Define basic terms including hinges, H-spread, step, adjacent value,
outside value, and far out value

Create a box plot

Create parallel box plots

Determine whether a box plot is appropriate for a given data set

We have already discussed techniques for visually
representing data (see histograms
and frequency polygons). In this
section, we present another important graph called a box
plot. Box plots are useful for identifying
outliers and for comparing distributions. We will explain box
plots with the help of data from an in-class experiment. As part of the "Stroop Interference Case Study," students
in introductory statistics were presented with a page containing
30 colored rectangles. Their task was to name the colors as quickly
as possible. Their times (in seconds) were recorded. We'll compare the
scores for the 16 men and 31 women who participated in the experiment
by making separate box plots for each gender. Such a display
is said to involve parallel
box plots.

There are several steps in constructing a box plot.
The first relies on the 25^{th}, 50^{th}, and
75^{th} percentiles in the distribution of scores. Figure
1 shows how these three statistics are used. For each gender, we
draw a box extending from the 25^{th} percentile to the
75^{th} percentile. The 50^{th} percentile is
drawn inside the box. Therefore,

the bottom of each box is the 25th percentile,

the top is the 75th percentile,

and the line in the middle is the 50th percentile.

The data for the women in our sample are shown
in Table 1.

Table 1. Women's times.

14
15
16
16
17

17
17
17
17
18

18
18
18
18
18

19
19
19
20
20

20
20
20
20
21

21
22
23
24
24

29

For these data, the 25^{th} percentile is
17, the 50^{th} percentile is 19, and the 75^{th}
percentile is 20. For the men (whose data are not shown), the
25^{th} percentile is 19, the 50^{th} percentile
is 22.5, and the 75^{th} percentile is 25.5.

Figure 1. The first step in creating box plots.

Before proceeding, the terminology in Table 2 is
helpful.

Table 2. Box plot terms and values for women's times.

Name

Formula

Value

Upper Hinge

75th Percentile

20

Lower Hinge

25th Percentile

17

H-Spread

Upper Hinge - Lower Hinge

3

Step

1.5 x H-Spread

4.5

Upper Inner Fence

Upper Hinge + 1 Step

24.5

Lower Inner Fence

Lower Hinge - 1 Step

12.5

Upper Outer Fence

Upper Hinge + 2 Steps

29

Lower Outer Fence

Lower Hinge - 2 Steps

8

Upper Adjacent

Largest value below Upper Inner Fence

24

Lower Adjacent

Smallest value above Lower Inner Fence

14

Outside Value

A value beyond an Inner Fence but not beyond an Outer
Fence

29

Far Out Value

A value beyond an Outer Fence

None

Continuing with the box plots, we put "whiskers" above
and below each box to give additional information about the spread
of the data. Whiskers are vertical lines that end in a horizontal
stroke. Whiskers are drawn from the upper and lower hinges to
the upper and lower adjacent values (24 and 14 for the women's
data).

Figure 2. The box plots with the whiskers drawn.

Although we don't draw whiskers all the way to outside
or far out values, we still wish to represent them in our box
plots. This is achieved by adding additional marks beyond the
whiskers. Specifically, outside values are indicated by small
"o's" and far out values are indicated by asterisks (*). In our
data, there are no far out values and just one outside value.
This outside value of 29 is for the women and is shown in Figure
3.

Figure 3. The box plots with the outside value shown.

There is one more mark to include in box plots (although
sometimes it is omitted). We indicate the mean score for a group
by inserting a plus sign. Figure 4 shows the result of adding
means to our box plots.

Figure 4. The completed box plots.

Figure 4 provides a revealing summary of the data.
Since half the scores in a distribution are between the hinges
(recall that the hinges are the 25^{th} and 75^{th}
percentiles), we see that half the women's times are between 17
and 20 seconds, whereas half the men's times are between 19 and 25.5. We
also see that women generally named the colors faster than the
men did, although one woman was slower than almost all of the
men. Figure 5 shows the box plot for the women's data with detailed
labels.

Figure 5. The box plot for the women's data with detailed labels.

Box plots provide basic information about a distribution. For example, a distribution with a positive skew would have a longer whisker in the positive direction than in the negative direction. A larger mean than median would also indicate a positive skew. Box plots are good at portraying extreme values and are especially good at showing differences between distributions. However, many of the details of a distribution are not revealed in a box plot, and to examine these details one should create a histogram and/or a stem and leaf display.

Statistical analysis programs may offer options
on how box plots are created. For example, the box plots in Figure
6 are constructed from our data but differ from the previous
box plots in several ways.

It does not mark outliers.

The means are indicated by green lines rather than
plus signs.

The mean of all scores is indicated by a gray line.

Individual scores are represented by dots. Since the scores
have been rounded to the nearest second, any given dot might
represent more than one score.

The box for the women is wider than the box for the men because
the widths of the boxes are proportional to the number of subjects
of each gender (31 women and 16 men).

Figure 6. Box plots showing the individual scores and the means.

Each dot in Figure 6 represents a group of subjects
with the same score (rounded to the nearest second). An alternative
graphing technique is to jitter the points.
This means spreading out different dots at the same horizontal
position, one dot for each subject. The exact horizontal position
of a dot is determined randomly (under the constraint that
different dots don’t overlap exactly). Spreading out the dots
helps you to see multiple occurrences of a given score. However, depending on the dot size and the screen resolution, some points may be obscured even if the points are jittererd. Figure
7 shows what jittering looks like.

Figure 7. Box plots with the individual scores jittered.

Different styles of box plots are best for different
situations, and there are no firm rules for which to use. When
exploring your data, you should try several ways of visualizing
them. Which graphs you include in your report should depend on
how well different graphs reveal the aspects of the data you consider
most important.