Section 3.5: The FiveNumber Summary and Boxplots
Objectives
By the end of this lesson, you will be able to...
 compute the fivenumber summary
 draw and interpret boxplots
For a quick overview of this section, feel free to watch this short video summary:
The FiveNumber Summary
The fivenumber summary of a set of data consists of the smallest data value, Q_{1}, the median, Q_{3}, and the largest value of the data.
Example 1
To illustrate, let's again look at those exam scores from Example 4 in Section 3.4.
48 
57  58  65  68  69  71  73  73 
74  75  77  78  78  78  79  80  85 
87  88  89  89  89  95  96  97  99 
Find the fivenumber summary.
From Example 4 in Section 3.4, we already know that Q_{1} = 71, median 78, and Q_{3} = 89. We only need the maximum and minimum, so the fivenumber summary is:
minimum = 48
Q_{1} = 71
median 78
Q_{3} = 89
maximum = 99
Boxplots
Using the fivenumber summary and the fences, we can create a new graph called a boxplot.
Drawing a Boxplot_{}
 Step 1: Determine the fivenumber summary and the lower and upper fences.
 Step 2: Draw a horizontal line and label it with an appropriate scale.
 Step 3: Draw vertical lines at Q_{1} , M, and Q_{3}. Enclose these vertical lines in a box.
 Step 4: Draw a line from Q1 to the smallest data value that is within the lower fence. Similarly, draw a line from Q3 to the largest value that is within the upper fence.
 Step 5: Any values outside the fences are outliers and are marked with an asterisk (*).
A typical boxplot will look something like this:
Example 2
To illustrate, let's again look at those exam scores from Example 4 in Section 3.4.
48 
57  58  65  68  69  71  73  73 
74  75  77  78  78  78  79  80  85 
87  88  89  89  89  95  96  97  99 
Take a moment and try to sketch a boxplot of this data set, following the description above.
Using the fivenumber summary from Example 1 above and the outlier calculation from Example 5 in Section 3.4, we have the following information:
minimum = 48
Q_{1} = 71
median 78
Q_{3} = 89
maximum = 99
Lower fence = 44
Upper fence =
116
A boxplot would then look something like this:
Technology
Here's a quick overview of how to create box plots in StatCrunch.

You can also visit the video page for links to see videos in either Quicktime or iPod format. 
Boxplots and Distribution Shape
The last thing we want to talk about in Chapter 3 is the relationship between the shape of a boxplot and the shape of the distribution.
In Section 2.2, we talked about distribution shape, showing the following four standards:
uniform 
symmetric (bellshaped) 
leftskewed 
rightskewed 
Let's now see how these are related to boxplots. Here's some information from your text:
Symmetric distributions
Distribution  Boxplot 
Q_{1} is equally far from the median as Q_{3} is 
The median line is in the center of the box 
The minimum is equally far from the median as the maximum is  The left whisker is equal in length to the right whisker 
Skewed left distributions
Distribution  Boxplot 
Q_{1} is further from the median as Q_{3} is 
The median line is to the right of center in the box 
The minimum is further from the median as the maximum is  The left whisker is longer than the right whisker 
Skewed right distributions
Distribution  Boxplot 
Q_{1} is closer to the median than Q_{3} is 
The median line is to the left of center in the box 
The minimum is closer to the median as the maximum is  The left whisker is shorter than the right whisker 
Source: Instructor Resources; Statistics: Informed Decisions Using Data
Author: Michael Sullivan III
© 2007, All right reserved.