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Introduction to Sampling Distribution

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Source: Graphs created by Katherine Williams

This tutorial introduces sampling distributions. When you're taking samples, you have one very large population. And you're taking your samples from it. Each sample you take, you're going to be grabbing a slightly different subset of the population. So each sample is going to be a little bit different.

Now for each sample, you're going to be able to compute different statistics for it. You can find the mean of that sample. You can find the mode, the median, the range, anything like that.

What you do with one sample, the mean of that one sample, might be different than the mean of the next one. Now when you do this, you're going to plot the distribution of those sample means. So when you're repeatedly taking samples from a distribution of any kind, then you would want to look at how those samples are distributed and look at how the means, in particular, are distributed.

I've used a small Java applet in order to show this and give you a better sense of what I'm talking about. Now this here shows a histogram for the grades for 256 students on a particular exam. The mean is 76 and the standard deviation is 15.5.

So our histogram shows that some students have low, some students have high scores, but most are falling in that middle range. Down at the bottom here shows our sample distribution histogram. So it's showing us the distribution of the sample means. In this case, our sample size is 5.

Some students have a low score. So when you take a sample, you can get a sample of five students who average out to be 60.

Others have a higher range. So you can take a sample of five students. And the histogram of the population shows that there's at least five students with the 90 to 100 range. So it is possible to get that mean. But a majority of the mean's found in this middle part here.

Now if we increase our sample size, so if we're no longer taking samples of size 5 but instead are moving up to a sample size of 20, we can see how that's going to change our sampling mean distribution. Now when we change that, our sampling mean distribution altered pretty significantly.

There's no longer students in those extremes. We no longer have samples that have means of 60 or 100. More of our samples have a mean of somewhere in that middle range, that 70 to 80, just narrowing in on that population mean of about 76.

Now if we go again and this time we increase it to a sample size of 40, we can predict that something similar is going to happen. And in fact, it does. Again, there's fewer of those sample means that have means that are in the 60's, the 100's, and instead more and more of our sample means have means closer to the population-- that's 76.

If we increase it one final time, we can see this again. So this time, we are increasing it to a sample size of 55. And even more of our samples have a mean around that 76. But again, not all of our samples are going to have a mean of exactly 76.

This has been your tutorial on sampling distributions.