Definition:Weighted Mean/Normalized

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 $1$.