On Bessel’s Correction: Unbiased Sample Variance, the Bariance, and a Novel Runtime-Optimized Estimator

Abstract

Bessel’s correction adjusts the denominator in the sample variance formula from n to n – 1 to ensure an unbiased estimator of the population variance. This paper provides rigorous algebraic derivations to reinforce the necessity of this correction. It further introduces the concept of Bariance, an alternative dispersion measure based on average pairwise squared differences that avoids reliance on the arithmetic mean. Building on this, we address practical concerns raised in Rosenthal’s article, which advocates for n-based estimates from a Mean Squared Error (MSE) perspective, particularly in pedagogical contexts and specific applied settings. Finally, the empirical component of this work, based on simulation studies, demonstrates that estimating the population variance via an algebraically optimized Bariance formula approach can yield a computational advantage. Specifically, the unbiased sample variance can be computed in linear time using the optimized Bariance estimator, resulting in shorter run-times while preserving statistical validity.