Geometric standard deviation

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then taking the natural logarithm of both sides results in

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

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

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

It can now be seen that ${\displaystyle \ln \mu _{\mathrm {g} }}$ 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

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

This simplifies to

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

The geometric version of the standard score is

$${\displaystyle z={{\ln x-\ln \mu _{\mathrm {g} }} \over \ln \sigma _{\mathrm {g} }}=\log _{\sigma _{\mathrm {g} }}\left({x \over \mu _{\mathrm {g} }}\right).}$$

If the geometric mean, standard deviation, and z-score of a datum are known, then the

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

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. ${\displaystyle \sigma _{\mathrm {g} }=\exp(\operatorname {stdev} (\ln A))}$.

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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