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In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter , which determines the "location" or shift of the distribution. Formally, this means that the probability density functions or probability mass functions in this class have the form

[citation needed]

Here, is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

Thus in the one-dimensional case if is increased, the probability density or mass function shifts rigidly to the right, maintaining its exact shape.

A location parameter can also be found in families having more than one parameter, such as location–scale families. In this case, the probability density function or probability mass function will be a special case of the more general form

where is the location parameter, θ represents additional parameters, and is a function parametrized on the additional parameters.

Additive noiseEdit

An alternative way of thinking of location families is through the concept of additive noise. If   is a constant and W is random noise with probability density   then   has probability density   and its distribution is therefore part of a location family.

See alsoEdit