Editing Weighted arithmetic mean (section)

====Frequency weights====
If the weights are ''frequency weights'' (where a weight equals the number of occurrences), then the unbiased estimator is:

:<math>
s^2\ = \frac {\sum\limits_{i=1}^N w_i \left(x_i - \mu^*\right)^2} {\sum_{i=1}^N w_i - 1}
</math>

This effectively applies Bessel's correction for frequency weights. For example, if values <math>\{2, 2, 4, 5, 5, 5\}</math> 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 <math>\{2, 4, 5\}</math> with corresponding weights <math>\{2, 1, 3\}</math>, and we get the same result either way.

If the frequency weights <math>\{w_i\}</math> are normalized to 1, then the correct expression after Bessel's correction becomes

:<math>s^2\ = \frac {\sum_{i=1}^N w_i} {\sum_{i=1}^N w_i - 1}\sum_{i=1}^N w_i \left(x_i - \mu^*\right)^2</math>

where the total number of samples is <math>\sum_{i=1}^N w_i</math> (not <math>N</math>). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if <math>w_i</math> has a different meaning other than frequency weight.

The estimator can be unbiased only if the weights are not [[Standard score|standardized]] nor [[Normalization (statistics)|normalized]], these processes changing the data's mean and variance and thus leading to a [[Base rate fallacy|loss of the base rate]] (the population count, which is a requirement for Bessel's correction).