square root can be called the standard error of the weighted mean (general case). Consequently, if all the observations have equal variance, , the weighted sample mean will have variance where . The variance attains its maximum value, , when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have , i.e., it degenerates into the standard error of the mean, squared. Note that because one can always transform non-normalized weights to normalized weights all formula in this section can be adapted to non-normalized weights by replacing all . Related concepts Weighted sample variance Typically when a mean is calculated it is important to know the variance and standard deviation about that mean. When a weighted mean is used, the variance of the weighted sample is different from the variance of the unweighted sample. The biased weighted sample variance is defined similarly to the normal biased sample variance : where for normalized weights. If the weights are frequency weights (and thus are random variables), it can be shown that is the maximum likelihood estimator of for iid Gaussian observations. For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N − 1 (see Bessel's correction). In the weighted setting, there are actually two different unbiased estimators, one for the case of frequency weights and another for the case of reliability weights. Frequency weights If the weights are frequency weights (where a weight equals the number of occurrences), then the unbiased estimator is: This effectively applies Bessel's correction for frequency weights. For example, if values are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample with corresponding weights , and we get the same result either way. If the frequency weights are normalized to 1, then the correct expression after Bessel's correction becomes where the total number of samples is (not ). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if has a different meaning other than frequency weight. Note that the estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing