Finding the Mode or Modal Value

The mode or modal value is the data value that appears most often in a set of data. To find the mode, we record all the different data values in the set, and how many times each one appears in our data set. The value that occurs the greatest number of times is the mode. It may help you to sort your data into increasing or decreasing order first, but it isn't necessary.

Example 1

Find the mode of the list of values:

Solution: Sorting these scores into increasing order makes it easy to see which one occurs most often. The sorted list is

and the value 18 occurs most often, so 18 is the mode.

Example 2

Find the mode of the following list of data values:

Solution: If we arrange these names in alphabetical order, the list becomes

The name Kate occurs twice, while the other names only occur once, so the modal value is Kate.

Modes Need Not Be Unique

Sometimes a set of data will have more than one mode. It is quite possible for two or more data values to occur the same number of times.

For example, in the list \(\{2,4,4,4,6,8,8,8,10,10,10\}\), three scores (\(4, 8\) and \(10\)) occur three times, but the rest of the scores occur only once.

So, this list has three modes: \(4, 8\) and \(6\).

A list that has two modes is called bimodal. A list that has more than two modes is called multimodal. The list above is multimodal.

Modal Classes

There are data sets in which each value occurs the same number of times. In this case, the mode is not a very useful measure. However, we can group data like this into classes. We may then be able to identify the mode of these classes, which is called the modal class.

Example 3

Consider the list \(\{2,3,5,7,11,15,18,21,22,23,24,25,26,31,32,33,40,41,50,52\}\).

Each value occurs only once, so the mode is no use here.

Let's try grouping the list into the classes 1-4,5-9,10-14,15-19,20-24,25-29,30-34,35-39,40-44,45-49,50-54.

Find the number of values in each class:

• 1-4: 2 values (2 and 3)
• 5-9: 2 values (5 and 7)
• 10-14: 1 value (11)
• 15-19: 2 values (15 and 18)
• 20-24: 4 values (21,22,23,24)
• 25-29: 2 values (25 and 26)
• 30-34: 3 values (31,32,33)
• 35-39: 0 values
• 40-44: 2 values (40 and 41)
• 45-49: 0 values
• 50-54: 2 values (50 and 52)

One class, 20-24 contains 4 values. The rest contain fewer values. So, the modal class is 20-24.

Dividing data up into classes and finding the modal class is also helpful in situations when the data set contains anomalies (something messes it up).

Example 4

Sam's English teacher wants to work out how long her students spend studying for a test on Shakespeare. She asks them to use tracking software to keep track of number of minutes they spend studying in the week before the test. When she collects the data, she notices that Sam seems to have spent far more time than anyone else studying for the test. When she questions Sam, he admits (typical Sam!) that he has spent much of the time with his favourite graphic novel tucked inside the covers of his English book. Mrs Fitzsimmons wonders how she can make use of her data. Sam is probably not the only one who was doing something else when he should have been studying, so even discarding Sam's data and taking an average is not likely to be useful.

The students have provided the following data set for analysis \(\{37, 38, 40, 42, 45, 52, 53, 54, 58, 59, 61, 83, 85, 106\}\). The \(106\) was Sam's "study" time. Grouping this data into 10 minute classes gives

• 30-39: 2 values (37 and 38)
• 40-49: 3 values (40, 42 and 45)
• 50-59: 5 values (52,53,54,58,59)
• 60-69: 1 value (61)
• 70-79: 0 values
• 80-89: 2 values (83 and 85)
• 90-99: 0 values
• 100-109: 1 value (106)

The 50-59 class occurs most often. This class is the modal class. So, Mrs Fitzsimmons can say that most of her students spent around 55 minutes studying for the Shakespeare test.