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Then we have a four and another four. So let me write that. So then we have the mean, or we have the absolute deviation of four from three, from the mean. And then plus, we have another four. We have this other four right up here. Four minus three. We take the absolute value, because once again, it's absolute deviation. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And then plus, we have another four. We have this other four right up here. Four minus three. We take the absolute value, because once again, it's absolute deviation. And then we divide it. And then we divide it by the number of data points we have. So what is this going to be? | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
We take the absolute value, because once again, it's absolute deviation. And then we divide it. And then we divide it by the number of data points we have. So what is this going to be? Two minus three is negative one, but we take the absolute value. It's just going to be one. Two minus three is negative one. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So what is this going to be? Two minus three is negative one, but we take the absolute value. It's just going to be one. Two minus three is negative one. We take the absolute value, and it's just going to be one. And you see that here visually. This point is just one away. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Two minus three is negative one. We take the absolute value, and it's just going to be one. And you see that here visually. This point is just one away. It's just one away from three. This point is just one away from three. Four minus three is one. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
This point is just one away. It's just one away from three. This point is just one away from three. Four minus three is one. Absolute value of that is one. This point is just one away from three. Four minus three, absolute value, that's another one. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Four minus three is one. Absolute value of that is one. This point is just one away from three. Four minus three, absolute value, that's another one. So you see in this case, every data point was exactly one away from the mean. And we took the absolute value so that we don't have negative ones here. We just care how far it is in absolute terms. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Four minus three, absolute value, that's another one. So you see in this case, every data point was exactly one away from the mean. And we took the absolute value so that we don't have negative ones here. We just care how far it is in absolute terms. So you have four data points. Each of their absolute deviations is four away, so the mean of the absolute deviations are one plus one plus one plus one, which is four, over four. So it's equal to one. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
We just care how far it is in absolute terms. So you have four data points. Each of their absolute deviations is four away, so the mean of the absolute deviations are one plus one plus one plus one, which is four, over four. So it's equal to one. So one way to think about it, it's saying on average, the mean of the distances of these points away from the actual mean is one. And that makes sense because all of these are exactly one away from the mean. Now let's see what results we get for this data set right over here. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So it's equal to one. So one way to think about it, it's saying on average, the mean of the distances of these points away from the actual mean is one. And that makes sense because all of these are exactly one away from the mean. Now let's see what results we get for this data set right over here. And I'll do it, let me actually get some space over here. At any point, if you get inspired, I encourage you to calculate the mean absolute deviation on your own. So let's calculate it. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Now let's see what results we get for this data set right over here. And I'll do it, let me actually get some space over here. At any point, if you get inspired, I encourage you to calculate the mean absolute deviation on your own. So let's calculate it. So the mean absolute deviation here, I'll write MAD, is going to be equal to, well let's figure out the absolute deviation of each of these points from the mean. So it's the absolute value of one minus three, that's this first one, plus the absolute deviation, so one minus three, that's the second one, and then plus the absolute value of six minus three, that's the sixth, and then we have the four, plus the absolute value of four minus three, and then we have four points. So one minus three is negative two, absolute value is two, and we see that here. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So let's calculate it. So the mean absolute deviation here, I'll write MAD, is going to be equal to, well let's figure out the absolute deviation of each of these points from the mean. So it's the absolute value of one minus three, that's this first one, plus the absolute deviation, so one minus three, that's the second one, and then plus the absolute value of six minus three, that's the sixth, and then we have the four, plus the absolute value of four minus three, and then we have four points. So one minus three is negative two, absolute value is two, and we see that here. This is two away from three. We just care about absolute deviation. We don't care if it's to the left or to the right. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So one minus three is negative two, absolute value is two, and we see that here. This is two away from three. We just care about absolute deviation. We don't care if it's to the left or to the right. Then we have another one minus three is negative two, but it's absolute value, so this is two, and that's this. This is two away from the mean. Then we have six minus three, absolute value of that's just going to be three, and that's this right over here. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
