Inverse-variance weighted mean ($\hat{\mu }={\sum }_i{w}_i{X}_i$)

For independent unbiased estimators $X_i$ of a common mean with known positive variances $v_i$, the linear unbiased estimator with smallest variance has weights $w_i=v_i^{-1}/{\sum }_j{v}_j^{-1}$ and variance $({\sum }_j{v}_j^{-1}{)}^{-1}$. Unbiasedness requires ${\sum }_i{w}_i=1$. Minimizing ${\sum }_i{v}_i{w}_i^2$ by a Lagrange multiplier gives the weights, and the strictly positive diagonal Hessian matrix proves a unique global minimum. This conclusion requires no normality assumption.

A weight $w_i=1/v_i$ gives more weight to a more precise estimate. In random-effects meta-analysis the variance of the corresponding marginal distribution is $v_i+{\tau }^2$, giving weight $(v_i+{\tau }^2{)}^{-1}$. A percentage weight divides the individual weight by the sum of all weights.