AP Statistics 1.4: Graphical Representations for One Categorical Variable

01 · Start here

The concept in 60 seconds

A categorical graph shows how observations are distributed across category labels. In a bar chart, a bar’s height or length represents a category’s count or share. In a pie chart, a slice represents that category’s share of the whole.

Read the scale before the shape. A bar labeled $20$ might mean $20$ students, $20\%$ of students, or a proportion of $0.20$ . The axis label tells you which.

For example, $24$ out of $60$ sampled students travelled by car. A count bar chart shows $24$ students. A percentage bar chart shows $40\%$ . A pie chart gives Car $40\%$ of the circle. These displays tell the same story about the same sample.

When comparing groups of different sizes, compare their relative frequencies if your question is about shares. More observations in a category do not automatically mean a larger proportion.

02 · One situation, deeper understanding

One travel-method table, three ways to display it

Continue the fictional Cedar High example. The school has $1{,}200$ enrolled students. For the Tuesday being studied, a sample of $60$ students records each student’s main home-to-school travel method. Each student has one recorded method.

One question, four category labels: Bus, Car, Walk and Bike. The illustration introduces the categories; the number of people drawn does not represent the sample frequencies.

Main travel method for $60$ sampled Cedar High students. Invented teaching data; the same totals as Topic 1.3.
Travel method	Frequency: students	Relative frequency	Percentage
Bus	$20$	$\frac{20}{60} = \frac{1}{3}$	$\approx 33.3\%$
Car	$24$	$\frac{24}{60} = 0.40$	$40.0\%$
Walk	$12$	$\frac{12}{60} = 0.20$	$20.0\%$
Bike	$4$	$\frac{4}{60} = \frac{1}{15}$	$\approx 6.7\%$
Total	$60$	$1$	$100\%$

Cedar High: main travel method — counts

Frequency bar chart · $60$ sampled students

Bus

20

Car

24

Walk

12

Bike

4

Number of sampled students · category axis: Main travel method

Read each category against the zero-based scale. Bus

20

; Car

24

; Walk

12

; Bike

4

. Sample size:

60

. Invented teaching data.

The bar lengths show the counts directly. Car is most common; Bike is least common. The gaps separate category labels. Bus, Car, Walk and Bike are not intervals on a numerical scale.

Cedar High: main travel method — percentages

Relative frequency bar chart · the same $60$ students

Bus

\approx 33.3\%

Car

40\%

Walk

20\%

Bike

\approx 6.7\%

Percentage of sampled students · category axis: Main travel method

Read each category against the zero-based scale. Bus

\approx 33.3\%

; Car

40\%

; Walk

20\%

; Bike

\approx 6.7\%

. Sample size:

60

. Invented teaching data.

Changing from counts to percentages changes the numbers on the scale, but not the ranking of categories within this sample. Divide every count by the same total, $60$ , and multiply by $100\%$ .

Cedar High: main travel method — parts of a whole

Pie chart · all $60$ sampled students form one whole

Bus: $\approx 33.3\%$ ( $20$ students)
Car: $40\%$ ( $24$ students)
Walk: $20\%$ ( $12$ students)
Bike: $\approx 6.7\%$ ( $4$ students)

The slices use exact proportions; displayed percentages are rounded where needed. The whole circle represents

100\%

of the sample. Invented teaching data.

Predict before reading on: Car has the longest bar and the largest slice. Does that mean a majority of these students travelled by car?

Check your prediction

No. Car accounts for $\frac{24}{60} = 40\%$ , which is less than half. A majority requires more than $50\%$ . “Largest category” and “majority” are different claims.

03 · Choose a useful display

Bar chart or pie chart: what does each show?

Frequency bar chart

Each bar represents a category. Its height or length shows the number of observations in that category.

Useful for: “How many sampled students travelled by car?” Answer: $24$ .

Relative frequency bar chart

Each bar shows a category’s proportion or percentage of the stated group.

Useful for: “What share of the sample travelled by car?” Answer: $0.40$ , or $40\%$ .

Pie chart

The whole circle represents $100\%$ of the group. Each slice’s area is proportional to its category’s relative frequency.

Useful for seeing how a small number of categories divide one whole.

Categorical distribution

The category labels and their counts or relative frequencies describe how the observations are distributed.

Useful descriptions name common categories, uncommon categories and differences in shares.

Which display should you choose?

A table makes exact values easy to look up. A bar chart makes categories easy to compare because the bars start from a common baseline. A pie chart emphasizes parts of a whole, but close-sized slices can be harder to compare accurately.

