11.1 Presenting Categorical Data Graphically

Categorical, or qualitative, data are pieces of information that allow us to classify the objects under investigation into various categories. We usually begin working with categorical data by summarizing the data into a frequency table.

Example 1

An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some color cars are more likely to get in accidents. To research this, they examine police reports for recent total-loss collisions. The data is summarized in the frequency table below.

import matplotlib.pyplot as plt

# The book's total-loss collision data. CHANGE a count, or add a color, and re-run.
data = {"Blue": 25, "Green": 52, "Red": 41, "White": 36, "Black": 39, "Grey": 23}

total = sum(data.values())
print("%-8s %10s %12s %9s" % ("Color", "Frequency", "Relative", "Pie angle"))
print("-" * 42)
for color, freq in data.items():
    print("%-8s %10d %11.1f%% %8.1f deg" % (color, freq, 100 * freq / total, 360 * freq / total))
print("-" * 42)
print("%-8s %10d %11.1f%%" % ("TOTAL", total, 100.0))
print()
print("Careful: this is a frequency table of CRASHES, not a risk table. To say")
print("green cars are risky you would also need to know how many green cars exist.")

pareto = dict(sorted(data.items(), key=lambda kv: -kv[1]))   # highest to lowest
fig, ax = plt.subplots(1, 3, figsize=(13, 3.6))

ax[0].bar(data.keys(), data.values(), color="#4472c4")
ax[0].set_title("Bar graph (as collected)")
ax[0].set_ylabel("Frequency")

ax[1].bar(pareto.keys(), pareto.values(), color="#ed7d31")
ax[1].set_title("Pareto chart (sorted)")
for i, v in enumerate(pareto.values()):
    ax[1].text(i, v + 1, str(v), ha="center", fontsize=8)

ax[2].pie(data.values(), labels=data.keys(), autopct="%1.1f%%", startangle=90)
ax[2].set_title("Pie chart")

# BASELINE TRAP: uncomment the next line to start the bar chart at 20 instead of 0
# ax[0].set_ylim(20, 55)   # same data, and green now looks 6x bigger than grey
plt.tight_layout()
plt.show()

Color	Frequency
Blue	25
Green	52
Red	41
White	36
Black	39
Grey	23

Sometimes we need an even more intuitive way of displaying data. This is where charts and graphs come in. There are many, many ways of displaying data graphically, but we will concentrate on one very useful type of graph called a bar graph. In this section we will work with bar graphs that display categorical data; the next section will be devoted to bar graphs that display quantitative data.

To construct a bar graph, we need to draw a vertical axis and a horizontal axis. The vertical direction will have a scale and measure the frequency of each category; the horizontal axis has no scale in this instance. The construction of a bar chart is most easily described by use of an example.

Notice that the height of each bar is determined by the frequency of the corresponding color. The horizontal gridlines are a nice touch, but not necessary. In practice, you will find it useful to draw bar graphs using graph paper, so the gridlines will already be in place, or using technology. Instead of gridlines, we might also list the frequencies at the top of each bar, like this:

The same vehicle-colour bar graph with the gridlines removed and the frequency printed above each bar instead: Blue 25, Green 52, Red 41, White 36, Black 39, Grey 23. Vertical axis Frequency, 0 to 55.

In this case, our chart might benefit from being reordered from largest to smallest frequency values. This arrangement can make it easier to compare similar values in the chart, even without gridlines. When we arrange the categories in decreasing frequency order like this, it is called a Pareto chart.

Example 4

In a survey[1], adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried “a great deal” about some selected concerns are summarized below.

Environmental Issue	Frequency
Pollution of drinking water	597
Contamination of soil and water by toxic waste	526
Air pollution	455
Global warming	354

Show solution

This data could be shown graphically in a bar graph:

To show relative sizes, it is common to use a pie chart.

Pie charts look nice, but are harder to draw by hand than bar charts since to draw them accurately we would need to compute the angle each wedge cuts out of the circle, then measure the angle with a protractor. Computers are much better suited to drawing pie charts. Common software programs like Microsoft Word or Excel, OpenOffice.org Write or Calc, or Google Docs are able to create bar graphs, pie charts, and other graph types. There are also numerous online tools that can create graphs[2].

Don’t get fancy with graphs! People sometimes add features to graphs that don’t help to convey their information. For example, 3-dimensional bar charts like the one shown below are usually not as effective as their two-dimensional counterparts.

Here is another way that fanciness can lead to trouble. Instead of plain bars, it is tempting to substitute meaningful images. This type of graph is called a pictogram.

Another distortion in bar charts results from setting the baseline to a value other than zero. The baseline is the bottom of the vertical axis, representing the least number of cases that could have occurred in a category. Normally, this number should be zero.

[1] Gallup Poll. March 5-8, 2009. http://www.pollingreport.com/enviro.htm

[2] For example: http://nces.ed.gov/nceskids/createAgraph/ or http://docs.google.com

[3]CNN/Opinion Research Corporation Poll. Dec 19-21, 2008, from http://www.pollingreport.com/civil.htm

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.