1.6 Using Percents to Describe Charts

Narrated lesson: Using Percents to Describe Charts

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Summarizing raw data is the first step we must take when we want to communicate the results of a study or experiment to a broad audience. However, even organized data can be difficult to read; for example, if a frequency table is large, it can be tough to compare the first row to the last row. As the old saying goes: a picture is worth a thousand words (or, in this case, summary statistics)! The methods we’ll use for creating visualizations will vary with the type of data. Let’s start by considering categorical data.

Visualizing Categorical Data

If the data we’re visualizing is categorical, then we want a quick way to represent graphically the relative numbers of units that fall in each category. When we create a frequency distribution, all we do is count the number of units in each category and record that number (this is the frequency of that category). Frequencies are nice when we’re organizing and summarizing data; they’re easy to compute, and they’re always whole numbers. But they can be difficult to understand for an outsider who’s being introduced to your data.

Let’s consider a quick example. Suppose you surveyed some people and asked for their favorite color. You communicated your results using a frequency distribution. Jerry is interested in data on favorite colors, so he reads your frequency distribution. The first row shows that twelve people indicated green was their favorite color. However, Jerry has no way of knowing if that’s a lot of people without knowing how many people total took your survey. Twelve is a pretty significant number if only twenty-five people took the survey, but it’s next to nothing if you recorded a thousand responses. For that reason, we will often summarize categorical data not with frequencies, but with proportions. The proportion of data that fall into a particular category is computed by dividing the frequency for that category by the total number of units in the data.

$Proportion of a category = \frac{Category frequency}{Total number of data units}$

Proportions can be expressed as fractions, decimals, or percentages.

Example 1 Finding Proportions

Suppose a teacher recorded the responses on the first question of a multiple choice quiz, with five possible responses (A, B, C, D, and E). The raw data was as follows:

A	A	C	A	B	B	A	E	A	C	A	A	A	C
E	A	B	A	A	C	A	B	E	E	A	A	C	C

Counting the responses in each category gives this frequency distribution:

Response to First Question	Frequency
A	14
B	4
C	6
D	0
E	4

$Proportion of a category equals Category frequency / Total number of data units$

Now, let's compute the proportions for each category.

Show solution

Step 1: In order to compute a proportion, we need the frequency (which we have in the table above) and the total number of units that are represented in our data. We can find that by adding up the frequencies from all the categories: $14 + 4 + 6 + 0 + 4 = 28$ .

Step 2: To find the proportions, we divide the frequency by the total. For the first category (“A”), the proportion is $\frac{14}{28} = \frac{1}{2} = 0.5 = 50 % .$ We can compute the other proportions similarly, filling in the rest of the table:

Response to First Question	Frequency	Proportion
A	14	$\frac{14}{28} = 50 %$
B	4	$\frac{4}{28} \approx 14.3 %$
C	6	$\frac{6}{28} \approx 21.4 %$
D	0	$\frac{0}{28} = 0 %$
E	4	$\frac{4}{28} \approx 14.3 %$

Step 3: Check your work: If you add up your proportions, you should get 1 (if you’re using fractions or decimals) or 100% (if you’re using percentages). In this case, $50 % + 14.3 % + 21.4 % + 0 % + 14.3 % = 100 % .$

Your Turn

In Your Turn 8.2, students in a statistics class were asked to provide their majors. Those results are again listed below:

Undecided	Biology	Biology	Sociology
Political Science	Sociology	Undecided	Undecided
Undecided	Biology	Biology	Education
Biology	Biology	Political Science	Political Science

You created a frequency distribution:

Major	Frequency
Biology	6
Education	1
Political Science	3
Sociology	2
Undecided	4

Now, find the proportions associated with each category. Express your answers as percentages.

Show solution

Major	Frequency	Proportion
Biology	6	37.5%
Education	1	6.3%
Political Science	3	18.8%
Sociology	2	12.5%
Undecided	4	25%

Note that these percentages add up to 100.1%, due to the rounding.

Step 1: To compute a proportion, you need the frequency which you have in the table, and the total number of units that are represented in the data.

Major	Frequency
Biology	6
Education	1
Political Science	3
Sociology	2
Undecided	4

Add the frequencies: 6 + 1 + 3 + 2 + 4 = 16

Step 2: To find the proportions, divide the frequency by the total.

