AP Statistics Two-Way Table Practice: Conditional Relative Frequency
To find a conditional relative frequency from a two-way table, divide the cell count by the total of the group named in the question. If the question asks about seniors, divide by the number of seniors, not by the table total. Joint, marginal and conditional relative frequencies share one table but use different denominators.
Try the question
A school asks 200 students whether they favor an earlier start time.
| Grade | Yes | No | Total |
|---|---|---|---|
| Juniors | 20 | 60 | 80 |
| Seniors | 60 | 60 | 120 |
| Total | 80 | 120 | 200 |
Among seniors, what proportion favor the earlier start?
- A. 0.50
- B. 0.30
- C. 0.75
- D. 0.40
Answer key: why each choice is right or wrong
✓ A. 0.50 — Correct
This earns the point because a conditional relative frequency divides by the total of the group named in the question: 60/120 = 0.50. Conditioning on seniors makes 120 the denominator.
✕ B. 0.30 — Incorrect
Tempting if you divide by the grand total (60/200) — but that is a joint relative frequency, not the proportion within seniors. Fix: divide by the number of seniors.
✕ C. 0.75 — Incorrect
Tempting if you divide by the number of yes responses (60/80) — but that gives the share of yes responses who are seniors, the reverse of what was asked. Fix: the group in the condition supplies the denominator.
✕ D. 0.40 — Incorrect
Tempting if you use the overall proportion saying yes (80/200) — but that is a marginal relative frequency for all students. Fix: restrict to the seniors' row.
The core idea
Joint, marginal and conditional relative frequency each divide by a different total from the same two-way table. Conditional relative frequencies within one row always sum to 100% on their own. That is what lets you compare groups of very different sizes side by side.
| Type | Numerator | Denominator |
|---|---|---|
| Joint | One cell | Table total |
| Marginal | Row or column total | Table total |
| Conditional | One cell | That cell's own row or column total |
How points are earned and lost
Earned: compute the relative frequency with the denominator that matches the question's wording, name the group that denominator describes, and compare conditional distributions across groups — not raw counts — to support an association conclusion.
Lost: dividing a cell by the grand total when the question asked “of the seniors,” which requires dividing by that row's own total.
The answer move
Write every percentage as a sentence: “Of the [group in the denominator], ___% are [category in the numerator].” If you cannot name the denominator group, you divided by the wrong total.
Weak answer vs. point-earning answer
A table of 150 students shows 90 seniors, 30 of them in band, and 60 juniors, 18 of them in band. What percentage of seniors are in band?
✕ Weak answer
30 out of the 150 total students are seniors in band, so 30/150 = 20% of seniors are in band.
✓ Point-earning answer
The question asks for a percentage “of the seniors,” so the denominator is the senior total (90), not the grand total (150): 30/90 ≈ 33.3% of seniors are in band. Dividing by 150 answers a different question — what percentage of all students are senior band members.
Before your next two-way table question
Underline the group named after “of the” or “among.” That group's own total is your denominator.
Review the topics behind this question: Topic 2.1: Two categorical variables and Topic 2.2: Summary statistics for two categorical variables. More practice: AP Statistics practice questions.
Frequently asked questions
What is a conditional relative frequency?
What is the difference between joint, marginal and conditional relative frequency?
How do I know which total to divide by?
How do I show association from a two-way table?
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