Definition: A sampling distribution shows how a statistic, such as a sample proportion or mean, varies across all possible samples of the same size from a population.
What is a sampling distribution?
Imagine taking many random samples of size n and computing the same statistic each time. The pattern of those values is the sampling distribution. It describes sample-to-sample variation, not the spread of individual data.
What are its center and spread?
For a sample proportion with true proportion p: the mean is p and the standard deviation is square root of p(1 – p)/n.
Worked example: suppose 40% of a town supports a bond, and you sample n = 100 people. The sample proportions center at 0.40 with standard deviation square root of (0.4 times 0.6/100) = square root of 0.0024, about 0.049.
Larger samples give less spread: with n = 400 the standard deviation drops to about 0.0245, half as large.
What do students mix up?
Confusing this with the distribution of the data in one sample. Mathbot says: one sample gives one dot; the sampling distribution is the whole pile of dots from many samples.