Major	Frequency	Proportion
Biology	6	$\frac{6}{{16}} = 37.5\%$
Education	1	$\frac{1}{{16}} = 6.25\%$
Political Science	3	$\frac{3}{{16}} = 18.75\%$
Sociology	2	$\frac{2}{{16}} = 12.5\%$
Undecided	4	$\frac{4}{{16}} = 25\%$

Step 3: Add up your proportions. You should get 100%.

37.5% + 6.25% + 18.75% + 12.5% + 25% = 100%

Now that we can compute proportions, let’s turn to visualizations. There are two primary visualizations that we’ll use for categorical data: bar charts and pie charts. Both of these data representations work on the same principle: If proportions are represented as areas, then it’s easy to compare two proportions by assessing the corresponding areas. Let’s look at bar charts first.

Bar Charts

A bar chart is a visualization of categorical data that consists of a series of rectangles arranged side-by-side (but not touching). Each rectangle corresponds to one of the categories. All of the rectangles have the same width. The height of each rectangle corresponds to either the number of units in the corresponding category or the proportion of the total units that fall into the category.

Example 2 Building a Bar Chart

In Example 1, we computed the following proportions:

Response to First Question	Frequency	Proportion
A	14	50%
B	4	14.3%
C	6	21.4%
D	0	0%
E	4	14.3%

Draw a bar chart to visualize this frequency distribution.

Show solution

Step 1: To start, we’ll draw axes with the origin (the point where the axes meet) at the bottom left:

A blank bar chart with horizontal and vertical axes. — **Figure 1.12**

Step 2: Next, we’ll place our categories evenly spaced along the bottom of the horizontal axis. The order doesn’t really matter, but if the categories have some sort of natural order (like in this case, where the responses are labeled A to E), it’s best to maintain that order. We'll also label the horizontal axis:

A bar chart. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis is blank. — **Figure 1.13**

Step 3: Now, we have a decision to make: Will we use frequencies to define the height of our rectangles, or will we use proportions? Let’s try it both ways. First, let’s use frequencies. Notice that our frequencies run from zero to 14; this will correspond to the scale we put on the vertical axis. If we put a tick mark for every whole number between 0 and 14, the result will be pretty crowded; let’s instead put a mark on the multiples of 3 or 5:

A bar chart. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing frequency ranges from 3 to 15, in increments of 3. — **Figure 1.14**

Step 4: Now, let’s draw in the first rectangle. The frequency associated with “A” is 14. So we’ll go to 14 on the vertical axis, and place a mark at that height above the “A” label:

A bar chart plots a horizontal dashed line. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing frequency ranges from 3 to 15, in increments of 3. The bar chart infers the following data. A: 14. — **Figure 1.15**

Step 5: Then, draw vertical lines straight down from the edges of your mark to make a rectangle:

A bar chart shows a vertical bar. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing frequency ranges from 3 to 15, in increments of 3. The bar chart infers the following data. A: 14. — **Figure 1.16**

Step 6: Finally, we can build the rest of the rectangles, making sure that the bases all have the same length of the base = width of the rectangle, and the rectangles don’t touch. Notice that, since the frequency for “D” is zero, that category has no rectangle (but we’ll leave a space there so the reader can see that there is a category with frequency zero). Here’s the result:

A bar chart plots frequency versus responses. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing frequency ranges from 3 to 15, in increments of 3. The bar graph infers the following data. A: 14; B: 4; C: 6; E: 4. — **Figure 1.17**

Step 7: That’s it! Now, let’s use proportions instead of frequencies. We'll label the vertical axis with evenly spaced numbers that run the full range of the percentages in our table: 0% to 50%. We can divide that into five equal parts (so that each has width 10%), and use that to label our vertical axis:

A bar chart. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing percent ranges from 10 percent to 50 percent, in increments of 10. — **Figure 1.18**

Step 8: Then, we can fill in the rectangles just as we did before. The height of the “A” rectangle is 50%, the “B” rectangle goes up to 14.3%, “C” goes to 21.4%, there is no rectangle for “D” (since its proportion is 0%), and the “E” rectangle also goes up to 14.3%:

A bar chart plots percent versus responses on question 1. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis representing percent ranges from 10 percent to 50 percent, in increments of 10. The graph infers the following data. A: 50; B: 14.3; C: 21.4; E: 14.3. Note: all values are approximate. — **Figure 1.19**

Step 9: Notice that the rectangles are basically identical in our two final bar charts. That’s no coincidence! For the same categories and a common nonzero total, frequencies and exact proportions have the same relative bar heights when plotted from zero on linear axes. The charts can look identical when their vertical scales are adjusted consistently. Here’s why: look at the bars for “B” and “C”. The frequencies for these are 4 and 6 respectively. Notice that 6 is 50% bigger than 4 (since $6 = 1.5 \times 4$ ), which means that the “C” bar will be 50% higher than the “B” bar. Now look at the same bars using proportions: since $\frac{6}{28} = 1.5 \times \frac{4}{28}$ , the bar for “C” will be 50% higher than the bar for “B.” The same relationships hold for the other bars, too.

