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Scale Parameter

A scale parameter sigma>0 stretches or contracts a statistical distribution. If Z has distribution function F_0 and X=mu+sigmaZ, then

F(x;mu,sigma)=F_0((x-mu)/sigma),

where mu is a location parameter. When a probability density function f_0 exists,

f(x;mu,sigma)=1/sigmaf_0((x-mu)/sigma).

The location parameter and any shape parameters are held fixed when varying sigma.

For the usual parametrization of the normal distribution, sigma is the standard deviation. This identification is not general. The Cauchy distribution, for example, has a scale parameter but no finite standard deviation.

See also

Cauchy Distribution, Location Parameter, Normal Distribution, Shape Parameter, Stable Distribution, Standard Deviation

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References

National Institute of Standards and Technology. "Location and Scale Parameters." §1.3.6.4 in NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm.

Cite this as:

Weisstein, Eric W. "Scale Parameter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScaleParameter.html

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