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 produce an unbiased estimator for the population variance. This paper includes rigorous derivations, geometric interpretations, and visualizations. It then introduces the concept of "bariance", an alternative pairwise distances intuition of sample dispersion without an arithmetic mean. Finally, we address practical concerns raised in Rosenthal’s article[1] advocating the use of \(n\)-based estimates from a more holistic \(MSE\)-based viewpoint for pedagogical reasons and in certain practical contexts. Finally, the empirical part using simulation reveals that the run-time of estimating population variance can be shortened when using an algebraically optimized “bariance“ approach to estimate an unbiased variance.