Location-Scale Family of Distributions

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The location-scale family of distributions includes many popular distributions like the normal distribution and uniform distribution.

location-scale family
The uniform distribution is a member of the location-scale family.

A random variable belongs to the location-scale family if its cumulative distribution
FX (x | a, b):= Pr(X ≤ | a, b) is a function only of (x – a)/b [1]:
location-scale family


If b = 1, the family is a location family, which is a subfamily of the location-scale family. If a = 0, the subfamily is a scale family with parameter b.

List of Location-Scale Family of Distributions

Many well-known statistical distributions belong to, or are closely related to, the location-scale family.

Several other distributions have special cases that belong to the family, including the beta distribution, chi-distribution, and power–function distribution. As an example, the four parameter beta distribution isn’t in the scale-location family because it has two “extra” parameters, c and d. However, if both c and d are set to 1, the distribution becomes the uniform distribution — which is a “true” member of the location-scale family.

Other distributions can be transformed or otherwise tweaked to join the family. For example, an exponential distribution that is reflected over the x-axis at x = a will be a location-scale distribution.

Log-Location Scale Family

A related family is the log-location scale family, which includes the lognormal distribution, the loglogistic distribution, and the Weibull distribution. A distribution is in the log-location scale family if Y = log(T) is a member of the location-scale family [2].


[1] The family of location–scale distributions. Retrieved December 15, 2021 from: http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf
[2] Location-Scale-Base Parametric Distributions.

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