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 Central Location

The Mode

Perhaps you have heard a statement like "The average number of automobiles owned by households in the United States is 1.37," and have been amused at the thought of a fraction of an automobile sitting in a driveway. In such a context the following measure for central location might make more sense.

Definition

The sample mode of a set of sample data is the most frequently occurring value.

The population mode is defined in a similar way, but we will not have occasion to refer to it again in this text.

On a relative frequency histogram, the highest point of the histogram corresponds to the mode of the data set. Figure 2.9 "Mode" illustrates the mode.

Figure 2.9 Mode

For any data set there is always exactly one mean and exactly one median. This need not be true of the mode; several different values could occur with the highest frequency, as we will see. It could even happen that every value occurs with the same frequency, in which case the concept of the mode does not make much sense.

EXAMPLE 8

Find the mode of the following data set.

$\begin{array}{llll}-1 & 0 & 2 & 0\end{array}$

Solution:

The value $\mathrm{0}$ is most frequently observed and therefore the mode is $\mathrm{0}$ .

EXAMPLE 9

Compute the sample mode for the data of Note 2.13 "Example 4".

Solution:

The two most frequently observed values in the data set are $\mathrm{1}$ and $\mathrm{2}$ . Therefore mode is a set of two values: $\{1,2\}$ .

The mode is a measure of central location since most real-life data sets have more observations near the center of the data range and fewer observations on the lower and upper ends. The value with the highest frequency is often in the middle of the data range.