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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 {A1, A2, ..., An} is denoted as μg, 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 A1, ..., An) = arithmetic (i.e. usual) SD of ln(A1), ..., ln(An).