Editing Geometric standard deviation

{{Short description|Statistical measure}}
{{more citations needed|date=May 2016}}
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 [[Dimensional analysis|dimension]] as the input values. Thus, the geometric standard deviation may be more appropriately called '''geometric SD factor'''.<ref>[http://www.graphpad.com/guides/prism/7/statistics/stat_the_geometric_mean_and_geometr.htm?toc=0&printWindow GraphPad Guide]</ref><ref>Kirkwood, T.B.L. (1993). [https://doi.org/10.3109/03639049309038775 "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">{{cite journal |last1=Kirkwood |first1=T.B.L. |title=Geometric means and measures of dispersion |journal=Biometrics |year=1979 |volume=35 |pages=908–9| jstor = 2530139}}</ref>

==Definition==
If the geometric mean of a set of numbers <math display="inline>{A_1, A_2, ..., A_n}</math> is denoted as {{nowrap|<math display="inline">\mu_\mathrm{g}</math>,}} 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>

==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>

==Geometric standard score==

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 distribution==
The geometric standard deviation is used as a measure of [[log-normal distribution|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. {{nowrap|<math>\sigma_\mathrm{g} = \exp(\operatorname{stdev}(\ln A))</math>.}}

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 interval]]s 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.

==References==
{{Reflist}}

==External links==
* [https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]

{{DEFAULTSORT:Geometric Standard Deviation}}
[[Category:Scale statistics]]
[[Category:Non-Newtonian calculus]]