# Location parameter

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

$f_}(x)=f(x-x_).$ [citation needed]

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

Thus in the one-dimensional case if $x_$ 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

$f_,\theta }(x)=f_{\theta }(x-x_)$ where $x_$ is the location parameter, θ represents additional parameters, and $f_{\theta }$ is a function parametrized on the additional parameters.

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