Editing Weighted arithmetic mean (section)

===Variance-defined weights===
{{main|Inverse-variance weighting}}
{{see also|Weighted least squares}}

For the weighted mean of a list of data for which each element <math>x_i</math> potentially comes from a different [[probability distribution]] with known [[variance]] <math>\sigma_i^2</math>, all having the same mean, one possible choice for the weights is given by the reciprocal of variance:

:<math>w_i = \frac{1}{\sigma_i^2}.</math>

The weighted mean in this case is:
:<math>\bar{x} = \frac{ \sum_{i=1}^n \left( \dfrac{ x_i}{\sigma_i^{2}} \right)}{\sum_{i=1}^n \dfrac{1}{\sigma_i^{2}}}
=\frac{\sum_{i=1}^n\left(x_i\cdot w_i\right)}{\sum_{i=1}^n w_i},</math>
and the ''standard error of the weighted mean (with inverse-variance weights)'' is:
:<math>\sigma_{\bar{x}} = \sqrt{\frac{ 1 }{\sum_{i=1}^n \sigma_i^{-2}}}
=\sqrt{\frac{1}{\sum_{i=1}^n w_i}},</math>

Note this reduces to <math> \sigma_{\bar{x}}^2 = \sigma_0^2/n</math> when all <math>\sigma_i = \sigma_0</math>.
It is a special case of the general formula in previous section,
:<math> \sigma^2_{\bar x} = \sum_{i=1}^n {w_i'^2 \sigma^2_i} = \frac{ \sum_{i=1}^n {\sigma_i^{-4} \sigma^2_i} }{\left(\sum_{i=1}^n \sigma_i^{-2}\right)^2}.</math>

The equations above can be combined to obtain:
:<math>\bar{x} = \sigma_{\bar{x}}^2 \sum_{i=1}^n \frac{x_i}{\sigma_i^2}.</math>

The significance of this choice is that this weighted mean is the [[maximum likelihood estimator]] of the mean of the probability distributions under the assumption that they are independent and [[normally distributed]] with the same mean.