Geometric standard deviation
Encyclopedia
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
. For such data, it may be preferred to the more usual standard deviation
.
then taking the natural logarithm
of both sides results in
The logarithm of a product is a sum of logarithms (assuming
is positive for all 
), so
It can now be seen that
is the arithmetic mean
of the set
, therefore the arithmetic standard deviation of this same set should be
This simplifies to
is
If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score
can be reconstructed by
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 interval
s 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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
. For such data, it may be preferred to the more usual standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
.
Definition
If the geometric mean of a set of numbers {A1, A2, ..., An} is denoted as μg, then the geometric standard deviation isDerivation
If the geometric mean isthen taking the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
of both sides results in
The logarithm of a product is a sum of logarithms (assuming
is positive for all ), soIt can now be seen that
is the arithmetic meanArithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the set
, therefore the arithmetic standard deviation of this same set should beThis simplifies to
Geometric standard score
The geometric version of the standard scoreStandard score
In statistics, a standard score indicates how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation...
is
If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score
Raw score
In statistics and data analysis, a raw score is an original datum that has not been transformed. This may include, for example, the original result obtained by a student on a test as opposed to that score after transformation to a standard score or percentile rank or the like.Often the conversion...
can be reconstructed by
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 interval
Confidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
s 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.