AP Statistics 1.3: Categorical Frequency Tables and Summary Statistics

01 · Start here

The concept in 60 seconds

A frequency table tells us how many observations fall in each category. A relative frequency table tells us what fraction of the recorded group falls in each category.

\text{Relative frequency}=\frac{\text{category count}}{\text{total count}}

\text{Percentage}=\text{relative frequency}\times100\%

For example, $20$ bus travellers out of $60$ students gives $\frac{20}{60} = \frac{1}{3}$ , or approximately $33.3\%$ .

A table is useful when you can turn it into a clear statement: “About $\frac{1}{3}$ of these $60$ sampled students travelled by bus.” Always name the category and the group behind the denominator.

One caution: the largest category is not automatically a majority. A majority means more than half of the group.

02 · One situation, deeper understanding

What is the main travel method at Cedar High?

Continue the fictional school scenario from Topics 1.1 and 1.2. Cedar High has $1{,}200$ enrolled students, and a sample of $60$ students records each student’s main home-to-school travel method on the Tuesday being studied.

Each sampled student has one recorded method. The completed sample contains $20$ Bus, $24$ Car, $12$ Walk and $4$ Bike responses. These invented totals describe all $60$ sample records; the $4$ preview rows shown in Topic 1.2 were only a small part of that sample.

Main travel method for the $60$ sampled Cedar High students. Invented teaching data.
Main travel method	Frequency: students	Relative frequency	Percentage
Bus	$20$	$\frac{20}{60} = \frac{1}{3}$ $\approx 0.3333$	$\approx 33.3\%$
Car	$24$	$\frac{24}{60} = 0.40$	$40.0\%$
Walk	$12$	$\frac{12}{60} = 0.20$	$20.0\%$
Bike	$4$	$\frac{4}{60} = \frac{1}{15}$ $\approx 0.0667$	$\approx 6.7\%$
Total	$60$	$1$ exactly, before rounding	$100\%$

The observational units are students. The categorical variable is main travel method. Counts and percentages summarize the category responses; they are not measurements of individual students’ travel durations.

Predict before reading on: Which travel method is most common? Is it used by a majority of these sampled students?

Check your prediction

Car is most common, with $24$ of $60$ students, or $40\%$ . It is not a majority because $40\%$ is less than $50\%$ . No single travel-method category has a majority in this sample.

03 · Understand the language

Key ideas and notation

Frequency

The number of observations in a category. It is also called a count.

Bus has a frequency of $20$ students.

Relative frequency

The category count divided by the total count. It describes a share of the group.

Bus has relative frequency $\frac{20}{60} = \frac{1}{3}$ .

Distribution

How observations are spread across the categories.

The travel-method distribution is described by the $4$ category counts or proportions.

Mode: most common category

The mode is the category with the highest frequency. Here the mode is Car; $24$ is its count. If categories tie for the highest count, report the tie rather than selecting one arbitrarily.

You may see $f$ used for a category’s frequency and $n$ for the sample total, giving relative frequency $\frac{f}{n}$ . For a particular sample category, its proportion can also be written $\hat{p}$ (“p-hat”), connecting this lesson to Topic 1.2.

Two columns do not automatically mean two variables. A table listing travel-method categories and their frequencies summarizes one categorical variable. The frequency column is the number of students with each recorded method.

04 · Build, calculate, check

How to build a frequency and relative frequency table

1. Identify the variableChoose the single categorical characteristic you are summarizing.

2. List the categoriesMake the category definitions clear so each record can be counted appropriately.

3. Count the observationsTally each record once for a single-choice categorical response.

4. Check the totalAdd the category counts and verify the number of recorded observations.

5. Divide each count by the totalUse the same denominator for every category in this distribution.

6. Label and interpretName the variable, group, counts and proportions. Add a contextual sentence.

Apply the steps to Cedar High

The categories have counts $20$ , $24$ , $12$ and $4$ . Their sum is $20 + 24 + 12 + 4 = 60$ . The denominator for each sample relative frequency is therefore $60$ , not the school’s population size of $1{,}200$ and not the number of categories, $4$ .

For Bus, divide $20$ by $60$ . For Car, divide $24$ by $60$ . Repeat for Walk and Bike. A combined table can show counts, fractions or decimals, and percentages together, as in the table above.

One category, three equivalent descriptions: count, proportion and percentage. The denominator stays tied to the whole recorded group. Icons are symbolic. View this visual at full size.

