The sampling distribution can be described by calculating its mean and standard error. The central limit theorem states that if the sample is large enough, its distribution will approximate that of the population you took the sample from. This means that if the population had a normal distribution, so will the sample. If you do not know the population distribution, it is generally assumed to be normal. You will need to know the standard deviation of the population in order to calculate the sampling distribution.
Add all of the observations together and then divide by the total number of observations in the sample. For example, a sample of heights of everyone in a town might have observations of 60 inches, 64 inches, 62 inches, 70 inches and 68 inches and the town is known to have a normal height distribution and standard deviation of 4 inches in its heights. The mean would (60+64+62+70+68) / 5 = 64.8 inches.
Add 1 / sample size and 1 / population size. If the population size is very large, all the people in a city for example, you need only divide 1 by the sample size. For the example, a town is very large, so it would just be 1 / sample size or 1/5 = 0.20.
Take the square root of the result from Step 2 and then multiply it by the standard deviation of the population. For the example, the square root of 0.20 is 0.45. Then, 0.45 x 4 = 1.8 inches. The sample's standard error is 1.8 inches. Together, the mean, 64.8 inches, and the standard error, 1.8 inches, describe the sample distribution. The sample has a normal distribution because the town does.
- calculator image by Szymon Apanowicz from Fotolia.com