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}\)