Check the whole table

Counts: nonnegative whole numbers, adding to the total number of observations.
Relative frequencies: each between $0$ and $1$ , adding to $1$ before rounding.
Percentages: each between $0\%$ and $100\%$ , adding to $100\%$ before rounding.

Rounding check: $3$ categories with $1$ observation each have exact relative frequencies $\frac{1}{3}$ , $\frac{1}{3}$ and $\frac{1}{3}$ . To one decimal place, their percentages are $33.3\%$ , $33.3\%$ and $33.3\%$ , totaling $99.9\%$ . The small difference comes from rounding. Keep exact fractions or extra decimals during calculations and round the final answers.

Make the denominator match the records

For this single-choice travel question, each student has one main method, so categories do not overlap and every recorded response belongs somewhere. If there are missing responses, state whether your denominator includes all selected students or only students with valid recorded responses.

A question such as “Which travel methods did you use this week? Select all that apply” has a different counting rule: one student may appear in several categories. In that situation, percentages of students can exceed $100\%$ in total. Do not use the single-choice total check without reading the question.

05 · Use numbers to answer questions

Proportions, percentages, ratios and supported claims

$\frac{20}{60}$ , $\frac{1}{3}$ , approximately $0.3333$ and approximately $33.3\%$ describe the same share of Cedar High’s sample. A part-to-whole ratio of $20:60 = 1:3$ carries the same information when clearly labeled “Bus students : All sampled students.”

Part to whole versus part to part

The ratio of Bus students to Car students is $20:24 = 5:6$ . This compares two categories. It is not the proportion of all sampled students who travelled by bus, because Walk and Bike students are not included in that comparison.

Read the labels before using a ratio:

\frac{20}{60}

gives the bus share of the full sample;

20:24

compares Bus with Car. Icons are symbolic. View this visual at full size.

Support a claim with the right comparison

“Car is most common.”

Supported for this sample: Car has the largest count, $24$ , and the largest relative frequency, $40\%$ .

“Most students travelled by car.”

If “most” means a majority, this is not supported: $\frac{24}{60} = 40\%$ , which is less than $50\%$ . Write “most common method” to avoid ambiguity.

“A majority did not travel by car.”

Supported for this sample: Bus, Walk and Bike total $20 + 12 + 4 = 36$ students; $\frac{36}{60} = 60\%$ .

“Car exceeds Walk by $20\,\text{percentage points}$ .”

Supported: $40\%-20\%=20\,\text{percentage points}$ . In counts, the difference is $24 – 12 = 12$ students.

A table of categorical labels does not call for the mean or median of those labels. Averaging numerical codes such as $\text{Bus}\mapsto\text{1}\text{ and }\text{Car}\mapsto\text{2}$ would depend on arbitrary coding. Use counts, proportions and clearly defined comparisons to describe this categorical distribution.

Keep the scope of your conclusion: These results describe the $60$ sampled students. The bus proportion $\frac{20}{60}$ is a statistic. It does not prove that exactly $\frac{1}{3}$ of all $1{,}200$ Cedar High students travelled by bus. A broader conclusion depends on how the sample was obtained and would need additional statistical reasoning.

06 · See the reasoning

Worked examples

Example 1: Build a table from raw category labels

Question: $10$ library visitors each choose one preferred reading category. Their responses are Fiction, Comics, Nonfiction, Fiction, Fiction, Comics, Nonfiction, Fiction, Nonfiction, Fiction. Construct a frequency and relative frequency table.

Name the units and variable. The units are these library visitors. The variable is preferred reading category.
Tally the responses. Fiction appears $5$ times, Nonfiction $3$ times and Comics $2$ times. Check: $5 + 3 + 2 = 10$ .
Divide each count by $10$ . The relative frequencies are $0.50$ , $0.30$ and $0.20$ .

Preferred reading category of $10$ library visitors. Invented teaching data.
Category	Frequency	Relative frequency	Percentage
Fiction	$5$	$\frac{5}{10} = 0.50$	$50\%$
Nonfiction	$3$	$\frac{3}{10} = 0.30$	$30\%$
Comics	$2$	$\frac{2}{10} = 0.20$	$20\%$
Total	$10$	$1.00$	$100\%$

Interpret: Fiction is the most common preference, chosen by $5$ of these $10$ visitors. Exactly $50\%$ is half, not a majority.

Example 2: Calculate a combined proportion and a ratio

Question: In the Cedar High sample, what proportion walked or biked? What is the ratio of Walk students to Bike students?

