Bessel's correction

Bessel’s correction is the convention of dividing by N−1 instead of N when computing sample variance and sample standard deviation. The correction compensates for the systematic under-estimation of population variance that results from using the sample mean (which is closer to the data than the unknown true population mean would be) as the centring point.

Named after Friedrich Bessel, the 19th-century German astronomer and mathematician. The mathematical proof: E[Σ(x − x̄)²] = (N−1)σ² where σ² is the true population variance and the expectation runs over all possible samples of size N. Dividing the observed sum by N−1 produces an unbiased estimator of σ².

At large N the correction is negligible (1/N vs 1/(N−1)). At small N it matters meaningfully. Our statistics calculator defaults to the Bessel-corrected form.