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

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 geometric interpretations and visualizations to reinforce the necessity of this correction. It further introduces the concept of Bariance an alternative dispersion measure based on pairwise squared differences that avoids reliance on the arithmetic mean. Building on this we address practical concerns raised in Rosenthals article [Rosenthal, 2015] 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 approach can yield a computational advantage. Specifically the unbiased Bariance estimator can be computed in linear time resulting in shorter runtimes while preserving statistical validity.