Measures of Central Location

This section elaborates on mean, median, and mode at the population level and sample level. This section also contains many interesting examples of range, variance, and standard deviation. Complete the exercises and check your answers.

Measures of Variability

The Range

The first measure of variability that we discuss is the simplest.

Definition

The range of a data set is the number R defined by the formula

$R=x_{\max }-x_{\min }$

where $x_{\max }$ is the largest measurement in the data set and $x_{\min }$ is the smallest.

EXAMPLE 10

Find the range of each data set in Table 2.1 "Two Data Sets".

Solution:

For Data Set I the maximum is 43 and the minimum is 38, so the range is $R=43-38=5$.

For Data Set II the maximum is 47 and the minimum is 33, so the range is $R=47-33=14$.

The range is a measure of variability because it indicates the size of the interval over which the data points are distributed. A smaller range indicates less variability (less dispersion) among the data, whereas a larger range indicates the opposite.