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That's our measure of dispersion there. And let's compare it to this data set over here. Let's compare it to the variance of this less dispersed data set. So let me scroll over a little bit. So we have some real estate, although I'm running out. Maybe I could scroll up here. There you go. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So let me scroll over a little bit. So we have some real estate, although I'm running out. Maybe I could scroll up here. There you go. So let me calculate the variance of this data set. So we already know its mean. So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
There you go. So let me calculate the variance of this data set. So we already know its mean. So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared. Remember that 10 is just the mean that we calculated. You have to calculate the mean first. Divided by, we have 1, 2, 3, 4, 5 squared differences. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So its variance of this data set is going to be equal to 8 minus 10 squared plus 9 minus 10 squared plus 10 minus 10 squared plus 11 minus 10 squared plus 12 minus 10 squared. Remember that 10 is just the mean that we calculated. You have to calculate the mean first. Divided by, we have 1, 2, 3, 4, 5 squared differences. So this is going to be equal to 8 minus 10 is negative 2 squared is positive 4. 9 minus 10 is negative 1 squared is positive 1. 10 minus 10 is 0 squared. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Divided by, we have 1, 2, 3, 4, 5 squared differences. So this is going to be equal to 8 minus 10 is negative 2 squared is positive 4. 9 minus 10 is negative 1 squared is positive 1. 10 minus 10 is 0 squared. You still get 0. 11 minus 10 is 1 squared. You get 1. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
10 minus 10 is 0 squared. You still get 0. 11 minus 10 is 1 squared. You get 1. 12 minus 10 is 2 squared. You get 4. Now what is this equal to? | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
You get 1. 12 minus 10 is 2 squared. You get 4. Now what is this equal to? All of that over 5. This is 10 over 5. So this is going to be, all right, this is 10 over 5, which is equal to 2. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Now what is this equal to? All of that over 5. This is 10 over 5. So this is going to be, all right, this is 10 over 5, which is equal to 2. So the variance here, let me make sure I got that right. Yes, we have 10 over 5. So the variance of this less dispersed data set is a lot smaller, the variance here. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So this is going to be, all right, this is 10 over 5, which is equal to 2. So the variance here, let me make sure I got that right. Yes, we have 10 over 5. So the variance of this less dispersed data set is a lot smaller, the variance here. The variance of this data set right here is only 2. So that gave you a sense. That tells you, look, this is definitely a less dispersed data set than that there. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So the variance of this less dispersed data set is a lot smaller, the variance here. The variance of this data set right here is only 2. So that gave you a sense. That tells you, look, this is definitely a less dispersed data set than that there. Now the problem with the variance is you're taking these numbers, you're taking the difference between them and the mean, then you're squaring it. It kind of gives you a bit of an arbitrary number. And if you're dealing with units, let's say if these are each negative, well, let's say they're distances. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
That tells you, look, this is definitely a less dispersed data set than that there. Now the problem with the variance is you're taking these numbers, you're taking the difference between them and the mean, then you're squaring it. It kind of gives you a bit of an arbitrary number. And if you're dealing with units, let's say if these are each negative, well, let's say they're distances. So this is negative 10 meters, 0 meters, 10 meters, this is 8 meters, so on and so forth. Then when you square it, you get your variance in terms of meters squared. It's kind of an odd set of units. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And if you're dealing with units, let's say if these are each negative, well, let's say they're distances. So this is negative 10 meters, 0 meters, 10 meters, this is 8 meters, so on and so forth. Then when you square it, you get your variance in terms of meters squared. It's kind of an odd set of units. So what people like to do is talk in terms of standard deviation. Which is just the square root of the variance. It's just the square root of the variance. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
It's kind of an odd set of units. So what people like to do is talk in terms of standard deviation. Which is just the square root of the variance. It's just the square root of the variance. Or the square root of sigma squared. And the symbol for the standard deviation is just sigma. So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
It's just the square root of the variance. Or the square root of sigma squared. And the symbol for the standard deviation is just sigma. So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters. The standard deviation of this first one up here, of this first data set, is going to be the square root of 200. Square root of 200 is what? The square root of 2 times 100, this is equal to 10 square roots of 2. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So now that we've figured out the variance, it's very easy to figure out the standard deviation of both of these characters. The standard deviation of this first one up here, of this first data set, is going to be the square root of 200. Square root of 200 is what? The square root of 2 times 100, this is equal to 10 square roots of 2. That's that first data set. Now the variance of the second data set is just going to be the square root of 2. Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
The square root of 2 times 100, this is equal to 10 square roots of 2. That's that first data set. Now the variance of the second data set is just going to be the square root of 2. Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2. Which is just 2. So the second data set has 1 tenth the standard deviation as this first data set. This is 10 roots of 2, this is just the root of 2. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Sorry, the standard deviation of the second data set is going to be the square root of its variance, which is 2. Which is just 2. So the second data set has 1 tenth the standard deviation as this first data set. This is 10 roots of 2, this is just the root of 2. So this is 10 times the standard deviation. And this, hopefully, will make a little bit more sense. Let's think about it. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
