Source: Charts, tables, and graphs created by the author
In this tutorial, you're going to learn about the center and variation of a sampling distribution. We're going to be using the mean to measure center, and the standard deviation to measure the variation.
So suppose we have our handy dandy spinner with three sectors that say 1, one that says a 2, two show a 3, and two show a 4. It's pretty easy to find the mean of the spinner from one spin. You just add up all the values, and divide by 8 because they're all equally likely sectors. So you end up with 2.375 as your mean.
The standard deviation of the spinner-- we won't go through the calculations-- but it's 1.218.
Now suppose we spun it four times to obtain an average. And then suppose we did it again and obtained another average, and another average. Consider every possible set of four outcomes. If we consider every possible scenario, and plot each scenario on this graph here, we can create what's called a sampling distribution of sample means.
So this distribution has a sample means of 1, for instance, where you got 1 on all four of the spins. Here you've got 4 on all four of the spins. But it looks like the data average is somewhere around the 2.25, or 2.5 region. Those are the most likely averages from four spins.
But, if you take a look real close, this distribution of sample means itself has a mean. It's somewhere here in the center. In fact, that number is 2.375. The thing is we've seen that 2.375 number before. It was the mean of the original spinner.
So the mean of the sampling distribution of sample means is the same as the mean of the original distribution for the spinner. Symbolically it looks like this. The mean of all of the sample means, x bar, is the same as the mean of the original distribution. Sometimes we call that the parent distribution.
By the way, this mean of the sampling distribution sometimes is called the grand mean.
Now this distribution also has a standard deviation. So recall that the standard deviation of the original distribution for the spinner was 1.218. This distribution also has a standard deviation. It's 0.609. Well, that's smaller than the original 1.218. It's not the same, but how does that 0.609 relate to 1.218.
If you think about it real hard, and look real closely, you can see that 0.609 is in fact, half of that number. So not only did the standard deviation get smaller, it's half as large as the original standard deviation was.
I wonder if that might happen again. This was for four spins. What is the distribution of sample means when we spin it nine times look like?
Well, first look at the shape. You can see that the extreme values are much less likely now, and things start moving towards the center. That might be indicative that the standard deviation got smaller yet again. But if we look at the mean of the distribution here for nine spins, it's actually the same as the previous mean of 2.375. Which was, again, the mean of the original spinner.
So, again, the mean doesn't change, but what about its standard deviation? Well, here it's even smaller, just like we thought it would be. And now it's 0.406 on either side of the mean. So what do we do with that information?
Let's look at the information that we have now. When we spun it just once the mean was 2.375, and the standard deviation was 1.218. When we spun it four times, the mean was 2.375. Standard deviation was 0.609, half of the original. When we spun it nine times, again the mean was 2.375. Standard deviation was 0.406. 1/3 of the standard deviation of the original distribution.
So we see that as the number of spins, as the trials increase, the standard deviation goes down. But it's not linear. It's not like this was 1/4 of the original standard deviation, or 1/9. It's proportional to the inverse of the square root of n.
So we divide the original standard deviation of 1.218 by the square root of sample size. 3 is the square root of 9. 2 is the square root of 4. But the mean of any sampling distribution, in this particular case with the spinner, will always be 2.375. The same as the mean of the original distribution.
So to recap. The mean of sampling distrib-- of a sampling distribution sample means, from samples of size n, is always going to be the same as the mean of the original distribution it came from. The parent distribution. Symbolically it looks like this. The mean of all the x bars is the same as the original mean from the parent distribution.
The standard deviation, on the other hand, gets smaller as the sample size increases. The more you sample, the more samples-- sorry. Not the more samples you take, but the larger the sample size. The more likely it is that the extreme values, like we saw-- The 4 and the 1, those get evened out, and pulled back towards the mean. And that's why the standard deviation decreases.
But it's also-- we're also able to quantify the decrease in standard deviation. Standard deviation for the sampling distribution is the standard deviation of the parent distribution, divided by the square root of sample size.
So we talked about the mean, which is the center of a sampling distribution, and the standard deviation, which is the variation for a sampling distribution. Good luck, and we'll see you next time.