Using Mode and Range to Describe a Data Set

Introduction: 

• In this chapter, we will learn about    

Mode: 

If you ask a shoe vendor the average shoe size he sells, he will give an answer with the size that he sells the most. That is the mode. 

The “mode” for a set of data is the value that occurs most often in the data set. A data set can have one mode, no mode, or more than one mode. 

Application of mode in real life: 

HR managers also calculate the mode of different positions in the company so that they can be aware of the most common position of employees at their company. 

Marketing managers also calculate the mode of the type of ad used, (e.g., web, digital media, TV, radio) so they can know which type of ads their company uses most often. 

Range: 

The “range” is the difference between the largest value and smallest value in a data set. 

Application of range in real life: 

The current year is 2021, and you were born in 2008. How old are you? Or how much time has passed? 

These types of questions can be answered with range. 

8.2.3 Use the mode to describe a data set 

Steps to calculate mode: 

Put the numbers in order from least to greatest.     

Count how many times each number occurs.              

The number that occurs the most is the mode!           

A data set can have one mode, no mode, or more than one mode. 

Example 1: 

The ages of eight employees at an insurance company are as follows. 

46.4, 29.3, 48.2, 35.1, 46.4, 39.5, 41.3, 25.2 

Find the mode of their ages. 

Solution: 

Put the ages of employees in order. 

25.2, 29.3, 35.1, 39.5, 41.3, 46.4, 46.4, 48.2 

Count how many times each number occurs. 

The value 46.4 occurs the most. 

Hence, the mode is 46.4. 

Example 2: 

Emma, a sixth-grader at Cambria Middle School, was thinking about joining the middle school track team. She read in the newspaper that Olympic athletes have lower resting heart rates than most people. She wondered about her own heart rate and wanted to compare it with other students.  

Heart rates are usually expressed as beats per minute (or bpm). Emma calculated her resting heart rate as 80 beats per minute. She asked her teacher, Ms. Christine, if she could collect the heart rates of the other students in her class. The following numbers are the resting heart rates (in beats per minute) for the 20 other students in Emma’s class. 

89, 85, 84, 79, 83, 85, 86, 88, 84, 81, 88, 85, 83, 83, 86, 82, 83, 86, 82, 84 

What heart rate occurred most often? 

Solution: 

The heart rate that occurred most often is nothing but, the mode of the data set. 

Put the heart rates of students in order. 

79, 81, 82, 82, 83, 83, 83, 83, 84, 84, 84, 85, 85, 85, 86, 86, 86, 88, 88, 89 

Count how many times each number occurs. 

Repeated once -79, 81, 89 

Repeated twice -82, 88 

Repeated thrice -84, 85, 86 

Repeated four times – 83. 

The value 83 occurs the most. 

Hence, the mode is 83. 

Emma’s heart rate is much below the heart rates of other students in the class. 

8.2.4 Use the range to describe a data set 

Steps to calculate range: 

Reorder the values of data set from smallest to largest.     

Subtract the first element from the last element.                  

Example1: 

Find the range for the data set = 2, 5, 10, 7, 5, 4, 3. 

Solution: 

Reordered values = 2, 3, 4, 5, 5, 7, 10 

Range = (10 – 2 )  

= 8 

Example 2: 

The average monthly high temperatures (in degrees Fahrenheit) f or Illinois and San Francisco are given in the table below. 

Find the rise in temperature in each city from winter to summer. 

Solution: 

Put the temperatures of Illinois in order. 

40, 43, 45, 55, 56, 67, 69, 79, 81, 85, 88, 89 

Range = (89 – 40) 

=49 

Put the temperatures of San Francisco in order. 

57, 57, 60, 62, 63, 63, 64, 65, 65, 68, 69, 70 

Range = (70 – 57) 

=13 

Concept Map: 

Exercise:

1. Fill in the blanks:

a. To find the _____ for a set of data, you subtract the smallest number from the largest number.

b. The _____ is the number that shows up the most times in a set of data.

2. Find the mode(s) and range of the following data:

5, 3, 2, 6, 5, 2, 5

3. Find the mode(s) and range of the following data:

24, 12, 10, 15, 10, 22, 12

4. Find the mode(s) and range of the following data:

116, 130, 120, 125, 140, 125

5. The heights, rounded to the nearest foot, of the Oak trees in a park are listed below.

27        13        8          52        26        42        48        52

Find the mode of their heights.

6. The heights (in inches) of eight rose plants are:

31, 45, 52, 40, 38, 41, 50, and 49

• What is the range of the rose plant heights?
• Find the mode(s) of the rose plant heights.

7. The populations (in millions) in 2000 on each of the six inhabited continents were:
803, 487, 348, 3686, 730, and 31

• What is the range of the populations?
• Find the mode(s) of the populations.

8. Paul and Kevin are having a friendly competition about the scores on their math quizzes. Both of their scores for the first five quizzes are given below:

Paul’s quiz scores: 18, 16, 19, 15, 17

Kevin’s quiz scores: 20, 20, 13, 12, 17

Find the mode of both sets of data.

9. The table shows the numbers of points scored by players on a throw ball team.

Points Scored

21       53        74        82        84        93

103     108     116     122     193

Find the range of the data.

10. The table below shows data about the students in three classes.

• What is the mode of the number of boys in the three classes given in the table?
• What is the mode of the number of girls in the three classes given in the table?

What have we learned:

• Find the mode and describe a data set.
• Find the range and describe a data set.