A simple illustration the weighted mean

A vector x of n random variables has been measured m times, the ith measurement resulting in an estimate of the mean value x, and of the covariance matrix St. A best estimate Jt of the pooled ( weighted ) average makes the sum of squared statistical distances to each x minimum. The scalar expression 

As a special case, replacing matrices and vectors by scalars, the case n = 1 reduces to the well-known weighted average 

From the definition of variance and correlation coefficient and equation (4.2.18), we find that the covariance matrix is 

The standard error matrix of the weighted average is calculated from equation (5.4.6) as 

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