# Why is the unbiased sample variance estimator so ubiquitous in science?

The bias-corrected sample variance $$\hat{\sigma}^2 = \dfrac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$$ is commonly used in natural sciences to estimate the variance $$\sigma^2$$ of a Gaussian random variable $$X$$ from an i.i.d. sample $$X_1, \dots, X_n$$.

One of the earliest things I learned in statistical inference classes is that bias is only one of many relevant properties when comparing estimators. My understanding is that mean square error (MSE) is generally a better measure of estimator performance than bias alone.

Why then is the minimum MSE estimator $$\hat{\sigma}_{MMSE}^2 = \dfrac{1}{n+1} \sum_{i=1}^n (X_i - \bar{X})^2$$ not the standard choice for this problem?

• Obviously, partly because unbiasedness is preferred over minmizing MSE. – BruceET Apr 11 '19 at 23:50

• Actually, one could perfectly well use the MMSE estimator in formulas for confidence intervals--but that would merely complicate them, because factors of $\sqrt{(n+1)/(n-1)}$ would need to be introduced! – whuber Nov 23 '19 at 17:33