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Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. All this is saying is, for each corresponding x ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find th... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data poin... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So in this particular situation, r is going to be equal to one over n minus one. We have four pairs, so it's gonna be one over three, and it's gonna be times a sum of the products of the z-scores. So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x samp... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
I'll do it like this. So the next one, it's going to be two minus two over 0.816, this is where I got the two from, and I'm subtracting from that the sample mean right over here, times, now we're looking at this two, two minus three over 2.160 plus, I'm happy there's only four pairs here, two minus two again, two minus... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so thi... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this. So we have one divided by three times five divided by 0.816 times 2.160, the zero won't make a difference, but I'll just write it down, and then I will close that parentheses, and let's ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be greater than or equal to negative one and less than or equal to one. If R is positive one, it means that an upward-sloping line can completely describe the relati... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to try and hand-draw a line here, and it does turn out that our least squares line will ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-sc... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just talked about, where an R of one will be strong positive correlation, R of negative one would be strong negative correlatio... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So the X sample mean is two. This is our X axis here. This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard deviations is this below the mean? And that turns out to be negative one over 0.816. That's what we have right over here. That's what this would have calculated. And then how many standard deviations for ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it,... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation bet... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the m... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going t... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point where, let's say X was b... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And we know a few other things about it. We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discre... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, ... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of ar... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certai... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a ... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is goin... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl f... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So gi... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value o... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
In this case, this would be equal to 20 ounces. But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the cas... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, tha... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variabl... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well,... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were ... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exer... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, s... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on. And what would be t... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
So this would be 10 right over here. That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, as... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know thi... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
Well, we know this. Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X s... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one. | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one. And that just happened to work out because we're dealing with the scenario where the variance, whe... | Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3 |
This random variable can take on values from one to five and has an equal probability of taking on any of these values from one to five. Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is ... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the ... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these wit... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be th... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
It is a normal distribution. They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's t... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that o... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard d... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
It's closer to 68.2%. For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know ... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure ou... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is g... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. And so that's going to be approximately 0.16. If you want a slightly more precise value, you could use a z-table. The area... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
If you want a slightly more precise value, you could use a z-table. The area below one standard deviation above the mean will be closer to about 84.1%, in which case this would be about 15.9%, or 0.159. But you can see that we got pretty close by knowing the general rule that it's roughly 68% between one standard devia... | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
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