We don't care if it's to the left or to the right. Then we have another one minus three is negative two, but it's absolute value, so this is two, and that's this. This is two away from the mean. Then we have six minus three, absolute value of that's just going to be three, and that's this right over here. We see this six is three to the right of the mean. We don't care whether it's to the right or the left, and then four minus three. Four minus three is one, absolute value is one, and we see that. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Then we have six minus three, absolute value of that's just going to be three, and that's this right over here. We see this six is three to the right of the mean. We don't care whether it's to the right or the left, and then four minus three. Four minus three is one, absolute value is one, and we see that. It is one to the right of three, and so what do we have? We have two plus two is four, plus three is seven, plus one is eight over four, which is equal to two. So the mean absolute deviation, let me write it down, it fell off over here. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Four minus three is one, absolute value is one, and we see that. It is one to the right of three, and so what do we have? We have two plus two is four, plus three is seven, plus one is eight over four, which is equal to two. So the mean absolute deviation, let me write it down, it fell off over here. Here for this data set, the mean absolute deviation is equal to two, while for this data set, the mean absolute deviation is equal to one. That makes sense. They have the exact same means. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So the mean absolute deviation, let me write it down, it fell off over here. Here for this data set, the mean absolute deviation is equal to two, while for this data set, the mean absolute deviation is equal to one. That makes sense. They have the exact same means. They both have a mean of three, but this one is more spread out. The one on the right is more spread out because on average, each of these points are two away from three, while on average, each of these points are one away from three. The means of the absolute deviations on this one is one. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
This right over here is a scratch pad on Khan Academy created by Khan Academy user Charlotte Auen. And what you see here is a simulation that allows us to keep sampling from our gumball machine and start approximating the sampling distribution of the sample proportion. So her simulation focuses on green gumballs, but we talked about yellow before. In the yellow gumballs, we said 60% were yellow, so let's make 60% here green. And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
In the yellow gumballs, we said 60% were yellow, so let's make 60% here green. And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample. So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
Now let's do another sample. So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then? | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew. And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
You're going to end up with a left skew. And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time? Well, notice, now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time? Well, notice, now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution. And the reason why that made sense, the standard deviation of your sample proportion, it is inversely proportional to the square root of n. And so that makes sense. So hopefully you have a good intuition now for the sample proportion, its distribution, the sampling distribution of the sample proportion, that you can calculate its mean and its standard deviation. And you feel good about it because we saw it in a simulation. | Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3 |
The possible colors are blue, yellow, white, red, orange, and green. How many four-color codes can be made if the colors cannot be repeated? To some degree, this whole paragraph in the beginning doesn't even matter. If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated? | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six. There's going to be six different possibilities for this slot right there. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six. There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left. If we think about all of the possibilities, all of the permutations, and permutations are when you think about all of the possibilities, and you do care about order, where you say that this is different than this. This is a different permutation than this. All of the different permutations here, when you pick four colors out of a possible of six colors, it's going to be six possibilities for the first one, times five for the second bucket, times four for the third bucket of third position, times three. | Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3 |
Let's say that you're curious about people's TV watching habits, and in particular, how much TV do people in the country watch? So what you're concerned with, if we imagine the entire country that we've already talked about, especially if we're talking about a country like the United States, but pretty much any country, is a very large population. In the United States, we're talking about on the order of 300 million people. So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. And then we're dividing not by the number of data points we have, but by 1 less than the number of data points we have. So this calculation, where we summed up all of this, and then we divided by 5, not by 6, this is the standard definition of sample variance. So I'll leave you there. | Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And I've taken some exercises from the Khan Academy exercises here, and I'm just gonna solve it on my scratch pad. The following data points represent the number of animal crackers in each kid's lunchbox. Sort the data from least to greatest, and then find the interquartile range of the data set. And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
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