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Introduction to Sampling Distribution

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This tutorial is going to introduce you to what a sampling distribution is. We're doing a sampling distribution of sample means for this particular sampling distribution. So before we even talk about what a sampling distribution is, we'll do an example and show you.

So consider the spinner shown above. You spin it four times and you obtain an average. So you spin it maybe one set of four. And you get a 2 the first time. And you get a 4 the second time. You get a 3 the third time. You get a 1 the fourth time. The mean is the average of 2, 4, 3, and 1-- 2 plus 4 plus 3 plus 1 is 10 divided by 4 is 2 and 1/2. So your first mean is 2 and 1/2.

But your mean won't be 2 and 1/2 every time. Suppose the next time you spin it four times. Second sample, you spin a 1, and then a 4, and then a 3, and then another 1. In this case, your average is 2.25. And I could do it again. Third sample, 4, 2, 4, 4, the average is 3 and 1/2. And the next one, the average was 2. The fifth time, the average was one and a half, and the sixth time the average was one and a fourth.

What we're going to do is we're going to take those sample means and we're going to place them on the x-axis. So we're going to draw out an axis. I'm having it go from 0 to 4. I can't average anything higher than four. In fact, I can't average anything lower than one either. But I'm going to take my average value, my mean of 2 and 1/2, and put a dot, like a dot plot on 2 and 1/2. And then I'll do the same thing with the next one. 2 and 1/4, put a dot there. And then 3 and 1/2, two, one and a half, one and a fourth.

I can keep doing this over and over and over again. I'm not going to do this just the six times. Ideally, we would like to do this hundreds or thousands of times. What we want to see is the distribution of all possible samples that could be taken of size four. So once we've enumerated every possible sample of size 4 from this spinner, then we'll be done. When you are done, the sampling distribution looks like this.

Every possible set of four outcomes. When we plot them all, it looks like this. Now notice, the lowest number you can get is one. The highest number you can get is four. This is when you've got four four's. This is when you've got all four of them being ones. Now notice that happens more often than all the numbers being four. Why? Because there were more ones on the spinner than there were fours.

Also notice because there were more ones that actually pulls the average down a bit. The highest number here-- the most frequent average is 2.25, not 2 and 1/2 half, which would be the exact middle between 1 and 4, to actually pull a little bit further down this way. So this distribution is skewed slightly to the right.

And so to recap, a sampling distribution is the distribution of all possible means that you could have for a given size. So we did a sampling distribution where we graphed all the possible means for samples of size 4. Sample is taken, mean is calculated, and you plot it with a dot. Then you take another sample, calculate the mean again, plot that. So it's a long and tedious process. In large populations, the sampling distribution consists of lots and lots and lots of points. And so we talked about a sampling distribution. It's a distribution of sample means. Good luck and we'll see you next time.