Geometric Mean
The [latex]n[/latex]-th root of the product of [latex]n[/latex] 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 [latex]\overline{x}_g[/latex] to designate the geometric mean of a distribution. Some authors also use G or [latex]\overline{x}^{G}[/latex].
The geometric mean of two numbers [latex]a[/latex] and [latex]b[/latex] is a number [latex]c[/latex] such that [latex]\dfrac{a}{c}[/latex] = [latex]\dfrac{c}{b}[/latex].
Therefore, [latex]c^{2}[/latex] = [latex]ab[/latex] and [latex]c[/latex] = [latex]\sqrt{ab}[/latex].
Example
Consider this distribution: 2, 2, 4, 5, 5, 7, 8, 10. The geometric mean [latex]\overline{x}_g[/latex] of this distribution is:[latex]\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}[/latex]