# Location parameter

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter ${\displaystyle x_{0}}$, 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

${\displaystyle f_{x_{0}}(x)=f(x-x_{0}).}$

Here, ${\displaystyle x_{0}}$ is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

Thus in the one-dimensional case if ${\displaystyle x_{0}}$ 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

${\displaystyle f_{x_{0},\theta }(x)=f_{\theta }(x-x_{0})}$

where ${\displaystyle x_{0}}$ is the location parameter, θ represents additional parameters, and ${\displaystyle f_{\theta }}$ is a function parametrized on the additional parameters.

An alternative way of thinking of location families is through the concept of additive noise. If ${\displaystyle x_{0}}$ is a constant and W is random noise with probability density ${\displaystyle f_{W}(w),}$ then ${\displaystyle X=x_{0}+W}$ has probability density ${\displaystyle f_{x_{0}}(x)=f_{W}(x-x_{0})}$ and its distribution is therefore part of a location family.