This is 10 roots of 2, this is just the root of 2. So this is 10 times the standard deviation. And this, hopefully, will make a little bit more sense. Let's think about it. This has 10 times more the standard deviation than this. And let's remember how we calculated it. Variance, we just took each data point, how far away from the mean, squared that, took the average of those. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Let's think about it. This has 10 times more the standard deviation than this. And let's remember how we calculated it. Variance, we just took each data point, how far away from the mean, squared that, took the average of those. Then we took the square root, really just to make the units look nice, but the end result is we said that that first data set has 10 times the standard deviation. As the second data set. So let's look at the two data sets. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Variance, we just took each data point, how far away from the mean, squared that, took the average of those. Then we took the square root, really just to make the units look nice, but the end result is we said that that first data set has 10 times the standard deviation. As the second data set. So let's look at the two data sets. This has 10 times the standard deviation. Which makes sense intuitively. I mean, they both have a 10 in here, but each of these guys, 9 is only 1 away from the 10, 0 is 1 away from the 10. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So let's look at the two data sets. This has 10 times the standard deviation. Which makes sense intuitively. I mean, they both have a 10 in here, but each of these guys, 9 is only 1 away from the 10, 0 is 1 away from the 10. So the standard deviation is 10 times the standard deviation. So the standard deviation, at least in my sense, is giving a much better sense of how far away, on average, we are from the mean. Anyway, hopefully you found that useful. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Or if we want to speak generally, into m different groups. What I want to do in this video is to figure out how much of this total sum of squares, how much of this is due to variation within each group versus variation between the actual groups. So first, let's figure out the total variation within the group. So let's call that the sum of squares within. So let's calculate the sum of squares within. And I'll do that in yellow. I actually already used yellow. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So let's call that the sum of squares within. So let's calculate the sum of squares within. And I'll do that in yellow. I actually already used yellow. So let's do this. Let me do blue. So the sum of squares within. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
I actually already used yellow. So let's do this. Let me do blue. So the sum of squares within. Let me make it clear. That stands for within. So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So the sum of squares within. Let me make it clear. That stands for within. So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means. So this is going to be equal to, let's start with these guys. So instead of taking the distance between each data point and the mean of means, I'm going to find the distance between each data point and that group's mean. Because we want to square the total sum of squares between each data point and their respective means. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So we want to see how much of the variation is due to how far each of these data points are from their central tendency, from their respective means. So this is going to be equal to, let's start with these guys. So instead of taking the distance between each data point and the mean of means, I'm going to find the distance between each data point and that group's mean. Because we want to square the total sum of squares between each data point and their respective means. So let's do that. So it's 3 minus, the mean here is 2 squared, plus 2 minus 2 squared, plus 1 minus 2 squared. Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Because we want to square the total sum of squares between each data point and their respective means. So let's do that. So it's 3 minus, the mean here is 2 squared, plus 2 minus 2 squared, plus 1 minus 2 squared. Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean. So plus 5 minus 4 squared. Plus 4 minus 4 squared. And then finally we have the third group. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Plus, I'm going to do this for all of the groups, but for each group, the distance between each data point and its mean. So plus 5 minus 4 squared. Plus 4 minus 4 squared. And then finally we have the third group. We're finding all of the sum of squares from each point to its central tendency within that. But we're going to add them all up. And then we find the third group. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And then finally we have the third group. We're finding all of the sum of squares from each point to its central tendency within that. But we're going to add them all up. And then we find the third group. So we have 5 minus 6 squared, plus 6 minus 6 squared, plus 7 minus 6 squared. And what is this going to equal? So this is going to be equal to, up here it's going to be 1 plus 0 plus 1. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And then we find the third group. So we have 5 minus 6 squared, plus 6 minus 6 squared, plus 7 minus 6 squared. And what is this going to equal? So this is going to be equal to, up here it's going to be 1 plus 0 plus 1. So that's going to be equal to 2 plus, and this is going to be equal to 1 plus 1 plus 0, so another 2. Plus, this is going to be equal to 1 plus 0 plus 1. 7 minus 6 is 1, squared is 1. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So this is going to be equal to, up here it's going to be 1 plus 0 plus 1. So that's going to be equal to 2 plus, and this is going to be equal to 1 plus 1 plus 0, so another 2. Plus, this is going to be equal to 1 plus 0 plus 1. 7 minus 6 is 1, squared is 1. So that's 2 over here. So this is going to be equal to, our sum of squares within, I should say, is 6. So one way to think about it, our total variation was 30. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
