Geometric standard deviation

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

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

If the geometric mean is

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

then taking the natural logarithm of both sides results in

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

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

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

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

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

This simplifies to

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

The geometric version of the standard score is

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

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

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

The geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean.[3] As the log-transform of a log-normal distribution results in a normal distribution, we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e. ${\displaystyle \sigma _{g}=\exp(\operatorname {stdev} (\ln(A)))}$.

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.

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