The Expected Sample Variance of Uncorrelated Random Variables with a Common Mean and Some Applications in Unbalanced Random Effects Models

There is a little-known but very simple generalization of the standard result that for uncorrelated random variables with common mean μ and variance 2 σ , the expected value of the sample variance is 2 σ . The generalization justifies the use of the usual standard error of the sample mean in possibly heteroscedastic situations, and motivates elementary estimators in even unbalanced linear random effects models. The latter both provides nontrivial examples and exercises concerning method-of-moments estimation, and also helps "demystify" the whole matter of variance component estimation. This is illustrated in general for the simple one-way context and for a specific unbalanced two-factor hierarchical data structure. The Expected Value of the Sample Variance It is completely standard in first courses in statistical theory at a variety of levels to prove that the expected value of the sample variance of independent identically distributed observations is the common variance. (See for example Wackerly, Mendenhall and Scheaffer (2002, page 372), Miller and Miller (2004, page 321), Wasserman (2004, page 52) and Casella and Berger (2002, page 213).) It is no harder to show something more general. Namely, there is the simple result below. * Financial support of the Deutsche Forschungsgemeinschaft (SFB 475, "Reduction of Complexity in Multivariate Data Structures") through the University of Dortmund and of the Los Alamos National Laboratory Statistical Sciences Group is gratefully acknowledged.