7 minus 6 is 1, squared is 1. So that's 2 over here. So this is going to be equal to, our sum of squares within, I should say, is 6. So one way to think about it, our total variation was 30. And based on this calculation, 6 of that 30 comes from variation within these samples. Now the next thing I want to think about is, how many degrees of freedom do we have in this calculation? How many kind of independent data points do we actually have? | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So one way to think about it, our total variation was 30. And based on this calculation, 6 of that 30 comes from variation within these samples. Now the next thing I want to think about is, how many degrees of freedom do we have in this calculation? How many kind of independent data points do we actually have? Well, for each of these, so over here, if you know, we have n data points in each one. In particular, n is 3 here. But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
How many kind of independent data points do we actually have? Well, for each of these, so over here, if you know, we have n data points in each one. In particular, n is 3 here. But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean. So in this case, for any of these groups, if you know 2 of these data points, you can always figure out the third. If you know these 2, you can always figure out the third if you know the sample mean. So in general, let's figure out the degrees of freedom here. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
But if you know n minus 1 of them, you can always figure out the nth one if you know the actual sample mean. So in this case, for any of these groups, if you know 2 of these data points, you can always figure out the third. If you know these 2, you can always figure out the third if you know the sample mean. So in general, let's figure out the degrees of freedom here. You have, for each group, when you did this, you had n minus 1 degrees of freedom. Remember, n is the number of data points you had in each group. So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So in general, let's figure out the degrees of freedom here. You have, for each group, when you did this, you had n minus 1 degrees of freedom. Remember, n is the number of data points you had in each group. So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1. Or you have, let me put it this way, you have n minus 1 for each of these groups. And there are m groups. So there's m times n minus 1 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So you have n minus 1 degrees of freedom for each of these groups, so it's n minus 1, n minus 1, n minus 1. Or you have, let me put it this way, you have n minus 1 for each of these groups. And there are m groups. So there's m times n minus 1 degrees of freedom. And in this case, in particular, each group, n minus 1 is 2, or in each case, you had 2 degrees of freedom. And there's 3 groups of that, so there are 6 degrees of freedom. There are 6 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So there's m times n minus 1 degrees of freedom. And in this case, in particular, each group, n minus 1 is 2, or in each case, you had 2 degrees of freedom. And there's 3 groups of that, so there are 6 degrees of freedom. There are 6 degrees of freedom. Let me write 6 degrees of freedom. In the future, we might do a more detailed discussion of what degrees of freedom mean and how to mathematically think about it. But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
There are 6 degrees of freedom. Let me write 6 degrees of freedom. In the future, we might do a more detailed discussion of what degrees of freedom mean and how to mathematically think about it. But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them. If you know them already, the third data point could actually be calculated from the other two. So we have 6 degrees of freedom over here. Now, that was how much of the total variation is due to variation within each sample. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
But the simplest way to think about it is really, truly independent data points, assuming you knew, in this case, the central statistic that we use to calculate the squared distance in each of them. If you know them already, the third data point could actually be calculated from the other two. So we have 6 degrees of freedom over here. Now, that was how much of the total variation is due to variation within each sample. Now let's think about how much of the variation is due to variation between the samples. And to do that, we're going to calculate. Let me get a nice color here. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Now, that was how much of the total variation is due to variation within each sample. Now let's think about how much of the variation is due to variation between the samples. And to do that, we're going to calculate. Let me get a nice color here. I think I've run out of all the colors. We'll call this sum of squares between. The b stands for between. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Let me get a nice color here. I think I've run out of all the colors. We'll call this sum of squares between. The b stands for between. So another way to think about it, how much of this total variation is due to the variation between the means, between the central tendency? That's what we're going to calculate right now. And how much is due to variation from each data point to its mean? | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
The b stands for between. So another way to think about it, how much of this total variation is due to the variation between the means, between the central tendency? That's what we're going to calculate right now. And how much is due to variation from each data point to its mean? So let's figure out how much is due to variation between these guys over here. So one way to think about it, for each of these data points, actually, let's think about just this first group. For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means? | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And how much is due to variation from each data point to its mean? So let's figure out how much is due to variation between these guys over here. So one way to think about it, for each of these data points, actually, let's think about just this first group. For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means? Well, it's going to be, so for this first guy up here, I'll just write it all out explicitly, the variation is going to be its sample mean. So it's going to be 2 minus the mean of means squared. And then for this guy, it's going to be the same thing. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
