Definition:Geometric Mean

This page is about the geometric mean. For other uses, see Definition:Mean.

Definition

Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers.

The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:

$\displaystyle G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$

That is, to find out the geometric mean of a set of $n$ numbers, multiply them together and take the $n$th root.

In the language of Euclid, the geometric mean of two magnitudes is called the mean proportional.

Thus the mean proportional of $a$ and $b$ is defined as that magnitude $c$ such that:

$a : c = c : b$

where $a : c$ denotes the ratio between $a$ and $c$.