Explainer: The Mode of a Data Set

In this explainer, we will learn how to find and interpret the mode of a data set.

The mode is an example of a measure of center (or measure of central tendency). If we have a set of numerical data, we would like to find a single number which can represent the whole data set or at least give us some information about typical values in the data set. There are a number of ways to describe typical values. For example, one way to describe a typical value is to see what value is in the middle; this is the median. The mean is another example. We could also describe a typical value by looking at the value that occurs most frequently; this is the mode.

The mode, mean, and median are all different examples of measures of center. We will only discuss the mode here.

Definition: The Mode

The mode of a set of data is the value which appears most often.

Let us look at an example of how to calculate the mode.

Example 1: Finding the Mode of a Data Set by Ordering

The following data points represent the number of goals scored by a player in 10 consecutive matches: 2,1,2,0,1,2,1,4,4,2.

What was the mode score?

Answer

We have to find the most common value from the data set 2,1,2,0,1,2,1,4,4,2.

We can do this by writing the data values in order to see which occurs the most.

The data value 2 appears the most times. Therefore, the mode is 2.

In the above example, it was easy to see which value was most common just by ordering them from least to greatest. When we have larger sets of data, however, this may not be a good method. Instead, we can use a tally chart or a frequency table to help us. We will do this in the next example.

Example 2: Finding the Mode of a Data Set by Making a Frequency Table

The table shows the number of books that 30 students read in a year. Find the mode number of books read.

1984886988
6631254383
44107587782

Answer

In this set of data, the data values are the number of books read. The mode is the most common number of books read.

To find the mode, we need to count how many times each data value appears. We can use a tally chart or a frequency table to record the results.

We can see that the data value 8 appeared more times than any other value. Therefore the most common number of books read was 8.

Now you have seen some examples of how to find the mode of a set of data. Let us summarize what we have learned.

How To: Find the Mode of a Data Set

Find the data value which appears most often. Some different ways to do this are the follwoing:

  1. Writing the data values (if they are numbers) in order from least to greatest and counting how often each value appears.
    For example, to find the mode of the data set 2, 5, 7, 3, 8, 5, 8, 2, 4, 8, 9, we could write the numbers in order.

    The number 8 appears the most times.
    The mode is 8.
  2. Drawing a tally chart to count the number of times each value occurs.
  3. Making a frequency table to record the number of times each value occurs.
    For example, to find the mode of the data set 2, 5, 7, 3, 8, 5, 8, 2, 4, 8, 9, we could draw a tally chart for a frequency table.

It is important to realize that the mode does not have to be a number. We can have data sets where the data values are not numbers. We will look at an example of this.

Example 3: Finding the Mode of a Nonnumerical Data Set

A bookstore sold 11 books by Haruki Murakami, 2 books by Henry Thoreau, and 6 books by Carl Jung. Determine the mode for this data.

Answer

The data values are the authors of the books.

If we were to list the data set, we would have a list of the authors of the books that the bookstore sold. If it helps, you can imagine making a tally chart for the data.

The mode is the most common data value; this is the author who sold the most books.

Therefore, the mode is Haruki Murakami.

Some data sets will have more than one mode; it is possible for more than one of the values to have the highest frequency. We will see this in the next example.

Example 4: Data Sets with More Than One Mode

Using the data in the table, find the modal temperature.

Temperature in FahrenheitTallyFrequency
60 12
6215
645
669
6815

Answer

The mode of the data values is the temperature, or temperatures, that occurred most frequently.

From the table, we can see that the highest frequency is 15 and that there are two different temperatures that both occurred 15 times.

Hence, the mode, or the modal temperatures, is 62F and 68F.

We will finish with one final example.

Example 5: Finding Missing Data Values Given the Mode

Amelia has the following data: 3, 6, 4, 5, 𝑚.

If the mode is 6, find the value of 𝑚.

Answer

The mode of the data set is the value that appears most often. Without the value 𝑚, the data set is 3,4,5,6 and all of the data values appear once.

Since we know that when we add the value 𝑚 into the data set, the mode is 6, this means that the most common value is 6. Hence, 6 must appear more than once in the data set: 3,4,5,6,𝑚.

Therefore, 𝑚=6.

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