For this first group, how much variation for each of these guys is due to the variation between this mean and the mean of means? Well, it's going to be, so for this first guy up here, I'll just write it all out explicitly, the variation is going to be its sample mean. So it's going to be 2 minus the mean of means squared. And then for this guy, it's going to be the same thing. His sample mean, 2 minus the mean of means squared plus, same thing for this guy, 2 minus the mean of means squared. Or another way to think about it, this is equal to 3 times 2 minus 4 squared, which is the same thing as 3. This is equal to 3 times 4 is equal to 12. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And then for this guy, it's going to be the same thing. His sample mean, 2 minus the mean of means squared plus, same thing for this guy, 2 minus the mean of means squared. Or another way to think about it, this is equal to 3 times 2 minus 4 squared, which is the same thing as 3. This is equal to 3 times 4 is equal to 12. And then we could do it for each of them. And actually, I want to find the total sum. So let me just write it all out, actually. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
This is equal to 3 times 4 is equal to 12. And then we could do it for each of them. And actually, I want to find the total sum. So let me just write it all out, actually. I think that might be an easier thing to do. Because I want to find, for all of these guys combined, the sum of squares due to the differences between the samples. So that's from the first sample, the contribution from the first sample. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So let me just write it all out, actually. I think that might be an easier thing to do. Because I want to find, for all of these guys combined, the sum of squares due to the differences between the samples. So that's from the first sample, the contribution from the first sample. And then from the second sample, you have this guy over here, 5, oh sorry, you don't want to calculate him. For this data point, the amount of variation due to the difference between the means is going to be 4 minus 4. It's going to be 4 minus 4 squared. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So that's from the first sample, the contribution from the first sample. And then from the second sample, you have this guy over here, 5, oh sorry, you don't want to calculate him. For this data point, the amount of variation due to the difference between the means is going to be 4 minus 4. It's going to be 4 minus 4 squared. Same thing for this guy. It's going to be 4 minus 4 squared. We're not taking it into consideration. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
It's going to be 4 minus 4 squared. Same thing for this guy. It's going to be 4 minus 4 squared. We're not taking it into consideration. We're only taking its sample mean into consideration. And then finally, plus 4 minus 4 squared. We're taking this minus this squared for each of these data points. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
We're not taking it into consideration. We're only taking its sample mean into consideration. And then finally, plus 4 minus 4 squared. We're taking this minus this squared for each of these data points. And then finally, we'll do that with the last group. Sample mean is 6. So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
We're taking this minus this squared for each of these data points. And then finally, we'll do that with the last group. Sample mean is 6. So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared. Now let's think about how many degrees of freedom we had in this calculation right over here. How many degrees of freedom? Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So it's going to be 6 minus 4 squared plus 6 minus 4 squared plus 6 minus 4 squared. Now let's think about how many degrees of freedom we had in this calculation right over here. How many degrees of freedom? Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means. If we know the mean of means, how much here is new information? Well, if you know the mean of the mean and you know two of these sample means, you can always figure out the third. If you know this one and this one, you can figure out that one. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Well, in general, I guess the easiest way to think about it is how much information did we have, assuming that we knew the mean of means. If we know the mean of means, how much here is new information? Well, if you know the mean of the mean and you know two of these sample means, you can always figure out the third. If you know this one and this one, you can figure out that one. If you know that one and that one, you can figure out that one. And that's because this is the mean of these means over here. So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
If you know this one and this one, you can figure out that one. If you know that one and that one, you can figure out that one. And that's because this is the mean of these means over here. So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here. So there's m minus 1 degrees of freedom here. Let me write that. There are m minus 1 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So in general, if you have m groups or if you have m means, there are m minus 1 degrees of freedom here. So there's m minus 1 degrees of freedom here. Let me write that. There are m minus 1 degrees of freedom. But with that said, well, in this case, m is 3. So we could say there's 2 degrees of freedom for this exact example. But actually, let's calculate the sum of squares between. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
There are m minus 1 degrees of freedom. But with that said, well, in this case, m is 3. So we could say there's 2 degrees of freedom for this exact example. But actually, let's calculate the sum of squares between. So what is this going to be? This is going to be equal to, I'll just scroll down, running out of space. This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
But actually, let's calculate the sum of squares between. So what is this going to be? This is going to be equal to, I'll just scroll down, running out of space. This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4. And then we have 3 4's over here. So it's 3 times 4 plus 3 times 0 plus, what is this? The difference between each of these, 6 minus 4 is 2, squared is 4. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
This is going to be equal to, this right here is 2 minus 4 is negative 2 squared is 4. And then we have 3 4's over here. So it's 3 times 4 plus 3 times 0 plus, what is this? The difference between each of these, 6 minus 4 is 2, squared is 4. So we have 3 times 4 plus 3 times 4. And we get 3 times 4 is 12 plus 0 plus 12 is equal to 24. So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