Combine the relevant counts. Walk and Bike are separate main-method categories, so $12 + 4 = 16$ students walked or biked.
Use the full sample for the proportion. $\frac{16}{60} = \frac{4}{15}$ $\approx 0.2667$ , or approximately $26.7\%$ .
Use the two category counts for their ratio. $\text{Walk}:\text{Bike}=12:4=3:1$ .

Interpret: About $26.7\%$ of the sampled students walked or biked. There were $3$ Walk students for every $1$ Bike student. These two results answer different questions.

Example 3: Decide whether a claim is supported

Question: A student says, “Car has the highest count, so a majority of the $60$ students travelled by car.” Evaluate the claim.

Check the count. Car’s count of $24$ is larger than the other category counts.
Check the majority threshold. A majority of $60$ requires more than $30$ students. Car has only $24$ .
Calculate the share. $\frac{24}{60} = 0.40 = 40\%$ , below $50\%$ .
Write a corrected claim. Car was the most common main travel method among these sampled students, but it was not used by a majority.

Reflect: “Largest category” compares categories with one another. “Majority” compares a category with half of the whole group.

07 · Say what your answer means

How to explain a table in context

Write a sentence that identifies the group, category and numerical evidence. When evaluating a claim, also state the comparison or threshold you used.

Weak answer	Improved answer	Why it is clearer
“Bus is $0.3333$ .”	“ $20$ of the $60$ sampled Cedar High students travelled mainly by bus that Tuesday, a proportion of $\frac{1}{3}$ , or about $33.3\%$ .”	Names the recorded group, method and denominator.
“Most are Car.”	“Car was the most common method in this sample, with $24$ students, or $40\%$ ; it did not have a majority.”	Distinguishes largest category from more than half.
“The ratio is $3:1$ .”	“The ratio of Walk students to Bike students was $12:4 = 3:1$ in the sample.”	Identifies both parts in the correct order.

Description frame: “[Category] accounted for [count] of the [total] recorded [units], or [percentage], in [context].”

Claim frame: “The data [support / do not support] this claim for [group] because [calculation] is [above / below / equal to] [relevant threshold].”

You do not need to mention every category in every answer. Choose the numbers that answer the question, and describe a whole distribution with its important comparisons rather than an unexplained list.

08 · Find and fix the mistake

Common mistakes and how to correct them

Dividing by the number of categories

Mistake: “There are $4$ methods, so Bus is $\frac{20}{4}$ .”

Fix: Use the total number of observations, $60$ . $4$ is the number of labels, not the number of sampled students.

Using the population as the denominator

Mistake: “The sample bus percentage is $\frac{20}{1{,}200} \times 100\%$ .”

Fix: These $20$ Bus responses came from the sample of $60$ . Use $\frac{20}{60}$ for the sample proportion.

Turning a part-to-part ratio into a whole-group percentage

Mistake: “ $\text{Bus}:\text{Car}=5:6$ , so $\frac{5}{6}$ of the whole sample travelled by bus.”

Fix: The whole sample also includes Walk and Bike. Its bus share is $\frac{20}{60}$ , not $\frac{20}{24}$ .

Calling the largest category a majority

Mistake: “Car is largest, so more than half used Car.”

Fix: Compare with $50\%$ . Car has $40\%$ ; it is largest without having a majority.

Mixing a proportion with a percentage

Mistake: “ $0.40 = 0.40\%$ .”

Fix: $0.40 = 40\%$ . A decimal proportion is converted to a percent by multiplying by $100\%$ .

Ignoring the response rules

Mistake: “Every survey’s percentages must total exactly $100\%$ .”

Fix: Check for rounding, missing responses and multiple selections before deciding that a table is wrong.

Try a correction: “There are $4$ categories and $20$ Bus students. Therefore Bus has relative frequency $\frac{20}{4} = 5$ , or $5\%$ , and a majority travelled by bus.”

Show the corrected reasoning

There are $60$ students in the sample. The bus relative frequency is $\frac{20}{60} = \frac{1}{3}$ $\approx 0.3333$ , or approximately $33.3\%$ . A relative frequency of $5$ would be impossible for a part of this whole. Bus does not have a majority because its percentage is below $50\%$ .

09 · Apply it yourself

Check your understanding: practice with solutions

These are original AP-style practice questions. Try each question before opening the solution. Use the hint when you need a starting point.

