Editing Weighted arithmetic mean (section)

==== Other notes ====

For uncorrelated observations with variances <math>\sigma^2_i</math>, the variance of the weighted sample mean is{{Citation needed|date=October 2018}}
: <math> \sigma^2_{\bar x} = \sum_{i=1}^n {w_i'^2 \sigma^2_i}</math>
whose square root <math>\sigma_{\bar x}</math> can be called the ''standard error of the weighted mean (general case)''.{{Citation needed|date=October 2018}}{{anchor|Standard error}}

Consequently, if all the observations have equal variance, <math>\sigma^2_i= \sigma^2_0</math>, the weighted sample mean will have variance
: <math> \sigma^2_{\bar x} =  \sigma^2_0 \sum_{i=1}^n {w_i'^2},</math>
where <math display="inline">1/n \le \sum_{i=1}^n {w_i'^2} \le 1</math>. The variance attains its maximum value, <math>\sigma_0^2</math>, 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 <math display="inline"> \sigma_{\bar x} = \sigma_0 / \sqrt {n} </math>, i.e., it degenerates into the [[standard error of the mean]], squared.

Because one can always transform non-normalized weights to normalized weights, all formulas in this section can be adapted to non-normalized weights by replacing all <math>w_i' = \frac{w_i}{\sum_{i=1}^n{w_i}}</math>.