Mean, Median, Mode, and Range to Describe a Data Set

Introduction: 

•  In this chapter, we will learn to calculate the mean, median, mode, and range and describe a data set using them.   

  Summarizing data:  

• When working with a large data set, it can be useful to represent the entire data set with a single value that describes the “middle” or “average” value of the entire set. In statistics, that single value is called the central tendency. The mean, median and mode are all ways to describe a single value. We can use mean, median and mode to summarize the data. The mean, median and mode are also called a measure of central tendency. 

8.2.5 Use the mean, median, mode, and range to describe a data set 

Example 1: 

Find the mean, median, mode and range of the data. 

126, 62, 144, 81, 144, 103 

Solution: 

The “mean” is computed by adding all of the values in the data together and dividing by the number of elements contained in the data set. 

Number of elements in data set is 6 

Mean = (126 + 62 + 144 + 81 + 144 + 103) / 6 

= 660/6 

= 110 

The “median” is the middle value of a set of ordered numbers. 

Step 1: 

Order the values from the least to the greatest. 

62, 81, 103, 126, 144, 144 

Step 2: 

Since there are 6 numbers, the (6 + 1)/2th position, which is the 3.5^th median can be computed by adding the 3rd and 4th terms in that group, which is then divided by 2. 

Hence,

103+126/ 2

= 229/ 2

The “mode” for a set of data is the value that occurs most often in the data set. 

Order the values from the least to the greatest. 

62, 81, 103, 126, 144, 144 

Count how many times each number occurs. 

Repeated once – 62, 81, 103, 126. 

Repeated twice – 144, 144. 

The value 144 occurs the most. 

Hence, the mode is 144. 

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

Reordered values = 62, 81, 103, 126, 144, 144 

Range = (144 – 62)  

= 82 

Example 2: 

The following table shows the known number of moons of different planets. 

What are the median, mode and range of these data? 

What is the mean number of moons for the eight planets, rounded to the nearest whole number? 

Solution: 

Order the values from least to greatest. 

0, 0, 1, 2, 13, 27, 50, 53 

Since there are 8 items, the (8 + 1)/2th position, which is the 4.5^th median can be computed by adding the 4th and 5th terms in that group, which is then divided by 2. 

Hence,

2+13/ 2

=15/2

The value 0 occurs the most. 

Hence, the mode is 0. 

Range = (53 – 0)  

= 53 

Mean = (0 + 0 + 1 + 2 + 13 + 27 + 50 + 53) / 8 

= 146/8 

= 18.25 

= 18 (Rounded to nearest whole number) 

Example 3: 

The average incubation periods (in days) of a House sparrow, Greater Roadrunner, Turkey vulture, Laysan albatross, Golden eagle, Wild turkey and Magellanic penguin are the following:  

11, 20, 39, 64, 40, 28, 40 

What do the mean, median, and mode tell you about the incubation periods? What does the range tell you about the incubation periods? 

Solution: 

Mean = 34.6 

Median = 39 

Mode = 40 

Range = 53 

The mean, median, and mode each give a measure of the typical incubation periods in days.  

Exercise:

1. Find the mean, median, mode and range of the data.
   14, 9, 20, 5, 17, 13
2. Find the mean, median, mode and range of the data.
   116, 130, 120, 125, 140, 125.
3. Find the mean, median, mode and range of the data.
   6, 86, 54, 72, 6, 33, 49, 22, 61, 14.
4. The following shows the number of hours of TV watched in a week of 6^th-grade students.

• What do the mean, median, and mode tell you about the number of hours of TV watched?
• What does the range tell you about the number of hours of TV watched?

5. The following table shows data about the students in three classes.

• What is the mean number of boys in the three classes?
• What is the mean number of girls in the three classes?
• What is the mode of the number of boys in the three classes?
• What is the median number of students in the three classes

6. The typical weight (in ounces) of human organs – the heart, brain, liver, spleen, kidney, lungs and intestine are the following:
11, 48, 48, 6, 11, 39, 70

• What do the mean, median, and mode tell you about the weight of human organs?
• What does the range tell you about the weight of human organs?

7. A zookeeper measured the length (in feet) of six American crocodiles and recorded the figures as follows:
10, 11, 13, 12, 15, 15

• What do the mean, median, and mode tell you about the length of crocodiles?
• What does the range tell you about the length of crocodiles?

8. The average length (in minutes) of a Major League Baseball game from the year 2000 to 2007 is the following:
179, 174, 172, 166, 166, 166, 166, 170

• What do the mean, median, and mode tell you about the duration of the games?
• What does the range tell you about the duration of the games?

9. What is the mean, median, mode, and range of the following data?
6, 28, 16, 32, 6, 16, 48, 29, 6, 35.

10. What is the mean, median, mode, and range of the following data?
12, 46, 32, 18, 26, 41, 46.

Concept Map: 

What have we learned:

• Calculation of the mean, median, mode, range and to describe a data set using them.