Geometric standard deviation

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 {"A"₁, "A"₂, ..., "A"_"n"} is denoted as μ_"g", then the geometric standard deviation is

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

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 "A"₁, ..., "A"_"n") = arithmetic (i.e. usual) SD of ln("A"₁), ..., ln("A"_"n").

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 logarithmtransformed values. By a simple set of logarithm transformations we see that thegeometric 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 ofdata 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.

External links

* [http://www.thinkingapplied.com/means_folder/deceptive_means.htm Deceptive Means]

ee also

*Geometric mean
*Log-normal distribution
*Natural logarithm