. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1⁄10. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function. So that is choice two. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function. So that is choice two. And then the lower bound. Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So that is choice two. And then the lower bound. Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero. From negative infinity to zero, where the mean is eight and the standard deviation is 10. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero. From negative infinity to zero, where the mean is eight and the standard deviation is 10. We press enter, and we get approximately 0.212, is approximately 0.212. Or you could say, what is the probability that the woman is taller than the man? Well, 0.212, or approximately, there's a 21.2% chance of that happening, a little better than one in five. | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So you need to shake the hand exactly once of every other person in the room so that you all meet. So my question to you is, if each of these people need to shake the hand of every other person exactly once, how many handshakes are going to occur? The number of handshakes that are going to occur. So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two? | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand. A could shake D's hand. Or B could shake, and I may just do what we calculated first, the 12, B could shake A's hand. B could shake C's hand. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
A could shake C's hand. A could shake D's hand. Or B could shake, and I may just do what we calculated first, the 12, B could shake A's hand. B could shake C's hand. B could shake, whoops. B could shake D's hand. And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
B could shake C's hand. B could shake, whoops. B could shake D's hand. And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand. and you could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here, and this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And you can say C could shake A's hand, C could shake B's hand, C could shake, C could shake D's hand. and you could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here, and this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, A, B is the same thing as B, A. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, A, B is the same thing as B, A. A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So, A, B is the same thing as B, A. A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B. B, D is the same thing as D, B. C, D is the same thing as D, C. And so we'd be left with, if we correct for the double counting, we're left with one, two, three, four, five, six combinations, six possible ways of choosing two from a pool of four, especially when you don't care about the order in which you choose them. Thank you. | Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
It can only take on a finite number of values, and I defined it as the number of workouts I might do in a week. And we calculated the expected value of our random variable X, which you could also denote as the mean of X, and we used the Greek letter mu, which we use for population mean. And all we did is it's the probability weighted sum of the various outcomes. And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09. So let's see if this makes sense. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09. So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four. So it'll look like that. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four. So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09. So 1.09 above the mean is going to get us close to 3.2. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09. So 1.09 above the mean is going to get us close to 3.2. And 1.09 below the mean is gonna get us close to one. And so this all, at least intuitively, feels reasonable. This mean does seem to be indicative of the central tendency of this distribution. | Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3 |
We have four different gum chewers, and they tell us how many bubbles each of them blew. So what I want to do is I want to figure out first the mean of the number of bubbles blown, and then also figure out how dispersed is the data. How much do these vary from the mean? And I'm going to do that by calculating the mean absolute deviation. So pause this video now. Try to calculate the mean of the number of bubbles blown. And then after you do that, see if you can calculate the mean absolute deviation. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And I'm going to do that by calculating the mean absolute deviation. So pause this video now. Try to calculate the mean of the number of bubbles blown. And then after you do that, see if you can calculate the mean absolute deviation. All right, so step one, let's figure out the mean. So the mean is just going to be the sum of the number of bubbles blown divided by the number of data points. So Manuela blew four bubbles. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And then after you do that, see if you can calculate the mean absolute deviation. All right, so step one, let's figure out the mean. So the mean is just going to be the sum of the number of bubbles blown divided by the number of data points. So Manuela blew four bubbles. So she blew four bubbles. Sophia blew five bubbles. Jada blew six bubbles. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So Manuela blew four bubbles. So she blew four bubbles. Sophia blew five bubbles. Jada blew six bubbles. And Tara blew one bubble. And we have one, two, three, four data points. So let's divide by four. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Jada blew six bubbles. And Tara blew one bubble. And we have one, two, three, four data points. So let's divide by four. And so this is going to be equal to 4 plus 5 is 9, plus 6 is 15, plus 1 is 16. So it's equal to 16 over 4, which is 16 divided by 4 is equal to 4. So the mean number of bubbles blown is 4. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So let's divide by four. And so this is going to be equal to 4 plus 5 is 9, plus 6 is 15, plus 1 is 16. So it's equal to 16 over 4, which is 16 divided by 4 is equal to 4. So the mean number of bubbles blown is 4. And I can do that. Let me actually do this with a bold line right over here. This is the mean number of bubbles blown. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So the mean number of bubbles blown is 4. And I can do that. Let me actually do this with a bold line right over here. This is the mean number of bubbles blown. So now what I want to do is I want to figure out the mean absolute deviation. I'll do it right over here. Mad. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
