probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...

. For such data, it may be preferred to the more usual

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

. Note that unlike the usual ''arithmetic'' standard deviation, the ''geometric'' standard deviation is a multiplicative factor, and thus is

dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...

, rather than having the same

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor.Kirkwood, T.B.L. (1993)
"Geometric standard deviation - reply to Bohidar"
Drug Dev. Ind. Pharmacy 19(3): 395-6. 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.

Definition

If the geometric mean of a set of numbers

\sigma_\mathrm = \exp \sqrt\,.

Derivation

If the geometric mean is

\mu_\mathrm = \sqrt /math>

then taking the

natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

of both sides results in

\ln \mu_\mathrm =  \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 so

\ln \mu_\mathrm =  \left \ln A_1 + \ln A_2 + \cdots + \ln A_n \right

It can now be seen that

\ln \mu_\mathrm

is the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the set therefore the arithmetic standard deviation of this same set should be

\ln \sigma_\mathrm = \sqrt\,.

This simplifies to

\sigma_\mathrm = \exp \sqrt\,.

Geometric standard score

The geometric version of the

standard score In statistics, the standard score or ''z''-score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores ...

z =  = \log _ \left(\right).

If the geometric mean, standard deviation, and z-score of a datum are known, then the

raw score Raw data, also known as primary data, are ''data'' (e.g., numbers, instrument readings, figures, etc.) collected from a source. In the context of examinations, the raw data might be described as a raw score (after test scores). If a scientist ...

can be reconstructed by

x = \mu_\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 the geometric 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 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 In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...

for details.

References

External links

Non-Newtonian calculus website
{{DEFAULTSORT:Geometric Standard Deviation Scale statistics Non-Newtonian calculus