Identifying the Mode of a Set of Numbers

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Definition: Mode

The mode of a set of numbers is the number with the highest frequency.

Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in the figure below.

An image of a calendar is shown. On Thursday the first, labeled New Year's Day, is written 2 mi. On Saturday the third is written 15 mi. On the 4th, 8 mi. On the 6th, 3 mi. On the 7th, 8 mi. On the 9th, 5 mi. On the 10th, 8 mi.

If we list the numbers in order it is easier to identify the one with the highest frequency.

\(2,3,5,8,8,8,15\)

Jolene ran \(8\) miles three times, and every other distance is listed only once. So the mode of the data is \(8\) miles.

How to Identify the mode of a set of numbers.

List the data values in numerical order.
Count the number of times each value appears.
The mode is the value with the highest frequency.

Example

The ages of students in a college math class are listed below. Identify the mode.

\(18,18,18,18,19,19,19,20,20,20,20,20,20,20,21,21,22,22,22,22,22,23,24,24,25,29,30,40,44\)

Solution

The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.

A table is shown with 2 rows. The first row is labeled 'Age' and lists the values: 18, 19, 20, 21, 22, 23, 24, 25, 29, 30, 40, and 44. The second row is labeled 'Frequency' and lists the values: 4, 3, 7, 2, 5, 1, 2, 1, 1, 1, 1, and 1.

Now look for the highest frequency. The highest frequency is \(7,\) which corresponds to the age \(20.\) So the mode of the ages in this class is \(20\) years.

Example

The data lists the heights (in inches) of students in a statistics class. Identify the mode.

\(56\)	\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)
\(60\)	\(62\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(70\)
\(60\)	\(63\)	\(63\)	\(64\)	\(66\)	\(66\)	\(67\)	\(74\)
\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)

Solution

List each number with its frequency.

A table is shown with 2 rows. The first row is labeled “Number” and lists the values: 56, 60, 61, 62, 63, 64, 65, 66, 67, 70, and 74. The second row is labeled “Frequency” and lists the values: 1, 2, 2, 1, 5, 4, 3, 5, 6, 1, and 1.

Now look for the highest frequency. The highest frequency is \(6,\) which corresponds to the height \(67\) inches. So the mode of this set of heights is \(67\) inches.

Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

Identifying the Mode of a Set of Numbers