Weighted mean

If

Error creating thumbnail: Unable to save thumbnail to destination

are several independent unbiased measurements of a physical quantity

Error creating thumbnail: Unable to save thumbnail to destination

and if the measurements have the standard deviations

Error creating thumbnail: Unable to save thumbnail to destination

 , then the weighted mean or weighted average

Error creating thumbnail: Unable to save thumbnail to destination

is an unbiased estimator of

Error creating thumbnail: Unable to save thumbnail to destination

if the weights w₁ are independent of the X₁. The variance of

Error creating thumbnail: Unable to save thumbnail to destination

is

Error creating thumbnail: Unable to save thumbnail to destination

The minimal variance

Error creating thumbnail: Unable to save thumbnail to destination

is obtained with the weights

Error creating thumbnail: Unable to save thumbnail to destination

If the individual standard deviations are all equal to

Error creating thumbnail: Unable to save thumbnail to destination

, the weighted average, with these weights, reduces to the ' (

Error creating thumbnail: Unable to save thumbnail to destination

Mean) arithmetic mean

Error creating thumbnail: Unable to save thumbnail to destination

with the variance

Error creating thumbnail: Unable to save thumbnail to destination

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

Error creating thumbnail: Unable to save thumbnail to destination

Convolution).