The mathematical expectation of sample variance: a general approach

The expected value of sample variance based on independently and identically distributed normal observations is well known and is often calculated by deriving its sampling distribution. However, the sampling distribution is difficult for other distributions, and more so if the observations are neither independently nor identically distributed. We demonstrate that the expected value in such a general situation depends on the second moment of the difference of pairs of its constituent random variables. We also prove, for this situation, an expression for expected variance that depends on the average of variances of observations, variation among true means, and the average of covariances of pairs of observations. Many special cases are expressed as corollaries to illustrate ideas. Some examples that provide insights in mathematical statistics are considered. An application to textile engineering is presented.