# Handbook of Descriptive Statistics/Measures of Statistical Variability/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. 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

## DerivationEdit

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

Thus

- ln(geometric SD of
*A*_{1}, ...,*A*_{n}) = arithmetic (i.e. usual) SD of ln(*A*_{1}), ..., ln(*A*_{n}).

- ln(geometric SD of