A sampling distribution is the probability distribution of a statistic, obtained under the assumption that we can generate many different random samples from the population. The sampling distribution depends on the population distribution and the sample size.
Examples
Sample mean. Suppose we have a random sample of size N, {Xj, j=1,...,N} from a normal distribution with mean µ and variance σ2. The sample mean is a statistic (a function of the random sample):
The sampling distribution of is a normal distribution since the statistic is the sum of independent normal random variables. The population mean of
is µ and the population standard deviation is
.
Sample Proportion. The sampling distribution of the sample proportion, , is binomial with a mean equal to the population proportion, π, and a standard deviation equal to
. If N is large enough, the sampling distribution can be approximated by the normal distribution.