Graphing Qualitative Variables
- Create a frequency table
- Determine when pie charts are valuable and when they are not
- Create and interpret bar charts
- Identify commmon graphical mistakes
When Apple Computer introduced the iMac computer
in August 1998, the company wanted to learn whether the iMac was
expanding Apples market share. Was the iMac just attracting
previous Macintosh owners? Or was it purchased by newcomers to
the computer market, and by previous Windows users who were switching
over? To find out, 500 iMac customers were interviewed. Each customer
was categorized as a previous Macintosh owners, a previous Windows
owner, or a new computer purchaser. This section examines graphical
methods for displaying the results of the interviews. Well
learn some general lessons about how to graph data that fall into
a small number of categories. A later section will consider how
to graph numerical data in which each observation is represented
by a number in some range. The key point about the qualitative
data that occupy us in the present section is that they do not
come with a pre-established ordering (the way numbers are ordered).
For example, there is no natural sense in which the category of
previous Windows users comes before or after the category of previous
iMac users. This situation may be contrasted with quantitative
data, such as a persons weight. People of one weight are
naturally ordered with respect to people of a different weight.
All of the graphical methods shown in this section
are derived from frequency tables. Table 1 shows a frequency table
for the results of the iMac study; it shows the frequencies of
the various response categories. It also shows the relative frequencies,
which are the proportion of responses in each category. For example,
the relative frequency for "none" of 0.17 = 85/500.
|Table 1. Frequency Table for the
The pie chart in Figure 1 shows the results
of the iMac study. In a pie chart, each category is represented
by a slice of the pie. The area of the slice is proportional to
the percentage of responses in the category. This is simply the
relative frequency multiplied by 100. Although most iMac purchasers
were Macintosh owners, Apple was encouraged by the 12% of purchasers
who were former Windows users, and by the 17% of purchasers who
were buying a computer for the first time.
|Figure 1. Pie chart of iMac purchases illustrating
frequencies of previous computer ownership.
Pie charts are effective for displaying the relative
frequencies of a small number of categories. They are not recommended,
however, when you have a large number of categories. Pie charts
can also be confusing when they are used to compare the outcomes
of two different surveys or experiments. In an influential book
on the use of graphs, Edward Tufte asserted "The only worse
design than a pie chart is several of them."
Here is another important point about pie charts.
If they are based on a small number of observations, it can be
misleading to label the pie slices with percentages. For example,
if just 5 people had been interviewed by Apple Computers, and
3 were former Windows users, it would be misleading to display
a pie chart with the Windows slice showing 60%. With so few people
interviewed, such a large percentage of Windows users might easily
have accord since chance can cause large errors with small samples.
In this case, it is better to alert the user of the pie chart
to the actual numbers involved. The slices should therefore be
labeled with the actual frequencies observed (e.g., 3) instead
of with percentages.
Bar charts can also be used to represent frequencies
of different categories. A bar chart of the iMac purchases is
shown in Figure 2. Frequencies are shown on the Y axis and the
type of computer previously owned is shown on the X axis. Typically
the Y-axis shows the number of observations rather than the percentage
of observations in each category as is typical in pie charts.
|Figure 2. Bar chart of iMac purchases as
a function of previous computer ownership.
Often we need to compare the results of different surveys, or of
different conditions within the same overall survey. In this case,
we are comparing the "distributions" of responses between
the surveys or conditions. Bar charts are often excellent for
illustrating differences between two distributions. Figure 3
shows the number of people playing card games at the Yahoo web
site on a Sunday and on a Wednesday on a day in the Spring of
2001. We see that there were more players overall on Wednesday
compared to Sunday. The number of people playing Pinochle was
nonetheless the same on these two days. In contrast, there were
about twice as many people playing hearts on Wednesday as on
Sunday. Facts like these emerge clearly from a well-designed
|Figure 3. A bar chart
of the number of people playing different card games on Sunday
The bars in Figure 3 are oriented horizontally
rather than vertically. The horizontal format is useful when you
have many categories because there is more room for the category
labels. Well have more to say about bar charts when we consider
numerical quantities later in the section Bar
Some graphical mistakes to avoid
Dont get fancy! People sometimes add features
to graphs that dont help to convey their information. For
example, 3-dimensional bar charts like the one shown in Figure
4 are usually not as effective as their two-dimensional counterparts.
|Figure 4. A three-dimensional
version of Figure 2.
Here is another way that fanciness can lead to trouble. Instead
of plain bars, it is tempting to substitute meaningful images.
For example, Figure 6 presents the iMac data using pictures of
computers. The heights of the pictures accurately represent the
number of buyers, yet Figure 6 is misleading because the viewer's
attention will be captured by areas. This can exaggerate the size
differences between the groups. In terms of percentages, the ratio
of previous Macintosh owners to previous Windows owners is about
6 to 1. But the ratio of the two areas in Figure 6 is about 35
to 1. A biased person wishing to hide the fact that many Windows
owners purchased iMacs would be tempted to use Figure 6 instead
of Figure 2! Edward
Tufte coined the term "lie factor" to refer to the
ratio of the size of the effect shown in a graph to the size of
the effect shown in the data. He suggests that lie factors greater
than 1.05 or less than 0.95 produce unacceptable distortion.
|Figure 6. A redrawing
of Figure 2 with a lie factor greater than 8.
Another distortion in bar charts results from
setting the baseline to a value other than zero. The baseline
is the bottom of the Y-axis, representing the least number of
cases that could have occurred in a category. Normally, this number
should be zero. Figure 7 shows the iMac data with a baseline of
50. Once again, the difference in areas suggests a different story
than the true differences in percentages. The number of Windows-switchers
seems minuscule compared to its true value of 12%.
|Figure 7. A redrawing
of Figure 2 with a baseline of 50.
Finally, we note that it is a serious mistake
to use a line graph when the X-axis contains merely qualitative
variables. A line graph is essentially a bar graph with the tops
of the bars represented by points joined by lines (the rest of
the bar is suppressed). Figure 5 inappropriately shows a line
graph of the card game data from Yahoo. The drawback to Figure
5 is that it gives the false impression that the games are naturally
ordered in a numerical way.
|Figure 5. A line graph
of the number of people playing different card games on Sunday
Pie charts and bar charts can both be effective
methods of portraying qualitative data. Bar charts are better
when there are more than just a few categories and for comparing
two or more distributions. Be careful to avoid creating misleading