Introduction to sampling distributions | Sampling distributions | AP Statistics | Khan Academy

What we're going to do in this video is talk about the idea of a sampling distribution. Now, just to make things a little bit concrete, let's imagine that we have a population of some kind. Let's say it's a bunch of balls; each of them has a number written on it. For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos.

So, for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even. Whatever these are, all of these are population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find.

And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, a sample of size n. Then we can calculate a statistic from that sample. Based on that sample, maybe, you know, we picked n balls from there, and so from that, we can calculate a statistic that is used to estimate this parameter.

But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter.

An interesting question is, what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is.

So let's make this even a little bit more concrete. Let's imagine where our population—I'm going to make this a very simple example. Let's say our population has three balls in it: one, two, three, and they're numbered one, two, and three. It's very easy to calculate. Let's say the parameter that we care about right over here is the population mean, and that, of course, is going to be 1 plus 2 plus 3, all of that over 3, which is 6 divided by 3, which is 2. So that is our population parameter.

But let's say that we wanted to take samples. Let's say samples of two balls at a time. Every time we take a ball, we'll replace it, so each ball we take is an independent pick. We're going to use those samples of two balls at a time in order to estimate the population mean. So, for example, this could be our first sample of size two. Let's say in that first sample I pick a one, and let's say I pick a two.

Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. For this sample of 2, it's going to be 1.5. Then I can do it again, and let's say I get a 1, and I get a 3. Well, now when I calculate the sample mean, the average of 1 and 3, or the mean of 1 and 3, is going to be equal to 2.

Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be, and then we can see the frequency of getting those sample means. So, let me draw a little bit of a table here. So, make a table right over here and let's see these are the numbers that we pick.

Remember, when we pick one ball, we'll record that number, then we'll put it back in, and then we'll pick another ball. So, these are going to be independent events, and it's going to be with replacement. And so, let's say we could pick a 1 and then a 1. We could pick a 1 then a 2, a 1 and a 3, we could pick a 2 and then a 1, we could pick a 2 and a 2, a 2 and a 3, we could pick a 3 and a 1, a 3 and a 2, or a 3 and a 3.

There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. So what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5, here it is 2, here it is 1.5, here it is 2, here it is 2.5, here it is 2, here it is 2.5, and then here it is three.

We can now plot the frequencies of these possible sample means that we can get, and that plot will be a sampling distribution of the sample means. So let's do that. Let me make a little chart right over here, a little graph right over here.

So these are the possible sample means we can get: a 1, we could get a 1.5, we could get a 2, we could get a 2.5, or we can get a 3. Now, let's see the frequency of it, and I will put that over here. So let's see how many ones out of our nine possibilities we have. How many were one? Well, only one of the sample means was one.

The relative frequency, if we just said the number, we could make this line go up one, or we could just say, "Hey, this is going to be one out of the nine possibilities," and so let me just make that. I'll call this right over here this is one ninth. Now, what about 1.5? Let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is 2 over 9.

And now, what about 2? Well, we can see there's one, two, three, so three out of the nine possibilities we got a two. So we could say this is two or we could say this is three ninths, which is the same thing, of course, as one third. So this right over here is 3 over 9.

And then, what about 2.5? Well, there's two 2.5s, so two out of the nine. Another way you could interpret this is that when you take a random sample with replacement of two balls, you have a 2/9 chance of having a sample mean of 2.5.

And then last but not least, right over here, there's one scenario out of the 9 where you get 2, 3, so 1/9. And so this right over here, this is the sampling distribution, sampling distribution for the sample mean for n equals 2 or for a sample size of 2.