Bar charts work vertically or horizontally. Horizontal bars are helpful when category names are long. For several groups, bar charts on a common percentage scale usually make small differences easier to judge than separate pie charts.

Pie-chart check: The categories must form nonoverlapping parts of the whole being displayed. A “select all that apply” survey can put one student in several categories, so its percentages of students may exceed $100\%$ . Use a suitably labeled bar chart rather than forcing those overlapping responses into a pie.

Category order has meaning only when the categories do

You can arrange unordered categories alphabetically, in a practical order, or by frequency. Reordering Bus and Car does not change the data. For ordered categories such as Low, Medium and High, keep the meaningful order. When comparing groups, use the same category order in each display.

Do not describe unordered travel-method categories as “left-skewed” or “right-skewed.” Moving the labels would change the apparent pattern. Means, medians and numerical spread are not appropriate summaries of these category labels.

04 · Build, label, check

How to construct a categorical graph

Build a bar chart in six steps

1. Start with a checked tableIdentify the variable, group and total. Verify the category counts.

2. Choose counts or sharesUse counts for “how many”; proportions or percentages for “what share.”

3. Label the graphGive it a useful title. Name the categories and label the numerical axis with its units.

4. Choose an honest scaleStart the bar-length scale at zero and use equal numerical intervals.

5. Draw separate barsUse equal bar widths and consistent gaps. Each bar ends at its category’s value.

6. Check and interpretMatch every bar to the table and write one supported statement in context.

For Cedar High, a count scale from $0$ to $30$ students accommodates the largest count, $24$ . The percentage chart uses $0\%$ to $50\%$ ; the largest category is $40\%$ . A percentage bar chart’s axis does not have to end at $100\%$ , as long as it starts at zero, is clearly labeled and includes all the bars.

Build a pie chart from proportions

First identify the whole and calculate each category’s share. Then give each category the same fraction of the circle. If drawing slices with a protractor, use:

\text{Slice angle}=\text{category proportion}\times360^\circ

For Car: $(\frac{24}{60}) \times 360^\circ = 144^\circ$ . Use the unrounded proportion to calculate the angle.

Exact slice angles for the Cedar High pie chart.
Category	Calculation	Slice angle
Bus	$(\frac{20}{60}) \times 360^\circ$	$120^\circ$
Car	$(\frac{24}{60}) \times 360^\circ$	$144^\circ$
Walk	$(\frac{12}{60}) \times 360^\circ$	$72^\circ$
Bike	$(\frac{4}{60}) \times 360^\circ$	$24^\circ$
Total	$120^\circ + 144^\circ + 72^\circ + 24^\circ$	$360^\circ$

Label the slices directly or provide a clear legend and percentages. State the sample size. A $40\%$ slice represents $24$ students when $n = 60$ , but $40$ students when $n = 100$ . The percentage alone does not tell you the count.

Final checks: Each count bar matches its count; each percentage bar matches its share; and the pie contains one complete circle. Exact proportions total $1$ and exact percentages total $100\%$ . Rounded labels can differ slightly from $100\%$ .

05 · Make a fair comparison

Compare groups using the right denominator

Suppose a separate fictional sample of $100$ Riverside High students records the same main travel method for the Tuesday being studied. Its counts are Bus $30$ , Car $40$ , Walk $20$ and Bike $10$ . We now have two distributions of the same categorical variable, one for each sample.

Invented sample comparison. Percentages use each school’s own sample total.
Method	Cedar High: $n = 60$	Riverside High: $n = 100$
Bus	$20$ students; $\approx 33.3\%$	$30$ students; $30\%$
Car	$24$ students; $40\%$	$40$ students; $40\%$
Walk	$12$ students; $20\%$	$20$ students; $20\%$
Bike	$4$ students; $\approx 6.7\%$	$10$ students; $10\%$

Compare travel-method shares, not just counts

Cedar High: $n = 60$ · Riverside High: $n = 100$ · same percentage scale

Bus

Cedar

\approx 33.3\%

Riverside

30\%

Car

Cedar

40\%

Riverside

40\%

Walk

Cedar

20\%

Riverside

20\%

Bike

Cedar

\approx 6.7\%

Riverside

10\%

Percentage within each school’s sample · category axis: Main travel method

Each category has two bars on the same zero-based scale. Each percentage uses its school’s own sample total. The data and exact values appear in the table above. Invented teaching data.

Counts answer one question; percentages answer another. Riverside has more sampled car travellers: $40$ versus $24$ . However, Car represents the same share of each sample: $\frac{40}{100} = \frac{24}{60} = 40\%$ . Both statements are true.

