Editing Weighted arithmetic mean (section)

===Correcting for over- or under-dispersion===
{{see also|#Weighted sample variance}}
Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that <math>\chi^2</math> is too large. The correction that must be made is

:<math>\hat{\sigma}_{\bar{x}}^2 = \sigma_{\bar{x}}^2 \chi^2_\nu </math>

where <math>\chi^2_\nu</math> is the [[reduced chi-squared]]:

:<math>\chi^2_\nu = \frac{1}{(n-1)} \sum_{i=1}^n \frac{ (x_i - \bar{x} )^2}{ \sigma_i^2 };</math>

The square root <math>\hat{\sigma}_{\bar{x}}</math> can be called the ''standard error of the weighted mean (variance weights, scale corrected)''.

When all data variances are equal, <math>\sigma_i = \sigma_0</math>, they cancel out in the weighted mean variance, <math>\sigma_{\bar{x}}^2</math>, which again reduces to the [[standard error of the mean]] (squared), <math>\sigma_{\bar{x}}^2 = \sigma^2/n</math>, formulated in terms of the [[sample standard deviation]] (squared),
:<math>\sigma^2 = \frac {\sum_{i=1}^n (x_i - \bar{x} )^2} {n-1}. </math>