# Geometric Mean

## Geometric Mean

The $$n$$-th root of the product of $$n$$ values in a distribution of a quantitative statistical characteristic.

### Notation

Because the geometric mean is a different measure from the arithmetic mean, we use the notation $$\overline{x}_g$$ to designate the geometric mean of a distribution.

Some authors also use G or $$\overline{x}^{G}$$.

The geometric mean of two numbers $$a$$ and $$b$$ is a number $$c$$ such that $$\dfrac{a}{c}$$ = $$\dfrac{c}{b}$$.
Therefore, $$c^{2}$$ = $$ab$$ and $$c$$ = $$\sqrt{ab}$$.

### Example

Consider this distribution: 2, 2, 4, 5, 5, 7, 8, 10.
The geometric mean $$\overline{x}_g$$ of this distribution is:

\begin{align}\overline{x}_g & = \sqrt [8] {2 \times 2 \times 4 \times 5 \times 5 \times 7 \times 8 \times 10} \\ & = \sqrt [8] {224\space000}\\ & \approx 4,66\\ \end{align}