1. Construct a table from raw responses

$12$ students each select one lunch: Sandwich, Salad, Pasta, Sandwich, Salad, Sandwich, Pasta, Salad, Sandwich, Pasta, Salad, Sandwich. Construct a frequency table and give each category’s percentage to one decimal place.

Hint

Tally each lunch separately. The three counts must add to $12$ . Divide each by $12$ before multiplying by $100\%$ .

Solution and reasoning

Sandwich appears $5$ times, Salad $4$ times and Pasta $3$ times. Their frequencies total $12$ .

Lunch	Frequency	Relative frequency	Percentage
Sandwich	$5$	$\frac{5}{12}$ $\approx 0.4167$	$\approx 41.7\%$
Salad	$4$	$\frac{4}{12} = \frac{1}{3}$	$\approx 33.3\%$
Pasta	$3$	$\frac{3}{12} = 0.25$	$25.0\%$

Sandwich is the most common lunch among these $12$ students, but $\frac{5}{12}$ is less than half, so it does not have a majority.

2. Choose the correct denominator

Use the Cedar High table. Which calculation gives the proportion of the sample whose main travel method was Bus?

$\frac{20}{4}$ , because there are $4$ travel categories.
$\frac{20}{1{,}200}$ , because $1{,}200$ students attend the school.
$\frac{20}{60}$ , because $60$ sampled students have recorded methods.
$\frac{20}{24}$ , because Car is the largest category.

Hint

The numerator and denominator must refer to the same recorded group: the $20$ Bus students are part of which group?

Solution and reasoning

C. The sample bus proportion is $\frac{20}{60} = \frac{1}{3}$ . The other denominators refer to the category count, the whole school, or just the Car category, and do not give the requested sample share.

3. Recover a missing frequency

A library classifies all $50$ visitors during an hour as Adult, Teen or Child, with one category per visitor. There are $22$ Adults, $18$ Teens and an unspecified number of Children. Find the Child frequency and its relative frequency. Check the three relative frequencies.

Hint

Subtract the two known category counts from $50$ . Every category’s denominator is still $50$ .

Solution and reasoning

$\text{Child frequency}=50-22-18=10$ visitors. The Child relative frequency is $\frac{10}{50} = 0.20$ , or $20\%$ . Adult is $\frac{22}{50} = 0.44$ and Teen is $\frac{18}{50} = 0.36$ . Check: $0.44 + 0.36 + 0.20 = 1$ .

4. Evaluate a majority claim

A student uses the Cedar High table to claim, “A majority of the sample travelled by car because Car is the most common category.” Is the claim supported? Write a correction with numerical evidence.

Hint

A majority requires more than half of the observations, regardless of the ranking of categories.

Solution and reasoning

The majority claim is not supported. Car accounts for $\frac{24}{60} = 40\%$ , which is below $50\%$ . A correct statement is: “Car was the most common method among the $60$ sampled students, but it was not used by a majority.”

5. Explain a ratio and a proportion

For Cedar High, calculate (a) the ratio Bus : Car and (b) the proportion of the whole sample travelling by Bus. Explain why the two answers have different denominators.

Hint

Part (a) names two categories. Part (b) names a category and the whole sample.

Solution and reasoning

(a) $\text{Bus}:\text{Car}=20:24=5:6$ . It compares Bus students with Car students. (b) The sample bus proportion is $\frac{20}{60} = \frac{1}{3}$ , approximately $33.3\%$ , because the whole sample includes all $4$ categories. Both are valid descriptions when their groups are clearly labeled; they answer different questions.

6. Handle missing responses honestly

A separate survey selects $30$ students. $27$ give a valid single travel-method response: $12$ Bus and $15$ Car. The other $3$ do not respond. Calculate the Bus relative frequency among valid responses, then state a limitation of the result.

Hint

The question specifies the group in the denominator. Do not silently treat missing responses as a travel category.

Solution and reasoning

Among the $27$ valid responses, Bus has relative frequency $\frac{12}{27} = \frac{4}{9}$ $\approx 0.4444$ , or about $44.4\%$ . This percentage describes the respondents. The travel methods of the $3$ nonrespondents are unknown, so $44.4\%$ should not be presented as the exact bus percentage for all $30$ selected students.

