One Categorical Variable: Tables and Summaries

A categorical data set can be hard to read one response at a time. A table brings the observations together without changing what was measured. Its rows should account for every observation, and its labels should make clear what the counts and proportions describe.

Build Frequency Tables

Suppose twelve students report their usual way to school:

Bus, Bike, Walk, Bus, Car, Bus, Bike, Walk, Bus, Car, Bike, Bus.

A frequency table records the number of observational units in each category. Here, count students, not letters in the category names. Tallying every response gives five bus, three bike, two walk, and two car responses. The frequencies sum to twelve, the number of recorded students. If a category has no observations, a zero-frequency row can still make the set of possible categories clear.

A relative frequency table divides each count by the total number of observations. It gives the proportion of units in the category. For bus, $5/12\approx0.4167$. The denominator is all twelve recorded students, not only students who took motorized transport.

Usual mode	Frequency, students	Relative frequency	Percentage
Bus	5	5/12	About 41.7%
Bike	3	3/12	25%
Walk	2	2/12	About 16.7%
Car	2	2/12	About 16.7%
Total	12	1	100% before rounding

Worked Example: Check Before You Report

The displayed percentages sum to 100.1% because three entries were rounded to one decimal place. The exact fractions sum to 1. This is a rounding effect, not a thirteenth student. Keep exact counts or fractions while calculating, then use an appropriate reported precision. A total of eleven counts, in contrast, would indicate that an observation was omitted or classified incorrectly.

Describe the Category Information

Percentages, relative frequencies, and ratios can communicate the same category information. Three bike users out of twelve students can be written as $3/12=0.25=25\%$, or as the count-to-total ratio 3:12, simplified to 1:4. State what the ratio compares. A bike-to-bus ratio of 3:5 answers a different comparison unless the rest of the table is also known.

The table supports claims about this sample. “Bus is the most common reported mode among these twelve students” follows from 5 being the largest frequency. “Half the students bike or walk” does not follow: there are 3+2=5 such students, or about 41.7%. A good description combines the category names, a numerical comparison, and the group represented by the data.

Worked Example: Counts and Proportions Together

Suppose another sample has 40 bus users out of 100 students. That sample contains more bus users than this twelve-student sample, but its bus proportion is 40%, slightly below 5/12. Counts describe how many observed units belong to a category. Proportions describe the share of a specified total. They are both useful, but a comparison of shares should not rest on counts alone when sample sizes differ.

To build a table, define non-overlapping category rules, classify each unit once, count every unit, check the total, and label proportions with their denominator. To describe it, choose a claim and point to the counts or relative frequencies that support it. The table makes the evidence visible; the contextual sentence explains why those numbers matter.

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.

Read the frequency

A sample of 20 customers chooses one drink: tea 8, coffee 7, water 5.

Part a

What does the tea frequency count? State its relative frequency and the denominator.

Your reasoning

Worked answer and rubric

Three ways to say the same share

Five of 25 sampled customers choose water.

Part a

Express this share as a relative frequency, percentage, and count-to-total ratio. Explain the common meaning.

Your reasoning

Worked answer and rubric

A claim needs the right total

Meal choice among 20 sampled diners.

Meal	Count
Wrap	9
Soup	6
Salad	5

Part a

Describe the most common choice with a numerical summary. Evaluate the claim “most diners chose wraps.”

Your reasoning

Worked answer and rubric

Compare shares

Sample A has 12 cycle commuters out of 30 people; sample B has 20 out of 80.

Part a

Describe the cycle shares in both samples and explain why the larger count does not show the larger share.

Your reasoning

Worked answer and rubric

Organize a new mode survey

Twenty park visitors report their usual trip mode, once each: Bus, Bike, Car, Bus, Walk, Bus, Bike, Car, Bus, Bike, Walk, Bus, Car, Bike, Bus, Walk, Bus, Bike, Car, Bus.

Part a

Construct a frequency table with all four categories and a total row. Explain what each frequency counts.

Your reasoning

Part b

Construct the corresponding relative frequency table. Check its total and explain the denominator.

Your reasoning

Part c

Describe the bike share as a fraction/proportion, percentage, and count-to-total ratio.

Your reasoning

Part d

Describe two useful features of the distribution with numerical summaries. Assess “a majority arrived by bus” and “equal numbers biked and walked.”

Your reasoning

Worked answer and rubric