This is the mean number of bubbles blown. So now what I want to do is I want to figure out the mean absolute deviation. I'll do it right over here. Mad. Mean absolute deviation. And what we want to do is we want to take the mean of how much do each of these data points deviate from the mean. And I know I just used the word mean twice in a sentence, so it might be a little confusing. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Mad. Mean absolute deviation. And what we want to do is we want to take the mean of how much do each of these data points deviate from the mean. And I know I just used the word mean twice in a sentence, so it might be a little confusing. But as we work through it, hopefully it'll make a little bit of sense. So how much does Manuela's, the number of bubbles she blew, how much does that deviate from the mean? Well, Manuela actually blew four bubbles, and four is the mean. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And I know I just used the word mean twice in a sentence, so it might be a little confusing. But as we work through it, hopefully it'll make a little bit of sense. So how much does Manuela's, the number of bubbles she blew, how much does that deviate from the mean? Well, Manuela actually blew four bubbles, and four is the mean. So her deviation, her absolute deviation from the mean is 0. How much did, actually let me just write this over here. So absolute deviation. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Well, Manuela actually blew four bubbles, and four is the mean. So her deviation, her absolute deviation from the mean is 0. How much did, actually let me just write this over here. So absolute deviation. That's AD. Absolute deviation from the mean. Manuela didn't deviate at all from the mean. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So absolute deviation. That's AD. Absolute deviation from the mean. Manuela didn't deviate at all from the mean. Now let's think about Sophia. Sophia deviates by 1 from the mean. We see that right there. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Manuela didn't deviate at all from the mean. Now let's think about Sophia. Sophia deviates by 1 from the mean. We see that right there. She's 1 above. Now we would say 1 whether it's 1 above or below, because we're saying absolute deviation. So Sophia deviates by 1. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
We see that right there. She's 1 above. Now we would say 1 whether it's 1 above or below, because we're saying absolute deviation. So Sophia deviates by 1. Her absolute deviation is 1. And then we have Jada. How much does she deviate from the mean? | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So Sophia deviates by 1. Her absolute deviation is 1. And then we have Jada. How much does she deviate from the mean? We see it right over here. She deviates by 2. She is 2 more than the mean. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
How much does she deviate from the mean? We see it right over here. She deviates by 2. She is 2 more than the mean. And then how much does Tara deviate from the mean? She is at 1, so that is 3 below the mean. So once again, this was 2. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
She is 2 more than the mean. And then how much does Tara deviate from the mean? She is at 1, so that is 3 below the mean. So once again, this was 2. This is 3. So she deviates. Her absolute deviation is 3. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So once again, this was 2. This is 3. So she deviates. Her absolute deviation is 3. And then we want to take the mean of the absolute deviations. That's the M in MAD, in mean absolute deviation. This is Manuela's absolute deviation. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Her absolute deviation is 3. And then we want to take the mean of the absolute deviations. That's the M in MAD, in mean absolute deviation. This is Manuela's absolute deviation. Sophia's absolute deviation. Jada's absolute deviation. Tara's absolute deviation. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
This is Manuela's absolute deviation. Sophia's absolute deviation. Jada's absolute deviation. Tara's absolute deviation. We want the mean of those. So we divide by the number of data points. And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Tara's absolute deviation. We want the mean of those. So we divide by the number of data points. And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half. Or let me just write it in all the different ways. We could write it as 3 halves or 1 and 1 half or 1.5, which gives us a measure of how much do these data points vary from the mean of 4. Now I know what some of you are thinking. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And we get 0 plus 1 plus 2 plus 3 is 6 over 4, which is the same thing as 1 and 1 half. Or let me just write it in all the different ways. We could write it as 3 halves or 1 and 1 half or 1.5, which gives us a measure of how much do these data points vary from the mean of 4. Now I know what some of you are thinking. Wait, I thought there was a formula associated with the mean absolute deviation. And it seems really complex. It has all of these absolute value signs and whatever else. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Now I know what some of you are thinking. Wait, I thought there was a formula associated with the mean absolute deviation. And it seems really complex. It has all of these absolute value signs and whatever else. Well, that's all we did. That was just a, when we write all those absolute value signs, that's just a fancy way of looking at each data point and thinking about, well, how much does it deviate from the mean, whether it's above or below? That's what the absolute value does. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