The difference between each of these, 6 minus 4 is 2, squared is 4. So we have 3 times 4 plus 3 times 4. And we get 3 times 4 is 12 plus 0 plus 12 is equal to 24. So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24. Now, let's put it all together. We said that the total variation, that if you look at all nine data points, is 30. Let me write that over here. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
So the sum of squares, or we could say the variation due to what's the difference between the groups, between the means, is 24. Now, let's put it all together. We said that the total variation, that if you look at all nine data points, is 30. Let me write that over here. So the sum of squares, the total sum of squares, is equal to 30. We figured out the sum of squares between each data point and its central tendency, its sample mean. We figured out it and we totaled it all up. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
Let me write that over here. So the sum of squares, the total sum of squares, is equal to 30. We figured out the sum of squares between each data point and its central tendency, its sample mean. We figured out it and we totaled it all up. We got 6. So the sum of squares within was equal to 6. And in this case, it was 6 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
We figured out it and we totaled it all up. We got 6. So the sum of squares within was equal to 6. And in this case, it was 6 degrees of freedom. And we also had 6 degrees of freedom. Or if we wanted to write it generally, there were m times n minus 1 degrees of freedom. And actually, for the total, we figured out we had m times n minus 1 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And in this case, it was 6 degrees of freedom. And we also had 6 degrees of freedom. Or if we wanted to write it generally, there were m times n minus 1 degrees of freedom. And actually, for the total, we figured out we had m times n minus 1 degrees of freedom. Actually, let me just write degrees of freedom in this column right over here. In this case, the number turned out to be 8. And then just now, we calculated the sum of squares between the samples is equal to 24. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And actually, for the total, we figured out we had m times n minus 1 degrees of freedom. Actually, let me just write degrees of freedom in this column right over here. In this case, the number turned out to be 8. And then just now, we calculated the sum of squares between the samples is equal to 24. And we figured out that it had m minus 1 degrees of freedom, which ended up being 2. Now, the interesting thing here, and this is why this kind of analysis of variance fits nicely together. And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And then just now, we calculated the sum of squares between the samples is equal to 24. And we figured out that it had m minus 1 degrees of freedom, which ended up being 2. Now, the interesting thing here, and this is why this kind of analysis of variance fits nicely together. And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares. So a way to think about it is that the total variation in this data right here can be described as the sum of the variation within each of these groups, when you take that total, plus the sum of the variation between the groups. And even the degrees of freedom work out. The sum of squares between had 2 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
And in future videos, we'll think about how we can actually test hypotheses using some of the tools that we're thinking about right now, is that the sum of squares within plus the sum of squares between is equal to the total sum of squares. So a way to think about it is that the total variation in this data right here can be described as the sum of the variation within each of these groups, when you take that total, plus the sum of the variation between the groups. And even the degrees of freedom work out. The sum of squares between had 2 degrees of freedom. The sum of squares within each of the groups had 6 degrees of freedom. 2 plus 6 is 8. That's the total degrees of freedom we had for all of the data combined. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
The sum of squares between had 2 degrees of freedom. The sum of squares within each of the groups had 6 degrees of freedom. 2 plus 6 is 8. That's the total degrees of freedom we had for all of the data combined. It even works if you look at the more general. So our sum of squares between had m minus 1 degrees of freedom. Our sum of squares within had m times n minus 1 degrees of freedom. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
That's the total degrees of freedom we had for all of the data combined. It even works if you look at the more general. So our sum of squares between had m minus 1 degrees of freedom. Our sum of squares within had m times n minus 1 degrees of freedom. So this is equal to m minus 1 plus mn minus m. These guys cancel out. This is equal to mn minus 1 degrees of freedom, which is exactly the total degrees of freedom we had for the total sum of squares. So the whole point of the calculations that we did in the last video and in this video is just to appreciate that this total variation over here, this total variation that we first calculated, can be viewed as the sum of these kind of two component variations. | ANOVA 2 Calculating SSW and SSB (total sum of squares within and between) Khan Academy.mp3 |
The normal distribution is arguably the most important concept in statistics. Everything we do, or almost everything we do in inferential statistics, which is essentially making inferences based on data points, is to some degree based on the normal distribution. And so what I want to do in this video and in this spreadsheet is to essentially give you as deep an understanding of the normal distribution as possible. And the rest of your life, if someone says, oh, we're assuming a normal distribution, you're like, oh, I know what that is. This is the formula, and I understand how to use it, et cetera, et cetera. So this spreadsheet, just so you know, is downloadable at www.khanacademy.org slash downloads slash, and if you just type that part in, you'll see everything that's downloadable. But then downloads slash normalintro.xls. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And the rest of your life, if someone says, oh, we're assuming a normal distribution, you're like, oh, I know what that is. This is the formula, and I understand how to use it, et cetera, et cetera. So this spreadsheet, just so you know, is downloadable at www.khanacademy.org slash downloads slash, and if you just type that part in, you'll see everything that's downloadable. But then downloads slash normalintro.xls. And then you'll get this spreadsheet right here. And I think I did this in the right standard. But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
