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

${\sigma }_g=\exp \left(\sqrt{\frac{\sum _{i=1}^n(\ln A_i-\ln {\mu }_g{)}^2}{n}}\right).(1)$

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

$\ln {\mu }_g=\frac{1}{n}\ln (A_1{A}_2\cdots A_n).$

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

$\ln {\mu }_g=\frac{1}{n}[\ln A_1+\ln A_2+\cdots +\ln A_n].$

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

$\ln {\sigma }_g=\sqrt{\frac{\sum _{i=1}^n(\ln A_i-\ln {\mu }_g{)}^2}{n}}.$

Thus

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