5.2: Describing Data

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Once we have collected data from surveys or experiments, we need to summarize and present the data in a way that will be meaningful to the reader. We will begin with graphical presentations of data and then explore numerical summaries of data.

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.

Definition: Frequency Table

A frequency table is a table with two columns. One column lists the categories, and another for the frequencies with which the items in the categories occur (how many items fit into each category).

Example $\PageIndex{1}$

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.

Table $\PageIndex{1}$
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. Here, we will work with bar graphs that display categorical data; later, we will be devoted to bar graphs that display quantitative data.

Definition: Bar Graph

A bar graph is a graph that displays a bar for each category with the length of each bar indicating the frequency of that category.

Example $\PageIndex{2}$

Using our car data from above, note the highest frequency is 52, so our vertical axis needs to go from 0 to 52, but we might as well use 0 to 55, so that we can put a hash mark every 5 units:

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:

Figure $\PageIndex{2}$: Car Data (Frequency listed at the top of each bar)

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.

Definition: Pareto Chart

A pareto chart is a bar graph ordered from highest to lowest frequency.

Example $\PageIndex{3}$

Transforming our bar graph from earlier into a Pareto chart, we get:

Example $\PageIndex{4}$

In a survey, 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.

Table $\PageIndex{2}$
Environmental Issue	Frequency
Pollution of drinking water	597
Contamination of soil and water by toxic waste	526
Air pollution	455
Global warming	354

This data could be shown graphically in a bar graph:

Figure $\PageIndex{4}$: Environmental worries data

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

Definition: Pie Chart

A pie chart is a circle with wedges cut of varying sizes marked out like slices of pie or pizza. The relative sizes of the wedges correspond to the relative frequencies of the categories.

Example $\PageIndex{5}$

For our vehicle color data, a pie chart might look like this:

Figure $\PageIndex{5}$: Pie chart of Vehicle color involved in total-loss collisions

Pie charts can often benefit from including frequencies or relative frequencies (percents) in the chart next to the pie slices. Often having the category names next to the pie slices also makes the chart clearer.

Figure $\PageIndex{6}$: Frequencies (percents) in the chart next to the pie slices

Example $\PageIndex{6}$

The pie chart below shows the percentage of voters supporting each candidate running for a local senate seat.

If there are 20,000 voters in the district, the pie chart shows that about 11% of those, about 2,200 voters, support Reeves.

Figure $\PageIndex{7}$: Voter preferences

Pie charts look nice but are harder to draw by hand than bar charts since we would need to compute the angle each wedge cuts out of the circle and 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.

Try it Now 1

Create a bar graph and a pie chart to illustrate the grades on a history exam below. A: 12 student, B: 19 students, C: 14 students, D: 4 students, F: 5 students.

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.

Figure $\PageIndex{8}$: 3-dimensional bar chart

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.

Definition: Pictogram

A pictogram is a statistical graphic in which the size of the picture is intended to represent the frequencies or size of the values being represented.

Example $\PageIndex{7}$

A labor union might produce the graph below to show the difference between the average manager salary and the average worker salary.

Looking at the picture, it would be reasonable to guess that the manager salaries is 4 times as large as the worker salaries – the area of the bag looks about 4 times as large. However, the manager salaries are in fact only twice as large as worker salaries, which were reflected in the picture by making the manager bag twice as tall.

Figure $\PageIndex{9}$: Salary representations

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.

Example $\PageIndex{8}$

Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December 2008. The difference in the vertical scale on the first graph suggests a different story than the true differences in percentages; the second graph makes it look like three times as many people oppose marriage rights as support it.

Figure $\PageIndex{10}$: Same-sex marriage graphs

Try it Now 2

A poll was taken asking people if they agreed with the positions of the 4 candidates for a county office. Does the pie chart present a good representation of this data? Explain.

Presenting Quantitative Data Graphically

Quantitative, or numerical, data can also be summarized into frequency tables.

