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_i, and now 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.