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| 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]]. 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.
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| ==Definition==
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| If the geometric mean of a set of numbers {''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub>} is denoted as μ<sub>''g''</sub>, then the geometric standard deviation is
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| :<math> \sigma_g = \exp \left( \sqrt{ \sum_{i=1}^n ( \ln { A_i \over \mu_g } )^2 \over n } \right). \qquad \qquad (1) </math>
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| ==Derivation==
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| If the geometric mean is
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| :<math> \mu_g = \sqrt[n]{ A_1 A_2 \cdots A_n }.\, </math>
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| then taking the [[natural logarithm]] of both sides results in
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| :<math> \ln \mu_g = {1 \over n} \ln (A_1 A_2 \cdots A_n). </math> | |
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| The logarithm of a product is a sum of logarithms (assuming <math>A_i</math> is positive for all <math>i</math>), so
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| :<math> \ln \mu_g = {1 \over n} [ \ln A_1 + \ln A_2 + \cdots + \ln A_n ].\, </math>
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| It can now be seen that <math> \ln \, \mu_g </math> is the [[arithmetic mean]] of the set <math> \{ \ln A_1, \ln A_2, \dots , \ln A_n \} </math>, therefore the arithmetic standard deviation of this same set should be
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| :<math> \ln \sigma_g = \sqrt{ \sum_{i=1}^n ( \ln A_i - \ln \mu_g )^2 \over n }.</math>
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| This simplifies to
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| :<math> \sigma_g = \exp{\sqrt{ \sum_{i=1}^n ( \ln { A_i \over \mu_g } )^2 \over n }}. </math>
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| ==Geometric standard score== | |
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| The geometric version of the [[standard score]] is
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| :<math> z = {{\ln ( x ) - \ln ( \mu_g )} \over \ln \sigma_g } = {\log _{\sigma_g} (x / \mu_g)}.\, </math>
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| If the geometric mean, standard deviation, and z-score of a datum are known, then the [[raw score]] can be reconstructed by
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| :<math> x = \mu_g {\sigma_g}^z. </math>
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| ==Relationship to log-normal distribution==
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| The geometric standard deviation is related to the [[log-normal distribution]].
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| The log-normal distribution is a distribution which is normal for the logarithm
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| transformed values. By a simple set of logarithm transformations we see that the
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| geometric standard deviation is the exponentiated value of the standard deviation of the log transformed values (e.g. exp(stdev(ln(''A''))));
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| As such, the geometric mean and the geometric standard deviation of a sample of
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| 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.
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| {{Unreferenced|date=November 2010}}
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| {{DEFAULTSORT:Geometric Standard Deviation}}
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| [[Category:Scale statistics]]
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| [[Category:Statistical terminology]]
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I am 33 years old and my name is Xavier Mordaunt. I life in Kierspe (Germany).
My weblog - how to get free castle clash gems