Definition:Weighted Mean/Normalized

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Definition

Let $S = \sequence {x_1, x_2, \ldots, x_n}$ be a sequence of real numbers.

Let $\map W x$ be a weight function to be applied to the terms of $S$.

Let the weights be normalized.

Then the weighted mean of $S$ can be expressed in the form:

$\ds \bar x := \sum_{i \mathop = 1}^n \map W {x_i} x_i$

as by definition of normalized weight function all the weights add up to $1$.