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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.
Suppose you have a spinner with the following sectors:
It's fairly easy to find the mean number spun from one spin. You just add up all the values, and divide by 8 because they're all equally likely sectors:
You end up with 2.375 as your mean. Using the standard deviation formula or Excel, the population standard deviation of the spinner is 1.218.
Now suppose you spun it four times to obtain an average. Next, you did it again and obtained another average, and then another average. You could eventually consider every possible set of four outcomes. If you consider every possible scenario and plot each scenario on a graph like the one below, you can create what's called a sampling distribution of sample means.
For this distribution, 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.
Now, this distribution of sample means itself has a mean. In the case of this distribution of sample means, the mean is somewhere in the center. In fact, the number is 2.375, which is the same as the mean of the original spinner. So, the mean of the distribution of sample means is the same as the mean of the original distribution for the spinner. Sometimes we call this the parent distribution or the grand mean.
Symbolically it looks like this:
Recall that the standard deviation of the original distribution for the spinner was 1.218. There is also a standard deviation of the distribution of sample means, shown below for 4 spins. The standard deviation, in this case, is 0.609.
How does that 0.609 relate to 1.218? Well, 0.609 is half of 1.218. The standard deviation on the distribution for four spins is only half as large as the original standard deviation was.
What is the distribution of sample means when you spin nine times? You'll notice that the mean of the distribution for nine spins is the same as the other means: 2.375. Next, look at the shape. You can see that the extreme values are much less likely now, and things start moving towards the center. This might be indicative of the standard deviation getting smaller yet again. In fact, the standard deviation is just 0.406 on either side of the mean.
Let's consider the three cases so far:
Source: THIS TUTORIAL WAS AUTHORED BY JONATHAN OSTERS FOR SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.