Mathematical Definition
Formally, the weighted mean of a non-empty set of data
with non-negative weights
is the quantity
which means:
Therefore data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).
The formulas are simplified when the weights are normalized such that they sum up to, i.e. . For such normalized weights the weighted mean is simply .
Note that one can always normalize the weights by making the following transformation on the weights . Using the normalized weight yields the same results as when using the original weights. Indeed,
The common mean is a special case of the weighted mean where all data have equal weights, . When the weights are normalized then
Read more about this topic: Weighted Mean
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