Weighted Mean - Mathematical Definition

Mathematical Definition

Formally, the weighted mean of a non-empty set of data

with non-negative weights

is the quantity

$\bar{x} = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i},$

which means:

$\bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}.$

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,

$\bar{x} = \sum_{i=1}^n w'_i x_i= \sum_{i=1}^n \frac{w_i}{\sum_{i=1}^n w_i} x_i = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}.$

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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