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:
- $\ds \bar x := \sum_{i \mathop = 1}^n \map W {x_i} x_i$
as by definition of normalized weight function all the weights add up to $1$.
Also see
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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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): weighted mean
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): weighted mean
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): mean
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): weighted mean
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): mean
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): weighted mean