The weighted sample mean, , is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one).

If the observations have expected values

then the weighted sample mean has expectation

In particular, if the means are equal, , then the expectation of the weighted sample mean will be that value,

Variance

Simple i.i.d. case

When treating

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