Editing Geometric standard deviation (section)

==Derivation==

If the geometric mean is

<math display="block">\mu_\mathrm{g} = \sqrt[n]{A_1 A_2 \cdots A_n}</math>

then taking the [[natural logarithm]] of both sides results in

<math display="block">\ln \mu_\mathrm{g} = {1 \over n} \ln (A_1 A_2 \cdots A_n).</math>

The logarithm of a product is a sum of logarithms (assuming <math display="inline">A_i</math> is positive for all {{nowrap|<math display="inline">i</math>),}} so

<math display="block">\ln \mu_\mathrm{g} = {1 \over n} \left[ \ln A_1 + \ln A_2 + \cdots + \ln A_n \right].</math>

It can now be seen that <math>\ln \mu_\mathrm{g}</math> is the [[arithmetic mean]] of the set {{nowrap|<math>\{ \ln A_1, \ln A_2, \dots , \ln A_n \}</math>,}} therefore the arithmetic standard deviation of this same set should be

<math display="block">\ln \sigma_\mathrm{g} = \sqrt{ {1\over n} \sum_{i=1}^n (\ln A_i - \ln \mu_\mathrm{g})^2 }\,.</math>

This simplifies to

<math display="block"> \sigma_\mathrm{g} = \exp \sqrt{ {1\over n}\sum_{i=1}^n \left( \ln {A_i \over \mu_\mathrm{g}} \right)^2 }\,.</math>