A measure of central tendency that represents the typical value of a set of numbers when those numbers are thought of as multiplicative in nature.
The geometric mean for a series of n numbers is the n-th root of their product. It is commonly used when the product of a set of numbers provides a better indication of its typical value than does the sum.
For example, a natural application of the geometric mean is calculating the average rate of return over several years of varied annual rates of return.
The geometric mean of a strictly positive random variable $X$ is $$ \text{GM}(X)=\exp\left(\mathbb{E}[\log(X)]\right). $$ For a discrete variable $X$ you can write the geometric mean as $\prod_i x_i^{p_i}$ where $p_i={p(X=x_i)}$. (Source)