But then downloads slash normalintro.xls. And then you'll get this spreadsheet right here. And I think I did this in the right standard. But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see. I literally copied and pasted this right here from Wikipedia. And I know it looks daunting. You have all these Greek letters there. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
But anyway, if you go on to Wikipedia, and if you were to type in normal distribution, or you were to do a search for normal distribution, let me actually get my pen tool going, this is what you would see. I literally copied and pasted this right here from Wikipedia. And I know it looks daunting. You have all these Greek letters there. But this is just the sigma right here. That is just the standard deviation of the distribution. We'll play with that a little bit in this chart and see what that means. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
You have all these Greek letters there. But this is just the sigma right here. That is just the standard deviation of the distribution. We'll play with that a little bit in this chart and see what that means. And well, I mean, you know what the standard deviation is in general, but this is the standard deviation of this distribution, which is a probability density function. And I encourage you to re-watch the video on probability density functions, because it's a little bit of a transition going from the binomial distribution, which is discrete, right? In the binomial distribution, say, oh, what is the probability of getting a 5? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
We'll play with that a little bit in this chart and see what that means. And well, I mean, you know what the standard deviation is in general, but this is the standard deviation of this distribution, which is a probability density function. And I encourage you to re-watch the video on probability density functions, because it's a little bit of a transition going from the binomial distribution, which is discrete, right? In the binomial distribution, say, oh, what is the probability of getting a 5? And you just kind of look at that histogram or that bar chart, and you say, oh, that's the probability. But in a continuous probability distribution, or a continuous probability density function, you can't just say, what is the probability of me getting a 5? You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
In the binomial distribution, say, oh, what is the probability of getting a 5? And you just kind of look at that histogram or that bar chart, and you say, oh, that's the probability. But in a continuous probability distribution, or a continuous probability density function, you can't just say, what is the probability of me getting a 5? You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5? You have to give it some range. And then your probability isn't given by just reading this graph. The probability is given by the area under that curve. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
You have to say, what is the probability of me getting between, let's say, a 4.5 and a 5.5? You have to give it some range. And then your probability isn't given by just reading this graph. The probability is given by the area under that curve. It would be given by this area. And for those of you all who know calculus, if P of x is our probability density function, it doesn't have to be a normal distribution, although it almost always, well, it often is a normal distribution. The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
The probability is given by the area under that curve. It would be given by this area. And for those of you all who know calculus, if P of x is our probability density function, it doesn't have to be a normal distribution, although it almost always, well, it often is a normal distribution. The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow. It'll actually be the integral from 4.5 to 5.5 of this probability density function, or of this probability density function, dx. So that's just the area of the curve. For those of you who don't know calculus yet, I encourage you to watch that playlist. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
The way you actually figure out the probability of, let's say, between 4.5 and 5.5, what is the probability, this is whatever, the odds of me getting between 4.5 and 5.5 inches of rain tomorrow. It'll actually be the integral from 4.5 to 5.5 of this probability density function, or of this probability density function, dx. So that's just the area of the curve. For those of you who don't know calculus yet, I encourage you to watch that playlist. But all this is, is saying the area of the curve from here to here. And actually, it turns out for the normal distribution, this isn't an easy thing to evaluate analytically. And so you do it numerically. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
For those of you who don't know calculus yet, I encourage you to watch that playlist. But all this is, is saying the area of the curve from here to here. And actually, it turns out for the normal distribution, this isn't an easy thing to evaluate analytically. And so you do it numerically. And you don't have to feel bad about doing it numerically, because, oh, how do I take the integral of this? There's actually functions for it. And you can even approximate it. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And so you do it numerically. And you don't have to feel bad about doing it numerically, because, oh, how do I take the integral of this? There's actually functions for it. And you can even approximate it. I mean, one way you could approximate it is you could use it the way you approximate integrals in general, where you could say, well, what is the area of this? What's roughly the area of this trapezoid? So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And you can even approximate it. I mean, one way you could approximate it is you could use it the way you approximate integrals in general, where you could say, well, what is the area of this? What's roughly the area of this trapezoid? So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base. Or you could just take the level of the, let me change colors, just because I think I'm overdoing it with the green, or you could just take the height of this line right here and multiply it by the base. And you'll get the area of this rectangle, which might be a pretty good approximation for the area under the curve, right, because you'll have a little bit extra over here, but you're going to miss a little bit over there. So it might be a pretty good approximation. