Weighted arithmetic mean

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the simple arithmetic mean.

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


 * $$\{x_1, x_2, \dots, x_n\},$$

(where x represents a set of mean values) with non-negative weights


 * $$\bar{x} = \frac{ \sum\limits_{i=1}^n w_i x_i}{\sum\limits_{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 $$1$$, i.e.:
 * $$ \sum_{i=1}^n {w_i'} = 1$$.

For such normalized weights the weighted mean is then:
 * $$\bar {x} = \sum_{i=1}^n {w_i' x_i}$$.

Note that one can always normalize the weights by making the following transformation on the original weights:
 * $$w_i' = \frac{w_i}{\sum_{j=1}^n{w_j}}$$.

Using the normalized weight yields the same results as when using the original weights:
 * $$\begin{align}

\bar{x} &= \sum_{i=1}^n w'_i x_i= \sum_{i=1}^n \frac{w_i}{\sum_{j=1}^n w_j} x_i = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{j=1}^n w_j} \\ & = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}. \end{align} $$ The ordinary mean $$\frac {1}{n}\sum_{i=1}^n {x_i}$$ is a special case of the weighted mean where all data have equal weights, $$w_i=1$$.