A bar chart plots frequency and percent versus responses on question 1. The horizontal axis representing response on question 1 ranges from A to E. The vertical axis on the left representing frequency ranges from 3 to 15, in increments of 3. The vertical axis on the right representing percent ranges from 10 percent to 50 percent, in increments of 10. The graph infers the following data. A: 14, 50 percent; B: 4, 14.3 percent; C: 6, 21.4 percent; E: 4, 14.3 percent. Note: all values are approximate. — **Figure 1.20**

Your Turn

The students in a statistics class were asked to provide their majors. The computed proportions for each of the categories are as follows:

Major	Frequency	Proportion
Biology	6	37.5%
Education	1	6.3%
Political Science	3	18.8%
Sociology	2	12.5%
Undecided	4	25%

Create a bar graph to visualize these data. Use percentages to label the vertical axis.

Show solution

A bar chart consists of a series of rectangles arranged side-by-side (but not touching). Each rectangle corresponds to one of the categories. All the rectangles have the same width. The height of each rectangle corresponds to either the number of units in the corresponding category or the proportion of the total units that fall into the category.

Step 1: Draw the axes with the origin at the bottom left.

Step 2: Next, place your categories evenly spaced along the bottom of the horizontal axis. The order does not matter. If they have some natural order, maintain that order. Label the horizontal axis.

You will have your five majors, along with the label “Majors.”

Step 3: Decide how to define the height of the rectangles, the frequency itself or the proportion. Mark the vertical axis with units appropriate to your choice.

Use “Percent” for your vertical axis. Label your tick marks every 5% up to a maximum of 40%.

Step 4: Draw in the rectangles. Place a horizontal mark above the labels at the height appropriate for each category.

Use the table to mark the height of your five bars.

Step 5: Draw vertical lines straight down from the edges of your mark to make a rectangle.

Step 6: Build the rest of the rectangles, making sure they all have the same width. If a category has a percentage of 0, leave a space. Do not let the rectangles touch.

In practice, most graphs are now made with computers. A spreadsheet application can help create charts; Google Sheets is one option for use in a browser.

Now that we’ve explored how bar graphs are made, let’s get some practice reading bar graphs.

Example 3 Reading Bar Graphs

The source graph shows 2020 model year cars available in the United States; use its plotted values for the following questions. This historical example does not establish the current car market, and the retained source gives no underlying model counts.

Figure 1.21 (data source: consumerreports.org/cars)

What proportion of available cars were sports cars?
What proportion of available cars were sedans?
Which categories of cars each made up less than 5% of the models available?

Show solution

The bar for sports cars goes up to 10%, so the proportion of models that are considered sports cars is 10%.
The bar corresponding to sedan goes up past 30% but not quite to 35%. It looks like the proportion we want is between 33% and 34%.
We’re looking for the bars that don’t make it all the way to the 5% line. Those categories are hatchback and wagon.

Your Turn

Use the source's retained bar graph of regions of U.S. higher-education institutions, excluding service academies. Read the values shown in this graph; this exercise does not establish the current distribution of institutions. The retained exercise supplies no dataset date or provenance.

Analyze the bar chart to answer the following questions.

Which region contains the largest number of institutions of higher learning?

Show solution

The chart shows the Southeast region has the highest percentage of around 25 percent.

Southeast

What proportion of all institutions of higher learning can be found in the Southwest?

Show solution

Just over 10%

Which regions each have under 5% of the total number of institutions of higher learning?

Show solution

Two regions have bars lower than the 5% line: Outlying Areas and Rocky Mountains.

Outlying Areas and Rocky Mtns.

Pie Charts

A pie chart consists of a circle divided into wedges, with each wedge corresponding to a category. The proportion of the area of the entire circle that each wedge represents corresponds to the proportion of the data in that category. Pie charts are difficult to make without technology because they require careful measurements of angles and precise circles, both of which are tasks better left to computers.

