Statistical Inference: A Short Course

3.4 The Mode

The mode of X is the value that occurs with the highest frequency; it is the most common or most probable value of X. Let us denote the population mode as “Mo;” the sample mode will appear as “mo.”

Example 3.4

For the sample data set X: 1, 2, 4, 4, 5, the modal value is mo = 4 (its frequency is two); for the variable Y: 1, 2, 4, 4, 5, 6, 6, 9, there are two modes—4 and 6 (Y is thus a bimodal variable); and for Z: 1, 2, 3, 7, 9, 11, there is no mode (no one value appears more often than any other).

For the properties of the mode:

1. The mode may or may not equal the mean and median.

2. The mode may not exist.

3. If the mode exists, it may not be unique.

4. The mode is not affected by outliers (for the data set X: 1, 2, 4, 4, 5, if 5 is replaced by 1000, the mode is still four).

5. The mode always corresponds to one of the actual values of a variable X

We next turn to measures of dispersion or variation within a data set. Here we are attempting to get a numerical measure of the spread or scatter of the values of a variable X along the horizontal axis.