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. For such data, it may be preferred to the more usual standard deviation.

Definition

If the geometric mean of a set of numbers {A₁, A₂, ..., A_n} is denoted as μ_g, then the geometric standard deviation is

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

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

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

This simplifies to

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

Geometric standard score

The geometric version of the standard score is

$z={\ln(x/\mu _{g}) \over \ln \sigma _{g}}.\,$

If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score can be reconstructed by

$x=\mu _{g}\sigma _{g}^{z}.$

Relationship to log-normal distribution

The geometric standard deviation is related to the log-normal distribution. The log-normal distribution is a distribution which is normal for the logarithm transformed values. By a simple set of logarithm transformations we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log transformed values (e.g. exp(stdev(ln(A))));

As such, the geometric mean and the geometric standard deviation of a sample of data from a log-normally distributed population may be used to find the bounds of confidence intervals analogously to the way the arithmetic mean and standard deviation are used to bound confidence intervals for a normal distribution. See discussion in log-normal distribution for details.