Bessel's correction

In statistics, Bessel's correction 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. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population standard deviation. However, the correction often increases the mean squared error in these estimations. This technique is named after Friedrich Bessel. In estimating the population variance from a sample when the population mean is unknown, the uncorrected sample variance is the mean of the squares of deviations of sample values from the sample mean (i.e. using a multiplicative factor 1/n). In this case, the sample variance is a biased estimator of the population variance. Multiplying the uncorrected sample variance by the factor gives an unbiased estimator of the population variance. In some literature, the above factor is called Bessel's correction. One can understand Bessel's correction as the degrees of freedom in the residuals vector (residuals, not errors, because the population mean is unknown): where is the sample mean. While there are n independent observations in the sample, there are only n − 1 independent residuals, as they sum to 0. For a more intuitive explanation of the need for Bessel's correction, see . Generally Bessel's correction is an approach to reduce the bias due to finite sample size. Such finite-sample bias correction is also needed for other estimates like skew and kurtosis, but in these the inaccuracies are often significantly larger. To fully remove such bias it is necessary to do a more complex multi-parameter estimation. For instance a correct correction for the standard deviation depends on the kurtosis (normalized central 4th moment), but this again has a finite sample bias and it depends on the standard deviation, i.e. both estimations have to be merged. There are three caveats to consider regarding Bessel's correction: It does not yield an unbiased estimator of standard deviation.