# Definition:Weighted Mean

## Definition

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

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

The **weighted mean** of $S$ is defined as:

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

This means that elements of $S$ with a larger weight contribute more to the **weighted mean** than those with a smaller weight.

If we write:

- $\forall i: 1 \le i \le n: w_i = \map W {x_i}$

we can write this **weighted mean** as:

- $\bar x := \dfrac {w_1 x_1 + w_2 x_2 + \cdots + w_n x_n} {w_1 + w_2 + \cdots + w_n}$

From the definition of the weight function, none of the weights can be negative.

While some of the weights may be zero, not *all* of them can, otherwise we would be dividing by zero.

### Normalized Weighted Mean

Let the weights be normalized.

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

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

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

## Also see

When the weight function is defined as:

- $\forall i: 1 \le i \le n: \map W {x_i} = w$

where $w$ is constant, the formula simplifies to the arithmetic mean:

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

So it can be seen that the arithmetic mean is a special case of the **weighted mean**.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**weighted mean** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**mean** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**weighted mean**