In both samples, Car is the most common method and Walk accounts for $20\%$ . Cedar’s sample has a higher Bus percentage, approximately $33.3\%$ versus $30\%$ . Riverside’s sample has a higher Bike percentage, $10\%$ versus approximately $6.7\%$ .

Comparison checklist: Same variable and category definitions; each group’s own denominator; same category order; same numerical scale; and a sentence that names both groups.

Percentage points: describe the difference clearly

The Bike shares differ by approximately $10\%-6.7\%=3.3\,\text{percentage points}$ . A percentage-point difference subtracts two percentages. Saying “ $3.3\%$ higher” can be ambiguous because a relative percentage increase uses a different calculation.

These comparisons describe the recorded samples. A difference in sample bars alone does not establish a difference between all students at the two schools, and it does not explain why the samples differ. Sampling and inference determine what broader claims are justified.

06 · See the reasoning

Worked examples

Example 1: Construct two bar charts from one table

Question: Construct a count bar chart and a percentage bar chart for the Cedar High sample. Explain what changes and what stays the same.

Identify the data. The variable is main travel method for $60$ sampled students. The counts are Bus $20$ , Car $24$ , Walk $12$ and Bike $4$ .
Draw the count chart. Label the category axis “Main travel method” and the numerical axis “Number of sampled students.” Start at zero; draw separate equal-width bars ending at $20$ , $24$ , $12$ and $4$ .
Convert to percentages. Divide by $60$ , then multiply by $100\%$ . The values are approximately $33.3\%$ , $40\%$ , $20\%$ and $6.7\%$ .
Draw the percentage chart. Keep the categories; label the numerical axis “Percentage of sampled students.” Use a zero baseline and consistent intervals.
Explain the connection. The axis values change, but Car remains largest and Bike smallest. Each count has been multiplied by the same factor, $\frac{100}{60}$ .

Check your work: Compare your drawings with the two bar charts above. A $24$ -student Car bar and a $40\%$ Car bar represent the same observations.

Example 2: Use a pie chart to evaluate a claim

Question: A student says, “Car is a majority because its slice is largest.” Evaluate the claim, then find the Walk slice angle.

Read the share. Car is $\frac{24}{60} = 40\%$ of the sample.
Apply the meaning of majority. A majority is more than half. Since $40\% < 50\%$ , the claim is not supported.
Write a correction. “Car was the most common method among the sampled Cedar High students, but it was not used by a majority.”
Find the Walk angle. Walk’s proportion is $\frac{12}{60} = 0.20$ . Its angle is $0.20 \times 360^\circ = 72^\circ$ .

Reflect: $\frac{1}{5}$ of the observations corresponds to $\frac{1}{5}$ of the circle. The largest slice can still be smaller than half a circle.

Example 3: Compare samples of different sizes

Question: Riverside has $30$ sampled bus travellers and Cedar has $20$ . Does Bus represent a higher proportion at Riverside? Justify your answer.

Identify the totals. Riverside’s sample has $100$ students; Cedar’s has $60$ .
Calculate each sample share. Riverside: $\frac{30}{100} = 30\%$ . Cedar: $\frac{20}{60}$ $\approx 33.3\%$ .
Compare percentages. $33.3\%$ is greater than $30\%$ , so Cedar has the higher Bus proportion in these samples.
Support the conclusion. “Although Riverside’s sample includes more bus travellers, Bus makes up a higher share of Cedar’s sample: approximately $33.3\%$ versus $30\%$ , a difference of about $3.3$ percentage points.”

Reflect: A count comparison alone answers “how many,” not “what proportion.”

07 · Say what the graph supports

How to justify a claim in context

A strong answer includes a claim, numerical evidence and the group being described. If comparing samples, name both groups and compare the same category using the same type of value.

Weak answer	Improved answer
“Car is biggest.”	“Car was the most common main travel method among the $60$ sampled Cedar High students, with $24$ students, or $40\%$ .”
“Riverside uses cars more.”	“Riverside’s sample included more car travellers, $40$ versus $24$ , but Car represented $40\%$ of each sample.”
“Cedar is $3.3$ higher.”	“The Bus percentage was about $3.3$ percentage points higher in Cedar’s sample: approximately $33.3\%$ versus $30\%$ .”
“Most Cedar students drive.”	“Car was the largest category in Cedar’s sample, at $40\%$ . The graph records travel by car, not whether the student was driving, and does not by itself establish a claim about all enrolled students.”

Writing frame: “Among [recorded group], [category] accounted for [count or percentage], so [supported claim].”

