One Categorical Variable: Bar and Pie Charts

A graph should let a reader see the category information without guessing the units or denominator. Bar and pie charts represent the same one-variable distribution in different ways. Choosing a graph does not change which observations belong to the sample.

Construct Bar and Pie Charts

A bar chart, also called a bar graph, uses one bar per category. The bar’s height or length represents its frequency, a count, or its relative frequency, a proportion. Label the categories and the numerical scale. Separate bars because the categories are distinct; their ordering does not represent a continuous measurement scale. Use a zero baseline so lengths faithfully show the values.

Consider 20 school-trip reports: bus 8, bike 5, car 4, and walk 3. A frequency bar chart has heights 8, 5, 4, and 3 students. A relative-frequency chart uses 0.40, 0.25, 0.20, and 0.15. The tallest category stays bus in either chart.

A pie chart uses one slice per category. A slice’s area as a fraction of the whole represents the relative frequency. All slices together must equal one complete circle, or 100% of its area. For a hand construction, multiply each category proportion by 360 degrees: bus 144°, bike 90°, car 72°, walk 54°. The angles sum to 360°. Use the same circle for every category and label its shares; decorative three-dimensional perspectives can make the displayed areas misleading.

Worked Example: Construction Checks

The bike bar has height 5 students, and the bike slice occupies 5/20=25% of the circle. Drawing five equally sized slices for five bike students while using one slice for each other category would not follow the category-area rule. Similarly, a bar labeled “5%” would confuse its frequency with its percentage. Check which quantity the graph’s scale actually represents.

Justify a Graphical Claim

Graphs reveal information that can support claims in context. The figures show bus as the most common single mode because its bar is tallest and its slice largest. They also show that bus is not a majority: its 40% slice is less than half the circle. To justify a claim, identify the relevant category values and explain the comparison. A graph’s attractive appearance is not evidence.

Compare Categorical Distributions

Use frequency or relative frequency tables, bar charts, and pie charts to compare two or more data sets for the same categorical variable. Keep category definitions consistent. If sample sizes differ, compare relative frequencies for shares; otherwise the bigger sample may have taller count bars merely because more people were observed.

Worked Example: A Second Neighborhood

A second sample of 40 students reports bus 12, bike 8, car 8, and walk 12. Its bus count is larger than 8, but its bus share is 12/40=30%, below the first sample’s 40%. Its walk share is 30%, compared with 15% in the first sample. Side-by-side relative-frequency bars or two labeled pies display those differences. A useful comparison is “Walking makes up twice the share in the second sampled neighborhood.” It concerns these samples and this variable; it does not establish why neighborhoods differ.

Independent Practice

Use the information in each question to make your own calculation, representation, or argument. Explain what your result means in its setting.

Original generated practice for the Fall 2026 course.

What is the bar measuring?

Six of 24 sampled customers choose tea.

Part a

Construct the tea bar in both a frequency and relative-frequency chart: label its category, numerical axis, baseline, and endpoint.

Your reasoning

Worked answer and rubric

A quarter of the whole

A survey of 40 households records 10 whose main heating type is electric.

Part a

Construct the electric slice of a pie chart, stating its fraction of area and angle. State the total-area condition for the full pie.

Your reasoning

Worked answer and rubric

Largest is not half

Part a

Justify or reject “more than half of the sampled students use a bus.” Cite graphical values.

Your reasoning

Worked answer and rubric

Same category, different sample sizes

Two samples measure the same usual-trip-mode variable.

Sample	Bus	Bike	Car	Walk	Total
A	8	5	4	3	20
B	12	8	8	12	40

Part a

Compare the bus and walk shares in the two distributions. Explain a graph choice that makes the comparison fair.

Your reasoning

Worked answer and rubric

Construct and compare library-use graphs

Independent visitor samples, one main purpose reported per visitor.

Main purpose	North branch	South branch
Borrow books	18	12
Study	12	24
Use computers	6	18
Attend event	4	6
Total	40	60

Part a

Construct North branch frequency and relative-frequency bar charts on paper. Label every category, scale, and axis; provide all bar endpoints with your response.

Your reasoning

Part b

Construct a North branch pie chart. Give each slice percentage and central angle and check the whole.

Your reasoning

Part c

Use your graphical representations to justify whether borrowing is a majority and whether computers plus events account for one-quarter of North visitors.

Your reasoning

Part d

Construct a South relative-frequency representation and compare the two distributions. Include all category shares and two contextual differences.

Your reasoning

Worked answer and rubric