In
probability theory
Probability theory 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 expressing it through a set ...
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
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, 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 coor ...
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 is denoted as μ
''g'', then the geometric standard deviation is
:
Derivation
If the geometric mean is
:
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
:
Geometric standard score
The geometric version of the
standard score is
:
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
:
Relationship to log-normal distribution
The geometric standard deviation is used as a measure of
log-normal
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a 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 normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
for details.
References
External links
Non-Newtonian calculus website
{{DEFAULTSORT:Geometric Standard Deviation
Scale statistics
Non-Newtonian calculus