Comparison frame: “For [same category], [group A] had [percentage] and [group B] had [percentage]. Therefore, [specific comparison], by [difference] percentage points.”

08 · Find and fix the mistake

Common graph mistakes and how to correct them

Ignoring the axis label

Mistake: Reading a $20\%$ bar as $20$ students.

Fix: Check whether the graph shows counts, proportions or percentages. Use the sample size if converting between them.

Cutting off the zero baseline

Mistake: Bars for counts $24$ and $20$ begin at $18$ , making one bar appear $3$ times as long.

Fix: Start the bar scale at zero. The actual count ratio is $\frac{24}{20} = 1.2$ , not $3$ .

Comparing counts as if they were shares

Mistake: Saying $\frac{30}{100}$ is greater than $\frac{20}{60}$ because $30 > 20$ .

Fix: Compare $30\%$ with approximately $33.3\%$ . Each denominator matters.

Forcing overlapping responses into a pie

Mistake: Making pie slices from a multiple-choice survey whose percentages of students total $150\%$ .

Fix: A pie must partition one whole. Use a bar chart with clearly stated counting rules for overlapping categories.

Making pictures distort the values

Mistake: Enlarging a bus icon in both height and width to represent twice the count, or using a tilted 3D pie.

Fix: Use a flat, labeled graph. Doubling both dimensions quadruples an icon’s area; perspective can also make slices appear unequal.

Confusing bar charts with histograms

Mistake: Treating Bus, Car, Walk and Bike as numerical intervals.

Fix: Categorical bars represent separate labels. Histograms represent quantitative values grouped into intervals; you will study them in Topic 1.5.

Try a correction: “Riverside’s $40$ car travellers prove cars are more popular there than at Cedar, where only $24$ students travelled by car.”

Show the corrected reasoning

The car counts differ, but the percentages are equal: $\frac{40}{100} = \frac{24}{60} = 40\%$ . The data show equal Car shares in the two samples. The counts alone do not establish that Car has a higher population proportion at Riverside.

09 · Apply it yourself

Check your understanding: practice with solutions

These are original AP-style practice questions. Sketch graphs on paper where requested, including a title, category labels and a numerical scale. Try each question before opening its answer.

1. Construct a bar chart

$40$ students each select one after-school activity: Art $18$ , Sport $14$ or Music $8$ . Construct a frequency bar chart. Then give Art’s percentage and explain whether it is a majority.

Hint

The numerical axis should show students and start at zero. For Art’s percentage, divide $18$ by $40$ and multiply by $100\%$ .

Solution and reasoning

A correct chart has separate, equal-width bars ending at Art $18$ , Sport $14$ and Music $8$ , with a consistent zero-based count scale.

After-school activity among $40$ sampled students

Frequency bar chart · one activity per student

Art

18

Sport

14

Music

8

Number of sampled students · category axis: Activity

Read each category against the zero-based scale. Art

18

; Sport

14

; Music

8

. Sample size:

40

. Invented teaching data.

Art’s percentage is $(\frac{18}{40}) \times 100\% = 45\%$ . Art is the largest category, but it is not a majority because $45\%$ is below $50\%$ .

2. Build pie-chart slices

Using the activity counts from Question 1, calculate the percentage and slice angle for Art, Sport and Music. Check that the angles make a whole circle.

Hint

For each category, calculate $\frac{\text{count}}{40}$ . Multiply by $100\%$ for its percentage or by $360^\circ$ for its slice angle.

Solution and reasoning

Art: $\frac{18}{40} = 45\%$ , angle $162^\circ$ . Sport: $\frac{14}{40} = 35\%$ , angle $126^\circ$ . Music: $\frac{8}{40} = 20\%$ , angle $72^\circ$ . Check: $45\% + 35\% + 20\% = 100\%$ , and $162^\circ + 126^\circ + 72^\circ = 360^\circ$ .

3. Read a relative frequency scale

A relative frequency bar chart shows Tea at $0.35$ , Coffee at $0.50$ and Water at $0.15$ for $80$ recorded customers. How many selected Tea? Explain what the Tea bar represents.

Hint

A proportion of $0.35$ is $35\%$ , not $0.35$ people. Multiply the share by the total to find the count.

Solution and reasoning

$\text{Tea count}=0.35\times80=28$ customers. The Tea bar means that $35\%$ of these $80$ customers selected Tea. Its height represents a relative frequency, not a count of $35$ customers.

4. Spot a misleading baseline

A graph compares category counts $24$ and $20$ , but its bar-length axis starts at $18$ . One visible bar is $6$ units long and the other is $2$ units long. A reader concludes that the first count is $3$ times the second. Explain the problem and give the correct count ratio.

