The geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean. If the mean of a set of numbers {A1, A2, ... , An} is denoted as , 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, 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
.
Exponentiating both sides results in equation (1). Q.E.D.

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

See also: natural logarithm.