Geometric standard deviation explained

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 dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor.^{[1] [2]} 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.^[3]

Definition

If the geometric mean of a set of numbers

Derivation

If the geometric mean is

$$\backslash mu\_\backslash mathrm\; =\; \backslash sqrt[n]$$

then taking the natural logarithm of both sides results in

$$\backslash ln\; \backslash mu\_\backslash mathrm\; =\; \backslash ln\; (A\_1\; A\_2\; \backslash cdots\; A\_n).$$

The logarithm of a product is a sum of logarithms (assuming $A\_i$ is positive for all so

$$\backslash ln\; \backslash mu\_\backslash mathrm\; =\; \backslash left[\backslash ln\; A\_1\; +\; \backslash ln\; A\_2\; +\; \backslash cdots\; +\; \backslash ln\; A\_n\; \backslash right].$$

It can now be seen that

ln \mu _{g} is the arithmetic mean of the set therefore the arithmetic standard deviation of this same set should be

$$\backslash ln\; \backslash sigma\_\backslash mathrm\; =\; \backslash sqrt\backslash ,.$$

This simplifies to

$$\backslash sigma\_\backslash mathrm\; =\; \backslash exp\; \backslash sqrt\backslash ,.$$

Geometric standard score

The geometric version of the standard score is

$$z\; =\; =\; \backslash log\; \_\; \backslash left(\backslash right).$$

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

$$x\; =\; \backslash mu\_\backslash mathrm\; ^z.$$

Relationship to log-normal distribution

The geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean. As the log-transform of a log-normal distribution results in a normal distribution, we see that thegeometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e.

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

• Non-Newtonian calculus website

Notes and References

1. http://www.graphpad.com/guides/prism/7/statistics/stat_the_geometric_mean_and_geometr.htm?toc=0&printWindow GraphPad Guide
2. Kirkwood, T.B.L. (1993). "Geometric standard deviation - reply to Bohidar". Drug Dev. Ind. Pharmacy 19(3): 395-6.
3. Kirkwood . T.B.L. . Geometric means and measures of dispersion . Biometrics . 1979 . 35 . 908–9. 2530139.