# 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. Note that unlike the usual *arithmetic* standard deviation, the *geometric* standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values.

## Definition

If the geometric mean of a set of numbers {*A*_{1}, *A*_{2}, ..., *A*_{n}} is denoted as μ_{g}, then the geometric standard deviation is

## Derivation

If the geometric mean is

then taking the natural logarithm of both sides results in

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

It can now be seen that is the arithmetic mean of the set , therefore the arithmetic standard deviation of this same set should be

This simplifies to

## Geometric standard score

The geometric version of the standard score is

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

## Relationship to log-normal distribution

The geometric standard deviation was proposed for the first time by Kirkwood (1979) as a measure of log-normal dispersion analogously to the geometric mean.^{[1]} 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. .

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.

## References

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