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So you could figure out the area of that trapezoid, taking the average of that point and that point and multiplying it by the base. Or you could just take the level of the, let me change colors, just because I think I'm overdoing it with the green, or you could just take the height of this line right here and multiply it by the base. And you'll get the area of this rectangle, which might be a pretty good approximation for the area under the curve, right, because you'll have a little bit extra over here, but you're going to miss a little bit over there. So it might be a pretty good approximation. That's actually what I do in the other video, just to approximate the area under the curve and give you a good sense that the normal distribution is what the binomial distribution becomes, essentially, if you have many, many, many, many trials. And what's interesting about the normal distribution, just so you know, I don't know if I already mentioned this, already, this right here, this is the graph. And this is just another word. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So it might be a pretty good approximation. That's actually what I do in the other video, just to approximate the area under the curve and give you a good sense that the normal distribution is what the binomial distribution becomes, essentially, if you have many, many, many, many trials. And what's interesting about the normal distribution, just so you know, I don't know if I already mentioned this, already, this right here, this is the graph. And this is just another word. People might talk about the central limit theorem, but this is really kind of one of the most important or interesting things about our universe, central limit theorem. I won't prove it here, but it essentially tells us, and you could kind of understand it by looking at the other video where we talk about flipping coins, and if we were to do many, many, many flips of coins, right, those are independent trials of each other. And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And this is just another word. People might talk about the central limit theorem, but this is really kind of one of the most important or interesting things about our universe, central limit theorem. I won't prove it here, but it essentially tells us, and you could kind of understand it by looking at the other video where we talk about flipping coins, and if we were to do many, many, many flips of coins, right, those are independent trials of each other. And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution. And what's interesting about that is each of those trials, in the case of flipping a coin, each trial is a flip of the coin, each of those trials don't have to have a normal distribution. So we could be talking about molecular interactions. And every time compound x interacts with compound y, what might result doesn't have to be normally distributed. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And if you take the sum of all of your flips, if you were to give yourself one point if you got ahead every time, and if you would take the sum of them, as you approach an infinite number of flips, you approach the normal distribution. And what's interesting about that is each of those trials, in the case of flipping a coin, each trial is a flip of the coin, each of those trials don't have to have a normal distribution. So we could be talking about molecular interactions. And every time compound x interacts with compound y, what might result doesn't have to be normally distributed. But what happens is if you take a sum of a ton of those interactions, then all of a sudden the end result will be normally distributed. And this is why this is such an important distribution. It shows up in nature all of the time. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And every time compound x interacts with compound y, what might result doesn't have to be normally distributed. But what happens is if you take a sum of a ton of those interactions, then all of a sudden the end result will be normally distributed. And this is why this is such an important distribution. It shows up in nature all of the time. And if people are trying to kind of, if you do take data points from something that is very, very complex, and that it is the sum of arguably many, many, almost infinite individual independent trials, it's a pretty good assumption to assume the normal distribution. We'll do other videos where we talk about when it is a good assumption and when it isn't a good assumption. But anyway, just to digest this a little bit, and let me actually rewrite it. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
It shows up in nature all of the time. And if people are trying to kind of, if you do take data points from something that is very, very complex, and that it is the sum of arguably many, many, almost infinite individual independent trials, it's a pretty good assumption to assume the normal distribution. We'll do other videos where we talk about when it is a good assumption and when it isn't a good assumption. But anyway, just to digest this a little bit, and let me actually rewrite it. This is what you'll see on Wikipedia, but this could be rewritten as 1 over sigma times the square root of 2 pi times, exp is just e to that power, so it's just e to this whole thing over here, minus x minus the mean squared over 2 sigma squared. This is a standard deviation. Standard deviation squared is just the variance. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
But anyway, just to digest this a little bit, and let me actually rewrite it. This is what you'll see on Wikipedia, but this could be rewritten as 1 over sigma times the square root of 2 pi times, exp is just e to that power, so it's just e to this whole thing over here, minus x minus the mean squared over 2 sigma squared. This is a standard deviation. Standard deviation squared is just the variance. And just so you know how to use this, you're like, oh wow, there's so many Greek letters here, what do I do? This tells you the height of the normal distribution function. Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Standard deviation squared is just the variance. And just so you know how to use this, you're like, oh wow, there's so many Greek letters here, what do I do? This tells you the height of the normal distribution function. Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something. I don't know. Well, no, that's not a good one. Let's say it's people's heights above 5' 9". | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Let's say that this is the distribution of people's, I don't know, how far north they live from my house or something. I don't know. Well, no, that's not a good one. Let's say it's people's heights above 5' 9". Let's say that this was 5' 9", and not 0. Right? What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
