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.
Consider the task of estimating the population variance from a sample when the population mean is unknown. The sample variance is calculated as the mean of the squared 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 biased 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 intuitively 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 samples, there are only n − 1 independent residuals, as they sum to 0.
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