Is a weighted average, assuming the weights sum up to one, always smaller than the unweighted arithmetic average? As if the weights are smaller than one the weighted mean is a convex combination of the individual points. Is there any condition on the individual observation for the statement to hold true? Would be great if you could provide a mathematical proof. Thank you!
[Math] Is weighted average always smaller or equal than arithmetic average
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Best Answer
This is not normally true, as examples in the comments show. However, there is a natural condition which will make it true: the condition that the weights are in the opposite order to the numbers, i.e. smaller numbers get larger weights. This then follows from Chebyshev's sum inequality.
More generally, if weights and numbers are negatively correlated then the weighted mean will be less than the arithmetic mean. However, this is not a very deep statement, since that is basically how "negatively correlated" is defined.