Unified approach to optimal estimation of mean and standard deviation from sample summaries

Abstract

Recently, various methods have been developed to estimate the sample mean and standard deviation when only the sample size, and other selected sample summaries are reported. In this paper, we provide a unified approach to optimal estimation that can be easily adopted when only some summary statistics are reported. We show that the proposed estimators have the lowest variance among linear unbiased estimators. We also show that in the most commonly reported cases, that is, when only a three-number or five-number summary is reported, the newly proposed estimators match the previously developed estimators. Finally, we demonstrate the performance of the estimators numerically.