Geometric standard deviation

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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. If the geometric mean of a set of numbers {A₁, A₂, ..., A_n} is denoted as μ_g, then the geometric standard deviation is

$\sigma_g = \exp \left( \sqrt{ \sum_{i=1}^n ( \ln A_i - \ln \mu_g )^2 \over n } \right). \qquad \qquad (1)$

1 Derivation
2 Geometric standard score
3 Relationship to log-normal distribution
4 See also

[edit] Derivation

If the geometric mean is

$\mu_g = \sqrt[n]{ A_1 A_2 \cdots A_n }.\,$

then taking the natural logarithm of both sides results in

$\ln \mu_g = {1 \over n} \ln (A_1 A_2 \cdots A_n).$

The logarithm of a product is a sum of logarithms (assuming $A i$ is positive for all $i$ ), so

$\ln \mu_g = {1 \over n} [ \ln A_1 + \ln A_2 + \cdots + \ln A_n ].\,$

It can now be seen that $\ln \, \mu_g$ is the arithmetic mean of the set $\{ \ln A_1, \ln A_2, \dots , \ln A_n \}$ , therefore the arithmetic standard deviation of this same set should be

$\ln \sigma_g = \sqrt{ \sum_{i=1}^n ( \ln A_i - \ln \mu_g )^2 \over n }.$

Thus

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

[edit] Geometric standard score

The geometric version of the standard score is

$z = {\ln ( x/\mu_g ) \over \ln \sigma_g }.\,$

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

$x = \mu_g \sigma_g^z.$

[edit] Relationship to log-normal distribution

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.

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Categories: Probability theory | Statistics