When a random sample is drawn, some summary of the values is usually computed. Any well-defined summary may be expressed mathematically as a function of an n-dimension vector. The domain of that function includes the support of the joint probability function of the sample. The function may be real-valued or vector-valued; thus the summary is a random variable (or random vector). This definition of a random variable as a function of others has been treated in lessons on transformation methods. Since a random sample has a simple probabilistic structure, the distribution of the statistic is usually not too difficult to find. Because the distribution of the statistic is derived from the sample, it is called the sampling distribution of the statistic. This distinguishes the probability distribution of the statistic from the distribution of the population from which the sample was drawn, which is the marginal distribution for any one of the random variables in the sample. In this lesson, we will discuss some properties of sampling distributions, especially for functions that are sums of random variables from a random sample.
