Handbook of Descriptive Statistics/Measures of Statistical Variability/Geometric Standard Deviation

In probability theory and statistics, the geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean. If the geometric mean of a set of numbers {A₁, A₂, ..., A_n} is denoted as μ_g, then the geometric standard deviation is

$\sigma _{g}=\exp \left({\sqrt {\sum _{i=1}^{n}(\ln A_{i}-\ln \mu _{g})^{2} \over n}}\right).\qquad \qquad (1)$

Derivation

If the geometric mean is

$\mu _{g}={\sqrt[{n}]{A_{1}A_{2}\cdots A_{n}}}.\,$

then taking the natural logarithm of both sides results in

$\ln \mu _{g}={1 \over n}\ln(A_{1}A_{2}\cdots A_{n}).$

The logarithm of a product is a sum of logarithms (assuming $A_{i}$ is positive for all $i$), so

$\ln \mu _{g}={1 \over n}[\ln A_{1}+\ln A_{2}+\cdots +\ln A_{n}].\,$

It can now be seen that $\ln \,\mu _{g}$ is the arithmetic mean of the set $\{\ln A_{1},\ln A_{2},\dots ,\ln A_{n}\}$, therefore the arithmetic standard deviation of this same set should be

$\ln \sigma _{g}={\sqrt {\sum _{i=1}^{n}(\ln A_{i}-\ln \mu _{g})^{2} \over n}}.$

Thus

ln(geometric SD of A₁, ..., A_n) = arithmetic (i.e. usual) SD of ln(A₁), ..., ln(A_n).