Pie charts are sometimes embellished with features like labels in the slices (which might be the categories, the frequencies in each category, or the proportions in each category) or a legend that explains which colors correspond to which categories. When making your own pie chart, you can decide which of those to include. The only rule is that there has to be some way to connect the slices to the categories (either through labels or a legend).

Example 4 Making Pie Charts

Use the data that follows to generate a pie chart.

**Table 1.9** (data source: www.consumerreports.org/cars)
Type	Percent	Type	Percent
SUV	43.6%	Minivan	5.5%
Sedan	33.6%	Hatchback	3.6%
Sports	10.0%	Wagon	3.6%

Show solution

First, reformat the supplied historical car data as six rows in two columns: Type and Percent, with one category and its supplied percentage on each row. The source’s displayed one-decimal percentages total 99.9%, rather than 100%. Rounding can cause this discrepancy, but the source gives no exact model counts with which to check the original proportions. This creates an approximate chart from the supplied values and does not recover exact underlying proportions. In Google Sheets on a computer, select these two columns including their headers and choose Insert, then Chart. Double-click the chart, open Setup, and select Pie chart under Chart type. The two-column format and steps follow Google’s pie-chart help and chart help, checked October 3, 2026; a later interface may differ.

A pie chart titled, 2020 model year cars available in the US. The circle graph infers the following data. SUV: 43.6 percentage; Sedan: 33.6 percentage; Sports: 10.0 percentage; Minivan: 5.5 percentage; Hatchback: 3.6 percentage; Wagon: 3.6 percentage. — **Figure 1.22** (data source: consumerreports.org/cars)

You can choose to use a legend to identify the categories, as well as label the slices with the relevant percentages.

Your Turn

In Your Turn 8.6, you created a bar chart using data on reported majors from students in a class. Here are those proportions again (sorted from largest to smallest):

Major	Proportion
Biology	37.5%
Undecided	25.0%
Political Science	18.8%
Sociology	12.5%
Education	6.3%

Create a pie graph using the exact frequencies for the 16 students: Biology 6, Undecided 4, Political Science 3, Sociology 2, and Education 1. The displayed percentages are rounded to one decimal place and total 100.1%; use the frequencies to size the wedges, and use the rounded percentages only as display labels.

Show solution

A pie chart consists of a circle divided into wedges, with each edge corresponding to a category. The proportion of the area of the entire circle that each wedge represents corresponds to the proportion of the data in that category. Use the exact frequencies, or the exact proportions in the table below, as chart data. The percentages printed on the illustrated pie are rounded display labels. In Google Sheets on a computer, select the data cells and choose Insert, then Chart. Double-click the chart, open Setup, and choose Pie chart under Chart type. These steps follow Google’s chart help, checked October 2, 2026; a later interface may differ.

Major	Proportion
Biology	37.5%
Education	6.25%
Political Science	18.75%
Sociology	12.5%
Undecided	25%

Your pie chart may look a bit different than the one in the solution and that’s fine. Just be sure that the relative size of the wedges is the same and that you have labels on your pie “slices.”

People in Mathematics

Florence Nightingale

Florence Nightingale (1820–1910) is best remembered today for her contributions in the medical field; after witnessing the horrors of field hospitals that tended to the wounded during the Crimean War, she championed reforms that encouraged sanitary conditions in hospitals. For those efforts, she is today considered the founder of modern nursing.

Figure 1.23 Florence Nightingale's significant contribution to the field of statistical graphics cannot be understated.Florence Nightingale's significant contribution to the field of statistical graphics cannot be understated. (credit: "Florence Nightingale" by Library of Congress Prints and Photographs Division/http://hdl.loc.gov/loc.pnp/pp.print, public domain)

Nightingale is also remembered for her contributions in statistics, especially in the ways we visualize data. She developed a version of the pie chart that is today known as a polar area diagram, which she used to visualize the causes of death among the soldiers in the war, highlighting the number of preventable deaths the British Army suffered in that conflict.

In 1858, Nightingale became the first woman member of the Statistical Society of London, now the Royal Statistical Society. She was later named an honorary member of the American Statistical Association. Nightingale's status as a revered pioneer in both nursing and statistics is a complex one, because some of her writings and opinions demonstrate a colonialist mindset and disregard for those who lost their lives and lands at the hands of the British. Her core statistical writings indicated that she felt superior to the Indigenous people she was treating. Members of both fields continue to debate her near-iconic role.

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