Weighted geometric mean

In statistics, given a set of data,

X = { x₁, x₂, ..., x_n}

and corresponding 'weights',

W = { w₁, w₂, ..., w_n}

the weighted geometric mean is

${\bar {x}}=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}=\quad \exp \left({\frac {1}{\sum _{i=1}^{n}w_{i}}}\;\sum _{i=1}^{n}w_{i}\ln x_{i}\right)$

If all the weights are equal, the weighted geometric mean is equal to the geometric mean.

Weighted versions of other means can also be calculated. Probably the best known weighted mean is the weighted arithmetic mean, usually simply called the weighted mean. Another example of a weighted mean is the weighted harmonic mean.