## measures of location

## measures of location

**measures of location** Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the *mean*, *median*, and *mode*.

The *mean* of a sample of *n* observations, denoted by *x̄*, is

The mean of a probability distribution, denoted by μ, is

for a discrete distribution and

for a continuous distribution; it is also called the *expectation* of *x*, denoted by *E*(*x*).

A *weighted mean* is used when members of a sample are known with different reliability. To each observation *x _{i}* corresponds a

*weight w*, and now

_{i}*x̄*is

If each observation is the mean of

*w*observations, the formulas for the weighted and unweighted means agree.

The

*median*is the value of

*x*exceeded by exactly half the sample or distribution. The median of a distribution is the value for which the cumulative distribution function,

*F*(

*x*), equals 0.5 (see probability distributions).

The

*mode*is the most commonly occurring value. For distributions in which the frequency function,

*f*(

*x*), has one or more local maxima, each maximum is called a mode.

These measures may be illustrated on the following sample of eight values of

*x*: 1,1,1,2,3,3,5,7

The mean is 2.875, the median is 2.5, and the mode is 1.