Definition:Geometric Mean

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This page is about Geometric Mean in the context of Algebra. For other uses, see Mean.


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:

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

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

Mean Proportional

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 $c$.

Also see

  • Results about geometric mean can be found here.