Weighted geometric mean
In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean. Given a sample x = ( x 1 , x 2 … , x n ) {\displaystyle x=(x_{1},x_{2}\dots ,x_{n})} and weights w = ( w 1 , w 2 , … , w n ) {\displaystyle w=(w_{1},w_{2},\dots ,w_{n})} , it is calculated as: x ¯ = ( ∏ i = 1 n x i w i ) 1 / ∑ i = 1 n w i = exp ( ∑ i = 1 n w i ln x i ∑ i = 1 n w i ) {\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}=\quad \exp \left({\frac {\sum _{i=1}^{n}w_{i}\ln x_{i}}{\sum _{i=1}^{n}w_{i}\quad }}\right)} The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.