Unit 2 · Topic 2.2 · about 20 minutes
Summary Statistics for Two Categorical Variables
Calculate joint, marginal and conditional relative frequencies from a two-way table, then use them to compare groups and justify a claim about association.
Predict first
At one high school, 56% of the students who were late to first period at least once last month live within a mile of the school. Does living close to school make a student more likely to be late?
| Late at least once | Never late | Total | |
|---|---|---|---|
| Within 1 mile | 80 | 420 | 500 |
| 1 to 3 miles | 36 | 164 | 200 |
| More than 3 miles | 28 | 72 | 100 |
| Total | 144 | 656 | 800 |
Three kinds of relative frequency
Every relative frequency from a two-way table is a count divided by a total. The only question is which total. Using the table above:
- A joint relative frequency is one cell divided by the total for the entire table. The share of all 800 students who live within a mile and were late is .
- A marginal relative frequency is a row total or a column total divided by the table total. The share of all students who were late is , and the share who live within a mile is . The name comes from the margins of the table, where those totals live.
- A conditional relative frequency restricts you to one category and divides by that category's total. Among students who live within a mile, the share who were late is . Among students who were late, the share who live within a mile is .
| Relative frequency | Numerator | Denominator | Example |
|---|---|---|---|
| Joint | One cell | Table total | |
| Marginal | A row or column total | Table total | |
| Conditional | One cell | Total of the row or column you restrict to |
The direction of a conditional matters
The last two numbers in that list use the same cell, 80, and tell different stories. answers "how often are students who live close late?" answers "where do the late students live?" The predict question gave you the second number and invited you to read it as the first.
Words like of, among and given tell you which group you are restricted to. That group's total goes in the denominator.
Using conditional relative frequencies to compare groups
To decide whether two categorical variables are associated, compute the conditional relative frequencies of one variable for each category of the other, then compare them. If distance and lateness were not associated, every distance group would have the same late rate, matching the overall marginal rate of 18%. They do not.
Percent late at least once, by distance from school
The late rate climbs with distance, from 16% to 28%, so distance and lateness are associated for these 800 students.
Worked exampleAnnual plan or monthly plan?
A gym tracked 1,000 new members for six months. Of the 400 who bought an annual plan, 316 were still members after six months. Of the 600 who paid month to month, 288 were still members. The owner claims that annual members are more likely to stay. Find (a) the proportion of all new members who bought an annual plan and stayed, (b) the proportion of all new members who stayed, and (c) whether the data support the owner's claim.
(a) Joint. Annual plan and stayed, out of all 1,000 members: .
(b) Marginal. Everyone who stayed, out of all 1,000 members: .
(c) Conditional, one for each plan. Annual: . Monthly: .
Compare and conclude in context. 79% of annual members were still members after six months, compared with 48% of monthly members. The data support the claim. They show an association, not a cause: people who already intend to keep going to the gym may be the ones who buy an annual plan.
(a) 0.316 (b) 0.604 (c) Yes. 79% of annual members stayed, compared with 48% of monthly members.
Check your understanding
A survey of 250 voters recorded each voter's age group and whether they plan to vote early. Of the 90 voters under 35, 27 plan to vote early. Of the 160 voters aged 35 or older, 72 plan to vote early. What proportion of all 250 voters are under 35 and plan to vote early? Give a decimal.
Use the same survey: 27 of the 90 voters under 35 and 72 of the 160 voters aged 35 or older plan to vote early. Which calculation gives the conditional relative frequency of planning to vote early among voters aged 35 or older?
In the same survey, 30% of voters under 35 and 45% of voters aged 35 or older plan to vote early. Which statement gives a correct conclusion together with a valid reason?
A survey of 800 students records how far each one lives from school and whether each was late at least once this month. Of the 100 students who live more than 3 miles away, 28 were late at least once, and 144 students in all were late at least once. Among the students who were late at least once, what proportion live more than 3 miles away?
A cafeteria manager claims that seniors are more likely than freshmen to buy a school lunch. Of 240 freshmen surveyed, 156 buy lunch. Of 300 seniors surveyed, 180 buy lunch. Which response gives the correct answer about the claim together with a valid reason?
Course alignment, for teachers
AP Statistics topic 2.2, Unit 2: Probability, Random Variables, and Probability Distributions.
- Skill 3.B: Calculate summary statistics, relative positions of points within a distribution, and predicted responses.
- Skill 4.A: Describe and compare tabular and graphical representations of data, as well as summary statistics.
- Skill 4.B: Justify a claim based on statistical calculations and results.