Example $\PageIndex{9}$

A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are:

19, 20, 18, 18, 17, 18, 19, 17, 20, 18, 20, 16, 20, 15, 17, 12, 18, 19, 18, 19, 17, 20, 18, 16, 15, 18, 20, 5, 0, 0

Table $\PageIndex{3}$
Score	Frequency
0	2
5	1
12	1
15	2
16	2
17	4
18	8
19	4
20	6

Using this table, it would be possible to create a standard bar chart from this summary as we did for categorical data:

Figure $\PageIndex{12}$: Standard bar chart from the above summary

However, since the scores are numerical values, this chart doesn’t really make sense; the first and second bars are five values apart, while the later bars are only one value apart. It would be better to treat the horizontal axis as a number line. This type of graph is called a histogram.

Definition: Histogram

A histogram is like a bar graph, but the horizontal axis is a number line.

Example $\PageIndex{10}$

For the values above, a histogram would look like:

Figure $\PageIndex{13}$: Histogram for above values

Notice that in the histogram, a bar represents values on the horizontal axis from that on the left hand-side of the bar up to, but not including, the value on the right hand side of the bar. So, for instance, the fact that there are four occurrences of 17 in the dataset is indicated by the bar, with height 4, between 17 and 18 in this histograph. Some people choose to have bars start at ½ values to avoid this ambiguity.

Unfortunately, not a lot of common software packages can correctly graph a histogram. About the best you can do in Excel or Word is a bar graph with no gap between the bars and spacing added to simulate a numerical horizontal axis.

If we have a large number of widely varying data values, creating a frequency table that lists every possible value as a category would lead to an exceptionally long frequency table and probably would not reveal any patterns. For this reason, it is common with quantitative data to group data into class intervals.

Definition: Class Intervals

Class intervals are groupings of the data. In general, we define class intervals so that:

Each interval is equal in size. For example, if the first class contains values from 120-129, the second class should include values from 130-139.
We have somewhere between 5 and 20 classes, typically, depending upon the number of data we’re working with.

Example $\PageIndex{11}$

Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263-121 = 142. We could create 7 intervals with a width of around 20 or 14 intervals with a width of around 10, or somewhere in between. Often we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number.

Table $\PageIndex{4}$
Interval	Frequency
120-134	4
135-149	14
150-164	16
165-179	28
180-194	12
195-209	8
210-224	7
225-239	6
240-254	2
255-269	3

A histogram of this data would look like:

Figure $\PageIndex{14}$: Histogram of the above data

In many software packages, you can create a graph similar to a histogram by putting the class intervals as the labels on a bar chart.

Figure $\PageIndex{15}$: Histogram created by putting class intervals on bar chart

Other graph types such as pie charts are possible for quantitative data. The usefulness of different graph types will vary depending upon the number of intervals and the type of data being represented. For example, a pie chart of our weight data is difficult to read because of the quantity of intervals we used.

Figure $\PageIndex{16}$: Pie chart of our weight data

Try it Now 3

The total cost of textbooks for the term was collected from 36 students. Create a histogram for this data.

$140 $160 $160 $165 $180 $220 $235 $240 $250 $260 $280 $285

$285 $285 $290 $300 $300 $305 $310 $310 $315 $315 $320 $320

$330 $340 $345 $350 $355 $360 $360 $380 $395 $420 $460 $460

When collecting data to compare two groups, it is desirable to create a graph that compares quantities.

Example $\PageIndex{12}$

The data below came from a task in which the goal is to move a computer mouse to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial.

Table $\PageIndex{5}$
Interval (milliseconds)	Frequency (small target)	Frequency (large target)
300-399	0	0
400-499	1	5
500-599	3	10
600-699	6	5
700-799	5	0
800-899	4	0
900-999	0	0
1000-1099	1	0
1100-1199	0	0

One option to represent this data would be a comparative histogram or bar chart in which bars for the small target group and large target group are placed next to each other.

Figure $\PageIndex{17}$: A comparative histogram or bar chart

Definition: Frequency Polygon

An alternative representation is a frequency polygon. A frequency polygon starts out like a histogram, but instead of drawing a bar, a point is placed in the midpoint of each interval at height equal to the frequency. Typically the points are connected with straight lines to emphasize the distribution of the data.

Example $\PageIndex{13}$

This graph makes it easier to see that reaction times were generally shorter for the larger target and that the reaction times for the smaller target were more spread out.

Reference

References (11)

Contributors and Attributions

Saburo Matsumoto
CC-BY-4.0