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, ..., A_n} is denoted as μ_g, then the geometric standard deviation is

$\backslash sigma\_g\; =\; \backslash exp\; \backslash left(\; \backslash sqrt\{\; \backslash sum\_\{i=1\}^n\; (\; \backslash ln\; A\_i\; -\; \backslash ln\; \backslash mu\_g\; )^2\; \backslash over\; n\; \}\; \backslash right).\; \backslash qquad\; \backslash qquad\; (1)$

Derivation

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If the geometric mean is

$\backslash mu\_g\; =\; \backslash sqrt*\{\; A\_1\; A\_2\; \backslash cdots\; A\_n\; \}.\backslash ,$

then taking the natural logarithm of both sides results in

$\backslash ln\; \backslash mu\_g\; =\; \{1\; \backslash over\; n\}\; \backslash ln\; (A\_1\; A\_2\; \backslash cdots\; A\_n).$

The logarithm of a product is a sum of logarithms, so

$\backslash ln\; \backslash mu\_g\; =\; \{1\; \backslash over\; n\}\backslash ln\; A\_1\; +\; \backslash ln\; A\_2\; +\; \backslash cdots\; +\; \backslash ln\; A\_n.\backslash ,$

It can now be seen that $\backslash ln\; \backslash ,\; \backslash mu\_g$ is the arithmetic mean of the set $\backslash \{\; \backslash ln\; A\_1,\; \backslash ln\; A\_2,\; \backslash dots\; ,\; \backslash ln\; A\_n\; \backslash \}$, therefore the arithmetic standard deviation of this same set should be

$\backslash ln\; \backslash sigma\_g\; =\; \backslash sqrt\{\; \backslash sum\_\{i=1\}^n\; (\; \backslash ln\; A\_i\; -\; \backslash ln\; \backslash mu\_g\; )^2\; \backslash over\; n\; \}.$

Thus

ln(geometric SD of A₁, ..., A_n) = arithmetic (i.e. usual) SD of ln(A₁), ..., ln(A_n).

Geometric standard score

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The geometric version of the standard score is

$z\; =\; \{\backslash ln\; (\; x/\backslash mu\_g\; )\; \backslash over\; \backslash ln\; \backslash sigma\_g\; \}.\backslash ,$

If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score can be reconstructed by

$x\; =\; \backslash mu\_g\; \backslash sigma\_g^z.$

Relationship to log-normal distribution

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The geometric standard deviation is related to the log-normal distribution. The log-normal distribution is a distribution which is normal for the logarithm transformed values. By a simple set of logarithm transformations we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log transformed values (e.g. exp(stdev(ln(A))));

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.

See also

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geometric mean, log-normal distribution, natural logarithm

Geometri simpangan baku