We have already discussed techniques for visually representing
data (see histograms
and frequency
polygons). In this section we present another important
method, called box plots. (We encountered a
simplified form of box plots in the introduction to this chapter.) 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. Students in Introductory
Statistics were presented with a page containing 30 colored
rectangles. Their task was to name the colors as quickly as
possible, and their times 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 25th, 50th, and 75th 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 25th percentile to the 75th
percentile. The 50th 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: Times (in seconds) for women to name the colors.
| 14 |
17 |
18 |
19 |
20 |
21 |
29 |
| 15 |
17 |
18 |
19 |
20 |
21 |
|
| 16 |
17 |
18 |
19 |
20 |
23 |
|
| 16 |
17 |
18 |
20 |
20 |
24 |
|
| 17 |
18 |
18 |
20 |
21 |
24 |
|
For these data, the 25th percentile is 17, the 50th percentile
is 19, and the 75th percentile is 20. For the men (whose data
are not shown), the 25th percentile is 19, the 50th percentile
is 22.5, and the 75th percentile is 25.5.
Before proceeding, the terminology in Table 2 is helpful.
Table 2: Terminology
| Name |
Formula |
Value for Women's Data |
| Upper Hinge |
75th percentile |
20 |
| Lower Hinge |
25th percentile |
17 |
| H-Spread |
Upper Hinge−Lower Hinge
Upper Hinge
Lower Hinge
|
3 |
| Step |
1.5H-Spread
1.5
H-Spread
|
4.5 |
| Upper Inner Fence |
Upper Hinge+1 Step
Upper Hinge
1 Step
|
24.5 |
| Lower Inner Fence |
Lower Hinge−1 Step
Lower Hinge
1 Step
|
12.5 |
| Upper Outer Fence |
Upper Hinge+2 Steps
Upper Hinge
2 Steps
|
29 |
| Lower Outer Fence |
Lower Hinge−2 Steps
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 (this value is on the fence, but not beyond) |
| Far Out Value |
A value beyond an Outer Fence |
None in these data |
Continuing with the box plots, we put "whiskers" above and below
each box, to give additional information about the spread of
data (Figure 2). Whiskers are
vertical lines that end in a horizontal stroke (the purpose of
the stroke is just to make the vertical lines more visible).
Whiskers are drawn from the upper and lower hinges to the upper
and lower adjacent values (24 and 14 for the women's data).
Although we don't draw whiskers all the way to outside or far
out values, we still wish to represent these outliers in our box
plots. This is achieved by adding additional marks beyond the
whiskers. Specifically, outside values are indicated by small
circles, and far out values are indicated by asterisks. In our
data, there are no far out values, and just one outside value.
The outside value of 29 is for the women, and is shown in Figure 3.
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 provides a revealing
summary of the data. Since half the scores in a distribution
are between the hinges (recall that the hinges are the 25th and
75th percentiles), we see that half the women's times are
between 17 and 20 whereas half the men's times are between 19
and 25. 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 boxplot for the women's data with detailed labels.
Here are some other examples of box plots.
Statistical analysis programs may offer options on how box
plots are created. For example, the box plot in Figure 6 is constructed from our data
but differs from the previous box plot in several ways.
- First, it does not mark outliers.
-
Second, the means are indicated by green lines rather than
plus signs.
-
The mean of all scores is indicated by a grey 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).
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 point is
determined randomly (under the constraint that different dots
don?t overlap). Spreading out the dots allows you to see
multiple occurrences of a given score. Figure 7 shows what jittering looks like.
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 graph you include in your report
should depend on how well different graphs reveal the aspects
of the data you consider most important.
percentiles
