A sampling distribution of sample means is a distribution that shows the means from all possible samples of a given size. Let’s start with an example of a sampling distribution.
Consider the spinner shown here:
Suppose you spin it four times to obtain an average. You get a 2 the first time, a 4 the second time, a 3 the third time, and a 1 the fourth time. The mean is the average of 2, 4, 3, and 1 is:
So, your first mean is 2.5.
Sample | Mean |
---|---|
S_{1} = {2, 4, 3, 1} | x̄_{1} = 2.50 |
However, your mean won't be 2.5 every time. Suppose you repeat this process five more times to get the following six samples:
Sample | Mean |
---|---|
S_{1} = {2, 4, 3, 1} | x̄_{1} = 2.50 |
S_{2} = {1, 4, 3, 1} | x̄_{2} = 2.25 |
S_{3} = {4, 2, 4, 4} | x̄_{3} = 3.5 |
S_{4} = {2, 2, 3, 1} | x̄_{4} = 2.00 |
S_{5} = {3, 1, 1, 1} | x̄_{5} = 1.50 |
S_{6} = {1, 1, 1, 2} | x̄_{6} = 1.25 |
So how can we represent all these distributions?
On the graph, the lowest number you can get is one, and the highest number you can get is four. On the far right of the graph is the point that represents a spin of 4 fours, {4, 4, 4, 4}. On the far left is the point that represents a spin of 4 ones, {1, 1, 1, 1}. Notice that 4 ones happens more than 4 fours. Why is that? If you take a look at the spinner, you'll see that there are more ones on the spinner than there are fours.
You can also notice that, since there are more ones, this actually pulls the average down a bit. The most frequent average is 2.25, not 2.5, which would be the exact middle between 1 and 4. Therefore, this distribution is skewed slightly to the right because the numbers on the spinner are not evenly distributed.
Source: Adapted from Sophia tutorial by Jonathan Osters.