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In statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, 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

${\displaystyle f_{\mu }(x)=f(x-\mu ).}$

Here, μ is called the location parameter.

In other words, when you graph the function, the location parameter determines where the origin will be located. If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.

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 have the form

${\displaystyle f_{\mu ,\theta }(x)=f_{\theta }(x-\mu )}$

where μ is the location parameter, θ represents additional parameters, and ${\displaystyle f_{\theta }}$ is a function of the additional parameters.

Additive noise

An alternative way of thinking of location families is through the concept of additive noise. If μ is an unknown constant and w is random noise with probability density ${\displaystyle f(w)}$, then ${\displaystyle x=\mu +w}$ has probability density ${\displaystyle f_{\mu }(x)=f(x-\mu )}$ and is therefore a location family.

See also

• Location test
• Invariant estimator
• Location-scale family
• Scale parameter

• Summary statistics
• Theory of probability distributions
• Statistical terminology