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The mode of a data set is the value that occurs most frequently. The mode is the least used measure of centrality. When items are produced via mass production, for example, clothes of certain sizes or rods of certain lengths, the modal value is of great interest. Note that in any data set, there may be no mode, or conversely, there may be multiple modes. We denote the mode of a data set by .

Example 2.5.6 (Finding a mode) Find the mode for the following data set:

3, 8, 5, 6, 10, 17, 19, 20, 3, 2, 11

Solution: In the data set of this example, each value occurs once except 3, which occurs twice. Thus, the mode for this set is

Example 2.5.7 (Data set with no mode) Find the mode for the following data set:

1, 7, 19, 23, 11, 12, 1, 12, 19, 7, 11, 23

Solution: Note that in this data set, each value occurs twice. Thus, this data set does not have any mode.

Example 2.5.8 (Tri‐modal data set) Find the mode for the following data set:

5, 7, 12, 13, 14, 21, 7, 21, 23, 26, 5

Solution: In this data set, values 5, 7, and 21 occur twice, and the rest of the values occur only once. Thus, in this example, there are three modes, that is,

These examples show that there is no mathematical relationship among the mean, mode, and median in the sense that if we know any one or two of these measures (i.e., mean, median, or mode), then we cannot find the missing measure(s) without using the data values. However, the values of mean, mode, and median do provide important information about the type or shape of the frequency distribution of the data. Although the shape of the frequency distribution of a data set could be of any type, in practice, the most frequently encountered distributions are the three types shown in Figure 2.5.1. The location of the measures of centrality as shown in Figure 2.5.1 provides the information about the shape of the frequency distribution of a given data.

Figure 2.5.1 Frequency distributions showing the shape and location of measures of centrality.