Scale parameter

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A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in with distribution function is said to belong to the same type as a fixed distribution with distribution function if . Here is the scale parameter and is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If and are the distribution functions of random variables and , respectively, then a transition from to (for ) represents a change in the unit of measurement.


The (possibly multi-dimensional) parameter in the family is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.


[a1] L. Breiman, "Statistics with a view towards applications" , Houghton Mifflin (1973) pp. 34–40
How to Cite This Entry:
Scale parameter. A.V. Prokhorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098