On the Variance of Weighted Means

Abstract

Consider a situation where an infinite population is subdivided into a large (infinite) number of sub-populations or strata. k strata are selected at random, and from the ith stratum (i = 1, 2, …, k)n i random observations x ij (j = 1, 2, …, n i ) are obtained. Let m be the mean of a random stratum and μ = E(m) the mean of the whole population. The aim of this paper is to estimate μ by the most suitable weighted average of the x ij , where the weights may or may not depend on the x ij . The model used is x ij = m i + e ij , where the m i = μ + a i are independent normal variates of mean μ and variance σ2 (m) The e ij are independent normal variates of mean zero and variances σ2 (m) independent of the m i . For example, m may be a dimension of a massproduced article, which varies from article to article and which can be measured without bias but with a random error e. Two special cases, σ2 (m) = 0 and all σ2 i = σ2 are given special attention.