Inverse-variance weighting

In statistics, inverse-variance weighting is a method of aggregating two or more random variables to minimize the variance of the weighted average. Each random variable is weighted in inverse proportion to its variance.

Given a sequence of independent observations yi with variances σi2, the inverse-variance weighted average is given by[1]

$\hat{y} = \frac{\sum_i y_i / \sigma_i^2}{\sum_i 1/\sigma_i^2} .$

The inverse-variance weighted average has the least variance among all weighted averages, which can be calculated as

$D^2(\hat{y}) = \frac{1}{\sum_i 1/\sigma_i^2} .$

If the variances of the measurements are all equal, then the inverse-variance weighted average becomes the simple average.

Inverse-variance weighting is typically used in statistical meta-analysis to combine the results from independent measurements.