Let's say it's people's heights above 5' 9". Let's say that this was 5' 9", and not 0. Right? What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function. And then you have to give it a range. You can't just say, how many people are exactly 5 inches taller than average. You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
What this tells you is, if you were to say, what percentage of people, or I guess, what is the probability, if you wanted to figure out what is the probability of finding someone who is roughly 5 inches taller than the average right here, what you would do is you would put in this number here, this 5, into x, and then you know the standard deviation, because you've taken a bunch of samples, you know the variance, which is the standard deviation squared, you know the mean, and you just put your x in there, and it'll tell you the height of the function. And then you have to give it a range. You can't just say, how many people are exactly 5 inches taller than average. You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average? You have to give it a little bit of range, because no one is exactly, or it's almost infinitely impossible to the atom to be exactly 5' 9". Even the definition of an inch isn't defined that particularly. So that's how you use this function. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
You would actually say, how many people are between 5.1 inches and 4.9 inches taller than the average? You have to give it a little bit of range, because no one is exactly, or it's almost infinitely impossible to the atom to be exactly 5' 9". Even the definition of an inch isn't defined that particularly. So that's how you use this function. I think this is so heavily used in, one, it shows up in nature, but in all of inferential statistics, I think it behooves you to become as familiar with this formula as possible. And I guess to make that happen, let me play around a little bit with this formula, just to kind of give you an intuition of how everything works out, et cetera, et cetera. So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So that's how you use this function. I think this is so heavily used in, one, it shows up in nature, but in all of inferential statistics, I think it behooves you to become as familiar with this formula as possible. And I guess to make that happen, let me play around a little bit with this formula, just to kind of give you an intuition of how everything works out, et cetera, et cetera. So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared. I've never seen it written this way, but it gives me a little intuition. That sigma squared, it's always written as sigma squared, but it's really just the variance. And the variance is what you calculate before you calculate the standard deviation. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
So if I were to take this, and I like to just maybe help you memorize it, this could be rewritten as, if we take the sigma into the square root sign, if we take the standard deviation in there, it becomes 1 over the square root of 2 pi sigma squared. I've never seen it written this way, but it gives me a little intuition. That sigma squared, it's always written as sigma squared, but it's really just the variance. And the variance is what you calculate before you calculate the standard deviation. So that's interesting. And then this top right here, this could be written as e to the minus 1 half times, and if we were to just take this, both of these things here are squared, so we could just say x minus the mean over sigma squared. And this kind of clarifies a little bit what's going on here a little bit better. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And the variance is what you calculate before you calculate the standard deviation. So that's interesting. And then this top right here, this could be written as e to the minus 1 half times, and if we were to just take this, both of these things here are squared, so we could just say x minus the mean over sigma squared. And this kind of clarifies a little bit what's going on here a little bit better. Because what's this? x minus sigma is the distance between whatever point we want to find, let's say we're here, x minus mu, mu is the mean, so that's here, so that's this distance. And this is the standard deviation, which is this distance. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And this kind of clarifies a little bit what's going on here a little bit better. Because what's this? x minus sigma is the distance between whatever point we want to find, let's say we're here, x minus mu, mu is the mean, so that's here, so that's this distance. And this is the standard deviation, which is this distance. So this in here tells me how many standard deviations I am away from the mean, and that's actually called the standard z-score I talked about in the other video. And then we square that, and then we take this to the minus 1 half, well, let me rewrite that. If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
And this is the standard deviation, which is this distance. So this in here tells me how many standard deviations I am away from the mean, and that's actually called the standard z-score I talked about in the other video. And then we square that, and then we take this to the minus 1 half, well, let me rewrite that. If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right? If you take something to an exponent and then take that to an exponent, you can just multiply these exponents. So likewise, this could be rewritten as, this is equal to 1 over the square root of 2 pi sigma squared, which is just the variance. And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition. | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half power, right? If you take something to an exponent and then take that to an exponent, you can just multiply these exponents. So likewise, this could be rewritten as, this is equal to 1 over the square root of 2 pi sigma squared, which is just the variance. And I'm just playing around with the formula, because I really want you to see all the ways that it, you know, maybe you'll get a little intuition. And I encourage you to email me if you see some insight on why this exists and all of that. But once again, I think it is cool that all of a sudden we have this other formula that has pi and e in it, and this is really just, you know, this is what the central, you know, so many phenomenon are described by this. And once again, pi and e show up together, right? | Introduction to the normal distribution Probability and Statistics Khan Academy.mp3 |
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