Geometric standard deviation
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.