Weighted mean

If are several independent unbiased measurements of a physical quantity and if the measurements have the standard deviations  , then the weighted mean or weighted average

is an unbiased estimator of if the weights w₁ are independent of the X₁. The variance of is

The minimal variance

is obtained with the weights

If the individual standard deviations are all equal to , the weighted average, with these weights, reduces to the '

( Mean) arithmetic mean

with the variance

If all X₁ are Gaussian random variables, so is the weighted mean ( Convolution).