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Shape parameter

Kind of numerical parameter of a parametric family of probability distributions

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated April 29, 2026
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In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributions that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the shape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does). For example, 'peakedness' refers to how round the main peak is.

01Estimation

Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite. Estimators of shape often involve higher-order statistics (non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as the L-moments. Maximum likelihood estimation can also be used.

02Examples

The following continuous probability distributions have a shape parameter:

By contrast, the following continuous distributions do not have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that (where they exist) the skewness and kurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters.

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Sources and credits

This article is adapted from the Wikipedia article “Shape parameter”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Skewness

Skewness in probability theory and statistics is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Similarly to kurtosis, it provides insights into shape-related characteristics of a distribution.

Kurtosis

Kurtosis, meaning 'curved, arching') refers to the degree of tailedness in the probability distribution of a real-valued, random variable in probability theory and statistics. Similar to skewness, kurtosis provides insight into specific characteristics of a distribution.

Location parameter

In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x 0 {\displaystyle x_{0}} , which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways: either as having a probability density function or probability mass function f {\displaystyle f(x-x_{0})} ; or having a cumulative distribution function F ( x − x 0 ) {\displaystyle F(x-x_{0})} ; or being defined as resulting from the random variable transformation x 0 + X {\displaystyle x_{0}+X} , where X {\displaystyle X} is a random variable with a certain, possibly unknown, distribution.