Displaying Categorical Data: Bar Graphs and Pie Charts

15 min read

Quick Reference

Try This First

A first-year seminar of 50 students reports the college each student plans to enter. The advising office records the results as a table of counts.

Before reading any definition, work these on paper.

1. Sketch five vertical bars, one per college, with each bar tall in proportion to its count. Which bar is tallest, and which is shortest?
2. Turn each count into a percent of the 50 students. (For Business, that is $\frac{14}{50}$ written as a percent.)
3. A classmate says “the five bars should touch, since this is a graph.” Predict whether that is correct, then keep reading.

You have just built a bar graph by hand and read its distribution. The pie chart is the second standard picture of the same table.

An Intuitive Introduction

A categorical variable sorts each individual into one of a few groups: the intended college above, a person’s blood type, a movie’s genre, a survey answer of “agree” or “disagree.” There is no meaningful order or arithmetic on the groups themselves. The number 18 in the Engineering row is a count of how many individuals fall in that group, not a measurement of any one student.

The distribution of a categorical variable is the list of groups together with how often each group occurs, either as a count or as a percent. Two pictures show this distribution.

• A bar graph answers “how big is each group compared to the others?” by drawing one bar per group, tall in proportion to that group’s count or percent.
• A pie chart answers “what share of the whole is each group?” by cutting a circle into slices, one slice per group, sized in proportion to that group’s percent.

Both pictures carry the same numbers. The choice between them is a choice of question. The bar graph is easy to read for ranking and comparison. The pie chart is easy to read for part-to-whole share, and it only makes sense when the groups together make up one whole.

Prerequisite Hub

Builds On

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Cross-Course Prerequisites

None. The arithmetic of counts and percents is in the review nodes above.

The Official Definitions

Bar graph. A bar graph displays the distribution of a categorical variable by drawing one bar for each category, with the height of each bar showing the count or the percent of individuals in that category. The bars are drawn with gaps between them because the categories are not measured on a numerical scale.

(Moore, McCabe, Craig, IPS 9th edition, Section 1.2, Displaying Distributions with Graphs; corroborated by OpenStax Introductory Statistics 2e, 2.1.)

Pie chart. A pie chart displays the distribution of a categorical variable as slices of a circle, with each slice sized in proportion to the percent of individuals in that category. A pie chart requires that the categories together make up some whole, since the slices must sum to 100 percent.

(Moore, McCabe, Craig, IPS 9th edition, Section 1.2, Displaying Distributions with Graphs.)

IPS Section 1.2 states no theorem here. The content is two definitions and the practice of building and reading the two displays correctly. The constraints to hold onto are the two clauses inside the definitions: a bar graph leaves gaps between bars (the categories are not points on a number line), and a pie chart needs the categories to partition one whole (the slices must sum to 100 percent).

Reading the Two Pictures Side by Side

Both pictures encode the same table, so the same data translates freely between the table, the bar graph, and the pie chart.

Worked Examples

Example 1: From a Count Table to a Bar Graph and a Pie Chart

A campus coffee stand records the size of every drink sold during one morning.

Predict first. Before any arithmetic, which size will have the tallest bar, and roughly what fraction of the circle will the Medium slice take? Medium has the largest count, so it gets the tallest bar, and 60 out of 120 is half, so the Medium slice should be about half the circle.

Step 1: Turn each count into a percent of the whole.

\[ \text{Small} = \frac{24}{120} = 0.20 = 20\%, \qquad \text{Medium} = \frac{60}{120} = 0.50 = 50\%, \qquad \text{Large} = \frac{36}{120} = 0.30 = 30\%. \]

Check the share: $20\% + 50\% + 30\% = 100\%$. The percents account for the whole, so a pie chart is valid here.

Step 2: Build the bar graph. Draw three bars, one per drink size, with gaps between them. The vertical axis can show counts (heights 24, 60, 36) or percents (heights 20, 50, 30). The shape of the picture is the same either way; only the axis label changes.

