Section 3.5: The Five-Number Summary and Boxplots

Objectives

By the end of this lesson, you will be able to...

1. compute the five-number summary
2. draw and interpret boxplots

For a quick overview of this section, feel free to watch this short video summary:

The Five-Number Summary

The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, 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 five-number summary.

From Example 4 in Section 3.4, we already know that Q1 = 71, median 78, and Q3 = 89. We only need the maximum and minimum, so the five-number summary is:

minimum = 48
Q1 = 71
median 78
Q3 = 89
maximum = 99

Boxplots

Using the five-number summary and the fences, we can create a new graph called a boxplot.

Drawing a Boxplot

• Step 1: Determine the five-number 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 Q1 , M, and Q3. 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 five-number summary from Example 1 above and the outlier calculation from Example 5 in Section 3.4, we have the following information:

minimum = 48
Q1 = 71
median 78
Q3 = 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.

 Enter or import the data. Select Graphics > Box Plot. Select the column(s) you want to create a box plot for. Click Next. Check "Use fences to identify outliers" and click Next. Enter any modifications and click Next. Choose a color scheme, if you wish, and click Create Graph! You can then choose Options > Copy to copy the box plot for use elsewhere.
 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 (bell-shaped) left-skewed right-skewed

Let's now see how these are related to boxplots. Here's some information from your text:

Symmetric distributions

 Distribution Boxplot Q1 is equally far from the median as Q3 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 Q1 is further from the median as Q3 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 Q1 is closer to the median than Q3 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