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 y_i with variances σ_i², 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.

See also

• Weighted least squares

References

1. Joachim Hartung, Guido Knapp, Bimal K. Sinha (2008). Statistical meta-analysis with applications. John Wiley & Sons. ISBN 978-0-470-29089-7.