Sample variance

In statistics, Bessel’s correction, named after Friedrich Bessel, is the use of n – 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample: it corrects the bias in the estimation of the population variance, and some (but not all) of the bias in the estimation of the population standard deviation.

That is, when estimating the population variance and standard deviation from a sample when the population mean is unknown, the sample variance is a biased estimator of the population variance, and systematically underestimates it. Multiplying the standard sample variance by n/(n – 1) (equivalently, using 1/(n – 1) instead of 1/n in the estimator’s formula) corrects for this, and gives an unbiased estimator of the population variance.

— Wikipedia on Bessel’s correction

The two estimators only differ slightly as can be seen, and for larger values of the sample size n the difference is negligible. While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E[s^2] is equal to the true variance of the sampled random variable; the use of the term n – 1 is called Bessel’s correction.

— Wikipedia on Sample variance

2012.05.16 Wednesday ACHK