# 5.3: The Mode

**At Grade**Created by: CK-12

**Learning Objectives**

- Understand the concept of mode.
- Identify the mode or modes of a data set for both quantitative and qualitative data.
- Describe a distribution of data as being unimodal, bimodal, or multimodal.
- Identify the mode of a set of data given in different representations.

Before class begins, bring out the blocks that you and your classmates chose from the pail for the lesson on mean. In addition, have the grid paper on display where each student in your class posted his or her number of blocks.

To begin the class, refer to the comments on the measures of central tendency that were recorded from the lesson on mean, when the brainstorming session occurred. Highlight the comments that were made with regard to the mode of a set of data and discuss this measure of central tendency with your classmates. Once the discussion has been completed, choose a handful of blocks like you did before, with your classmates doing the same.

The mode of the set of blocks can be given as a quantitative value or as a qualitative value. You and your classmates can tell from the grid paper which number of blocks was picked most. The chart below shows that 5 students each picked 5 blocks from the pail. This is a mode for quantitative data, since the answer is in the form of a number.

To extend the mode to include qualitative data, you and your classmates should each now determine the color or colors of block(s) that appear most often in each of your handfuls. To do this, group your blocks according to color and count them, and have your classmates do the same. There may be more than 1 color that occurs with the same highest frequency. The color(s) that appear most often for each handful of blocks is the mode for that particular handful.

The **mode** of a set of data is simply the value that appears most frequently in the set. If 2 or more values appear with the same frequency, each is a mode. The downside to using the mode as a measure of central tendency is that a set of data may have no mode or may have more than 1 mode. However, the same set of data will have only 1 mean and only 1 median. The word *modal* is often used when referring to the mode of a data set. If a data set has only 1 value that occurs most often, the set is called **unimodal**. Likewise, a data set that has 2 values that occur with the greatest frequency is referred to as **bimodal**. Finally, when a set of data has more than 2 values that occur with the same greatest frequency, the set is called **multimodal**. When determining the mode of a data set, calculations are not required, but keen observation is a must. The mode is a measure of central tendency that is simple to locate, but it is not used much in practical applications.

*Example 16*

The posted speed limit along a busy highway is 65 miles per hour. The following values represent the speeds (in miles per hour) of 10 cars that were stopped for violating the speed limit:

\begin{align*}76 \qquad 81 \qquad 79 \qquad 80 \qquad 78 \qquad 83 \qquad 77 \qquad 79 \qquad 82 \qquad 75\end{align*}

What is the mode?

*Solution:*

There is no need to organize the data, unless you think that it would be easier to locate the mode if the numbers were arranged from least to greatest. In the above data set, the number 79 appears twice, but all the other numbers appear only once. Since 79 appears with the greatest frequency, it is the mode of the data values.

Mode = 79 miles per hour

*Example 17*

The weekly wages of 7 randomly selected employees of Wendy’s were $98.00, $125.00, $75.00, $120.00, $86.00, $92.00, and $110.00, respectively. What is the mode of these wages?

*Solution:*

Each value in the above data set occurs only once. Therefore, this data has no mode.

*Example 18*

The ages of 12 randomly selected customers at a local coffee shop are listed below:

\begin{align*}23, 21, 29, 24, 31, 21, 27, 23, 24, 32, 33, 19\end{align*}

What is the mode of the above ages?

*Solution:*

The above data set has 3 values that each occur with a frequency of 2. These values are 21, 23, and 24. All other values occur only once. Therefore, this set of data has 3 modes.

Modes = 21, 23, and 24

Remember that the mode can be determined for qualitative data as well as quantitative data, but the mean and the median can only be determined for quantitative data.

*Example 19*

6 students attending a local swimming competition were asked what color bathing suit they were wearing. The responses were red, blue, black, pink, green, and blue.

What is the mode of these responses?

*Solution:*

The color blue was the only response that occurred more than once and is, therefore, the mode of this data set.

Mode = blue

When data is arranged in a frequency table, the mode is simply the value that has the highest frequency.

*Example 20*

The following table represents the number of times that 100 randomly selected students ate at the school cafeteria during the first month of school:

\begin{align*}&\text{Number of Times Eating in the Cafeteria} && 2 \quad 3 \quad 4 \quad \ \ 5 \quad \ 6 \quad \ 7 \quad \ 8\\
&\text{Number of Students} && 3 \quad 8 \quad 22 \quad 29 \quad 20 \quad 8 \quad 10\end{align*}

What is the mode of the numbers of times that a student ate at the cafeteria?

*Solution:*

The table shows that 29 students ate 5 times in the cafeteria. Therefore, 5 is the mode of the data set.

Mode = 5 times

**Lesson Summary**

You have learned that the mode of a data set is simply the value that occurs with the highest frequency. You have also learned that it is possible for a set of data to have no mode, 1 mode, 2 modes, or more than 2 modes. Observation is required to determine the mode of a data set, and this mode can be for either quantitative or qualitative data.

**Points to Consider**

- Is reference made to the mode in any other branch of statistics?
- Can the mode be useful when presenting graphical representations of data?

**Vocabulary**

- Bimodal
- The term used to describe the distribution of a data set that has 2 modes.

- Cumulative frequency
- The sum of the frequencies up to and including that frequency.

- Frequency distribution table
- A table that lists a group of data values, as well as the number of times each value appears in the data set.

- Mean
- A measure of central tendency that is determined by dividing the sum of all values in a data set by the number of values.

- Measures of central tendency
- Values that describe the center of a distribution. The mean, median, and mode are 3 measures of central tendency.

- Median
- The value of the middle term in a set of organized data. For a set of data with an odd number of values, it is the value that has an equal number of data values before and after it, or the middle value. For a set of data with an even number of values, the median is the average of the 2 values in the middle positions.

- Mode
- The value or values that occur with the greatest frequency in a data set.

- Multimodal
- The term used to describe the distribution of a data set that has more than 2 modes.

- Outliers
- Extreme values in a data set.

- Unimodal
- The term used to describe the distribution of a data set that has only 1 mode.