It has all of these absolute value signs and whatever else. Well, that's all we did. That was just a, when we write all those absolute value signs, that's just a fancy way of looking at each data point and thinking about, well, how much does it deviate from the mean, whether it's above or below? That's what the absolute value does. It doesn't matter if it's 3 below. We just say 3. If it's 2 above, we just say 2. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
That's what the absolute value does. It doesn't matter if it's 3 below. We just say 3. If it's 2 above, we just say 2. We don't put a positive or negative on it. But just so you're comfortable seeing how this is the exact same thing you would have done with the formula, let's do it that way as well. So the mean absolute deviation is going to be equal to, well, we'll start with Manuela. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
If it's 2 above, we just say 2. We don't put a positive or negative on it. But just so you're comfortable seeing how this is the exact same thing you would have done with the formula, let's do it that way as well. So the mean absolute deviation is going to be equal to, well, we'll start with Manuela. How many bubbles did she blow? She blew 4. From that, you subtract the mean of 4. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So the mean absolute deviation is going to be equal to, well, we'll start with Manuela. How many bubbles did she blow? She blew 4. From that, you subtract the mean of 4. Take the absolute value. That's her absolute deviation. And of course, this does evaluate to this 0 here. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
From that, you subtract the mean of 4. Take the absolute value. That's her absolute deviation. And of course, this does evaluate to this 0 here. Then you take the absolute value. Sophia blew 5 bubbles, and the mean is 4. Then you do that for Jada. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And of course, this does evaluate to this 0 here. Then you take the absolute value. Sophia blew 5 bubbles, and the mean is 4. Then you do that for Jada. Jada blew 6 bubbles. The mean is 4. And then you do it for Tara. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
Then you do that for Jada. Jada blew 6 bubbles. The mean is 4. And then you do it for Tara. Tara blew 1 bubble, and the mean is 4. And then you divide it by the number of data points you have. And let me make it very clear. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And then you do it for Tara. Tara blew 1 bubble, and the mean is 4. And then you divide it by the number of data points you have. And let me make it very clear. This right over here, this 4 is the mean. This 4 is the mean. So you're taking each of the data points, and you're seeing how far it is away from the mean. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
And let me make it very clear. This right over here, this 4 is the mean. This 4 is the mean. So you're taking each of the data points, and you're seeing how far it is away from the mean. You're taking the absolute value, because you just want to figure out the absolute distance. Now you see, or maybe you see, 4 minus 4, this is, let me do this in a different color, 4 minus 4, that is the 0. That is that 0 right over there. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So you're taking each of the data points, and you're seeing how far it is away from the mean. You're taking the absolute value, because you just want to figure out the absolute distance. Now you see, or maybe you see, 4 minus 4, this is, let me do this in a different color, 4 minus 4, that is the 0. That is that 0 right over there. 5 minus 4, absolute value of that, well, that's going to be, let me do this in a new color, this is just going to be 1. This thing is the same thing as that over there. And we were able to see that just by inspecting this graph or this chart. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
That is that 0 right over there. 5 minus 4, absolute value of that, well, that's going to be, let me do this in a new color, this is just going to be 1. This thing is the same thing as that over there. And we were able to see that just by inspecting this graph or this chart. And then 6 minus 4, absolute value of that, that's just going to be 2. That 2 is that 2 right over here, which is the same thing as this 2 right over there. And then finally, our 1 minus 4, that's negative 3, but the absolute value of that is just positive 3, which is this positive 3 right over there, which is the distance, this distance right over here. | Mean absolute deviation example Data and statistics 6th grade Khan Academy.mp3 |
So let's start again with a fair coin. And this time, instead of flipping it four times, let's flip it five times. So five flips of this fair coin. And what I want to think about in this video is the probability of getting exactly three heads. And the way I'm going to think about it is, if you have five flips, how many different equally likely possibilities are there? So you're going to have the first flip. Let me draw it over here. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
And what I want to think about in this video is the probability of getting exactly three heads. And the way I'm going to think about it is, if you have five flips, how many different equally likely possibilities are there? So you're going to have the first flip. Let me draw it over here. First flip, and there's two possibilities there. It could be heads or tails. Second flip, two possibilities there. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Let me draw it over here. First flip, and there's two possibilities there. It could be heads or tails. Second flip, two possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Second flip, two possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities. So it's 2 times 2 times 2 times 2 times 2. I hope I said that five times. So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
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