Hint

The visible lengths show each value minus $18$ , not the original values. Compare the counts themselves.

Solution and reasoning

The truncated baseline exaggerates the difference: $\frac{6}{2} = 3$ compares distances above $18$ . The actual count ratio is $\frac{24}{20} = 1.2$ . The first count is $1.2$ times the second. Redraw the bars from zero to represent the counts honestly.

5. Compare two samples fairly

In sample A, $18$ of $40$ students select Art. In sample B, $24$ of $80$ select Art. Which sample has more Art selections? Which has the higher Art proportion? Suggest a useful graph for comparing the shares.

Hint

Answer “how many” with counts and “what share” with proportions. Use each sample’s own total.

Solution and reasoning

Sample B has more Art selections: $24$ versus $18$ . Sample A has the higher Art share: $\frac{18}{40} = 45\%$ , compared with $\frac{24}{80} = 30\%$ in B, a difference of $15$ percentage points. A percentage bar chart with both samples on the same zero-based scale makes that comparison clear.

6. Choose a graph for overlapping categories

$100$ students select every club they attend. $60$ select Sport, $50$ Music and $40$ Art, with some students selecting several clubs. A student proposes pie slices of $60\%$ , $50\%$ and $40\%$ . Is that a valid pie chart of the students? What should be used instead?

Hint

The same student can contribute to more than one category. Check whether the proposed slices partition $100\%$ of the students.

Solution and reasoning

No. The percentages total $150\%$ because the student groups overlap, so they do not form disjoint parts of one circle representing the $100$ students. Use a bar chart of counts or percentages, clearly labeled as club attendance with multiple selections allowed. A different chart of the $150$ total selections would have a different denominator and answer a different question.

10 · Return here before a quiz

Quick revision notes

Count bar chart: $\text{bar height or length}=\text{category frequency}$ .
Relative frequency bar chart: $\text{bar height or length}=\text{category proportion or percentage}$ .
Read first: title, recorded group, sample size, categories and axis units.
Construct carefully: a zero baseline, equal numerical intervals, equal-width separate bars and clear labels.
Pie chart: each slice is a share of one whole; exact percentages total $100\%$ .
Pie angle: $\text{category proportion}\times360^\circ$ ; exact angles total $360^\circ$ .
Compare shares: divide by each group’s own total and use a common scale.
Difference in percentages: report percentage points when subtracting percentages.
Largest category: not necessarily a majority; majority means more than $50\%$ .
Justify a claim: name the category and group, give numerical evidence, and stay within what the data support.

Three questions to remember: What does the scale measure? What is the whole? Do my numbers support my claim?

11 · Put it together

Final understanding check

A school surveys $50$ students about their preferred class format. The single-choice responses are In-person $26$ , Online $15$ and Hybrid $9$ . Another school surveys $100$ students, with In-person $45$ , Online $35$ and Hybrid $20$ . All data are invented for practice.

Describe how to construct a percentage bar chart for the first sample, giving the three bar values.
Calculate the Hybrid slice angle for a pie chart of the first sample.
Is In-person a majority in the first sample? Justify with numerical evidence.
Compare the In-person shares in the two samples. Explain why comparing $26$ with $45$ alone would not answer this question.

Show a complete answer

1. Title the chart “Preferred class format among $50$ sampled students.” Label the categories In-person, Online and Hybrid and the numerical axis “Percentage of sampled students.” Use separate equal-width bars from zero ending at $52\%$ , $30\%$ and $18\%$ , with equal numerical intervals.

2. $\text{Hybrid angle}=\frac{9}{50}\times360^\circ=64.8^\circ$ .

3. Yes. In-person is $\frac{26}{50} = 52\%$ , which is more than $50\%$ , so it has a majority in the first sample.

4. The In-person shares are $52\%$ in the first sample and $45\%$ in the second. The first sample’s share is $7$ percentage points higher. The second sample has more In-person responses, $45$ versus $26$ , but also twice as many students. Counts alone do not compare shares.

Self-check: Did you label percentages, use each sample’s own denominator, and support every claim with a number? If yes, you have the main reasoning skills for this topic.

12 · Connect the ideas

Continue learning

Previous lesson: Topic 1.3: Tabular Representation and Summary Statistics for One Categorical Variable. Review counts and proportions when a graph’s scale is unclear.

Topic 1.5: Graphical Representations for One Quantitative Variable →

Next, move from category labels to numerical values and learn how quantitative graphs display their distributions.

Review graph construction · Review sample comparisons · Back to the lesson overview

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