Geometric standard deviation
Template:Short description Template:More citations needed In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor.<ref>GraphPad Guide</ref><ref>Kirkwood, T.B.L. (1993). "Geometric standard deviation - reply to Bohidar". Drug Dev. Ind. Pharmacy 19(3): 395-6.</ref> When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from (the geometric mean divided by the geometric SD factor) to (the geometric mean multiplied by the geometric SD factor), and one cannot add/subtract "geometric SD factor" to/from geometric mean.<ref name="Geometric means and measures of dispersion">Template:Cite journal</ref>
DefinitionEdit
If the geometric mean of a set of numbers <math display="inline>{A_1, A_2, ..., A_n}</math> is denoted as Template:Nowrap then the geometric standard deviation is
<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>
DerivationEdit
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 Template:Nowrap 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 Template:Nowrap 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>
Geometric standard scoreEdit
The geometric version of the standard score is
<math display="block"> z = {{\ln x - \ln \mu_\mathrm{g}} \over \ln \sigma_\mathrm{g}} = \log _{\sigma_\mathrm{g}} \left({x \over \mu_\mathrm{g}}\right).</math>
If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score can be reconstructed by
<math display="block">x = \mu_\mathrm{g} {\sigma_\mathrm{g}}^z.</math>
Relationship to log-normal distributionEdit
The geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean.<ref name="Geometric means and measures of dispersion" /> As the log-transform of a log-normal distribution results in a normal distribution, we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e. Template:Nowrap
As such, the geometric mean and the geometric standard deviation of a sample of data from a log-normally distributed population may be used to find the bounds of confidence intervals analogously to the way the arithmetic mean and standard deviation are used to bound confidence intervals for a normal distribution. See discussion in log-normal distribution for details.