Statistical Properties
The weighted sample mean, with normalized weights (weights summing to one) 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,
If the observations have expected values
then the weighted sample mean has expectation
Particularly, if the expectations of all observations are equal, then the expectation of the weighted sample mean will be the same,
For uncorrelated observations with standard deviations, the weighted sample mean has standard deviation
Consequently, when the standard deviations of all observations are equal, the weighted sample mean will have standard deviation . Here is the quantity
such that . It attains its minimum value for equal weights, and its maximum when all weights except one are zero. In the former case we have, which is related to the central limit theorem.
Note that due to the fact that one can always transform non-normalized weights to normalized weights all formula in this section can be adapted to non-normalized weights by replacing all by .
Read more about this topic: Weighted Mean
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