Boxplots

A boxplot, sometimes called a box and whisker plot, is a type of graph used to display patterns of quantitative data.

Boxplot Basics

A boxplot splits the data set into quartiles. The body of the boxplot consists of a "box" (hence, the name), which goes from the first quartile (Q1) to the third quartile (Q3).

Within the box, a vertical line is drawn at the Q2, the median of the data set. Two horizontal lines, called whiskers, extend from the front and back of the box. The front whisker goes from Q1 to the smallest non-outlier in the data set, and the back whisker goes from Q3 to the largest non-outlier.

Smallest non-outlier Q1 Q2   Q3 Largest non-outlier

. .

 -6 -4 -2 0 2 4 6 8 10 12 14 16

If the data set includes one or more outliers, they are plotted separately as points on the chart. In the boxplot above, two outliers are shown to the right of the second whisker.

How to Interpret a Boxplot

Here is how to read a boxplot. The median is indicated by the vertical line that runs down the center of the box. In the boxplot above, the median is between 4 and 6, around 5.

Additionally, boxplots display two common measures of the variability or spread in a data set.

• Range. If you are interested in the spread of all the data, it is represented on a boxplot by the horizontal distance between the smallest value and the largest value, including any outliers. In the boxplot above, data values range from about 0 (the smallest non-outlier) to about 16 (the largest outlier), so the range is 16. If you ignore outliers, the range is illustrated by the distance between the opposite ends of the whiskers - about 10 in the boxplot above.
• Interquartile range (IQR). The middle half of a data set falls within the interquartile range. In a boxplot, the interquartile range is represented by the width of the box (Q3 minus Q1). In the chart above, the interquartile range is equal to about 7 minus 3 or about 4.

And finally, boxplots often provide information about the shape of a data set. The examples below show some common patterns.

 2 4 6 8 10 12 14 16
Skewed right

 2 4 6 8 10 12 14 16
Symmetric

 2 4 6 8 10 12 14 16
Skewed left

Each of the above boxplots illustrates a different skewness pattern. If most of the observations are concentrated on the low end of the scale, the distribution is skewed right; and vice versa. If a distribution is symmetric, the observations will be evenly split at the median, as shown above in the middle figure.

Problem 1

Consider the boxplot below.

 2 4 6 8 10 12 14 16 18

Which of the following statements are true?

I. The distribution is skewed right.
II. The interquartile range is about 8.
III. The median is about 10.

(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III

Solution

The correct answer is (B). Most of the observations are on the high end of the scale, so the distribution is skewed left. The interquartile range is indicated by the length of the box, which is 18 minus 10 or 8. And the median is indicated by the vertical line running through the middle of the box, which is roughly centered over 15. So the median is about 15.