On Bessel’s Correction: Unbiased Variance, Degrees of Freedom, and the Sum of Pairwise Differences

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. We then introduce the concept of “bariance,” an alternative pairwise interpretation of sample dispersion. Finally, we address practical concerns raised in Rosenthal’s article advocating the use of \(n\)-based estimates from a MSE-based viewpoint for practical reasons and in certain contexts. Finally the empirical part using simulation reveals a shorter runtime for estimating population variance can be shortened using an optimized “bariance“ approach using scalar sums.