Step 3: Build the pie chart. A full circle is $360$ degrees. Each slice’s angle is its percent of $360$ degrees.

\[ \text{Small} = 0.20 \times 360^\circ = 72^\circ, \qquad \text{Medium} = 0.50 \times 360^\circ = 180^\circ, \qquad \text{Large} = 0.30 \times 360^\circ = 108^\circ. \]

Check the angles: $72^\circ + 180^\circ + 108^\circ = 360^\circ$. The Medium slice is a half circle, matching the prediction.

Read it back. Medium is the most common size (half of all drinks). Small is the least common (one fifth). The bar graph makes that ranking instant; the pie chart makes the “half of all sales” share instant.

Example 2: When a Pie Chart Is the Wrong Picture

A student surveys 30 classmates and asks each to name every streaming service they use. The counts come back like this.

Predict first. Can these numbers become a pie chart? Predict yes or no before reading on.

The sum of the counts is 55, but only 30 students were surveyed. Each student could name more than one service, so the counts overlap. The categories do not partition one whole: a single student is counted in two or three slices at once.

A pie chart is not valid here, because the slices would have to sum to more than 100 percent. The correct display is a bar graph, where one bar shows how many of the 30 students named each service. The bars can each rise as high as 30, and they do not need to add to 30, because the categories are not mutually exclusive.

This is the standard test before drawing a pie chart: ask whether each individual lands in exactly one category. If an individual can be counted twice, the pie chart fails and the bar graph is the honest picture.

Example 3: Reading a Distribution Off a Finished Bar Graph

A finished bar graph (percent on the vertical axis) reports the favorite season of a class. The four bars reach these heights.

Read the distribution. Summer is the most preferred season at 40 percent; Winter is the least at 10 percent.

Check the whole. $30\% + 40\% + 20\% + 10\% = 100\%$. The percents account for every student, so the same data could also be shown as a pie chart.

A comparison question. “Summer is preferred by 40 percent and Spring by 30 percent, so Summer is 10 percent more popular.” The phrase “10 percent more popular” is loose. The honest statement is that Summer’s share is 10 percentage points larger than Spring’s. The relative comparison ($\frac{40}{30} \approx 1.33$) would say Summer was chosen about 33 percent more often than Spring. Naming which comparison you mean keeps the reading correct.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can identify when a variable is categorical and therefore suited to a bar graph or pie chart.
• I can convert a table of counts into percents and confirm the percents sum to 100.
• I can explain why the bars in a bar graph are drawn with gaps between them.

Competent (Level 3-4):

• I can build a pie chart by converting each percent to a central angle out of 360 degrees.
• I can decide whether a table is valid for a pie chart by checking that the categories partition one whole.
• I can read a distribution off a finished bar graph or pie chart and name the largest and smallest groups.

Proficient (Level 5):

• I can translate freely among a count table, a bar graph, and a pie chart of the same data.
• I can distinguish a percentage-point gap (additive) from a relative change (multiplicative) and state which a comparison means.
• I can justify, with a specific example, why a given table fails the pie chart requirement.

Connections

Looking back:

• Categorical vs. Quantitative Variables establishes the variable type these displays require.
• Distribution of a Variable defines the distribution that the bar graph and pie chart picture.
• Proportions and Percentages supplies the count-to-percent arithmetic.

Looking ahead:

• Histograms carry the bar idea to a quantitative variable, where the bars touch because the axis is a number line.
• Describing a Distribution: Shape, Center, Spread moves from drawing a distribution to describing it in words.

Real-world connections:

• Election results are reported as a bar graph of seats by party or a pie chart of vote share.
• A budget is often shown as a pie chart, because each dollar belongs to exactly one category and the categories partition the whole.
• Survey questions of the “choose all that apply” kind produce overlapping categories, which is why their results appear as bar graphs rather than pie charts.

Resources

Last updated: 2026-06-16