10 · Return here before a quiz

Quick revision notes

Frequency: the count of observations in a category.
Relative frequency: $\frac{f}{n}$ , where $f$ is the category count and $n$ is the total observations in the stated group.
Percentage: $\text{relative frequency}\times100\%$ .
Part-to-whole ratio: can express the same information as a proportion, when the groups are labeled.
Part-to-part ratio: compares two selected counts; it is not automatically a share of the whole.
Single-choice total check: counts add to the observation total; exact proportions add to $1$ and exact percentages to $100\%$ .
Rounding: displayed proportions or percentages may have a small total discrepancy.
Mode: the most common category; report any tie for the highest count.
Majority: strictly more than $50\%$ , not merely the largest category.
Context: identify the category, recorded group and evidence behind your claim.

Memory check: What is being counted? What is the whole group? Have I used its total? What do the numbers let me claim about that group?

11 · Can you do this independently?

A final AP-style understanding check

$40$ sampled residents each name the one material they recycled most often last week. The responses are Plastic: $18$ , Paper: $12$ , Glass: $6$ and Metal: $4$ . Each resident gives one response, and none are missing.

Identify the observational unit and categorical variable. Construct a table with counts and relative frequencies.
Identify the most common category. Does it have a majority?
Evaluate the claim that a majority of these sampled residents named Plastic or Paper.
Give the ratio Plastic : Paper and explain how it differs from the Plastic proportion of the whole sample.

Show a model answer and self-check

(a) An observational unit is a sampled resident. The variable is the material that resident recycled most often last week.

Material	Frequency	Relative frequency	Percentage
Plastic	$18$	$\frac{18}{40} = 0.45$	$45\%$
Paper	$12$	$\frac{12}{40} = 0.30$	$30\%$
Glass	$6$	$\frac{6}{40} = 0.15$	$15\%$
Metal	$4$	$\frac{4}{40} = 0.10$	$10\%$
Total	$40$	$1.00$	$100\%$

(b) Plastic is the mode, with $18$ of $40$ responses, or $45\%$ . It does not have a majority, because $45\%$ is less than $50\%$ .

(c) The claim is supported for these sampled residents: $18 + 12 = 30$ named Plastic or Paper; $\frac{30}{40} = 0.75 = 75\%$ , above $50\%$ .

(d) $\text{Plastic}:\text{Paper}=18:12=3:2$ . This compares those two categories. Plastic’s share of the full sample is $\frac{18}{40} = 45\%$ , using all $40$ residents in the denominator.

Self-check: Did your counts total $40$ , your exact proportions total $1$ , and your claims name the sampled residents? Did you distinguish the largest category from a majority and label both parts of the ratio?

Before you move on, check that you can:

Build a categorical table from raw labels or given category counts.
Explain why each relative frequency uses its stated denominator.
Distinguish a count, a proportion, a percentage and a ratio.
Support a contextual claim with a calculation and an appropriate comparison.

12 · Connect the ideas

Continue learning

A frequency table organizes the categories and their shares. Next, use graphs to make those comparisons easier to see.

Next lesson

Topic 1.4: Graphical Representations for One Categorical Variable →

Connect frequency and relative frequency tables with bar charts and pie charts.

Previous: Topic 1.2 · Review table construction · Back to the lesson overview

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The concept in 60 seconds

What is the main travel method at Cedar High?

Key ideas and notation

Category

Frequency

Relative frequency

Distribution

Mode: most common category

How to build a frequency and relative frequency table

Apply the steps to Cedar High

Check the whole table

Make the denominator match the records

Proportions, percentages, ratios and supported claims

Part to whole versus part to part

Support a claim with the right comparison

“Car is most common.”

“Most students travelled by car.”

“A majority did not travel by car.”

“Car exceeds Walk by 20 percentage points20\,\text{percentage points}.”

Worked examples

Example 1: Build a table from raw category labels

Example 2: Calculate a combined proportion and a ratio

Example 3: Decide whether a claim is supported

How to explain a table in context

Common mistakes and how to correct them

Dividing by the number of categories

Using the population as the denominator

Turning a part-to-part ratio into a whole-group percentage

Calling the largest category a majority

Mixing a proportion with a percentage

Ignoring the response rules

Check your understanding: practice with solutions

1. Construct a table from raw responses

2. Choose the correct denominator

3. Recover a missing frequency

4. Evaluate a majority claim

5. Explain a ratio and a proportion

6. Handle missing responses honestly

Quick revision notes

A final AP-style understanding check

Before you move on, check that you can:

Continue learning

“Car exceeds Walk by $20\,\text{percentage points}$ .”