Compare conditional distributions · AP Statistics

Use conditional distributions to investigate association between categorical variables. One variable defines the groups being compared, and the other describes the outcome within each group. A grand-total percentage may be useful, but it does not by itself compare the groups.

Build the method

A two-way table records each person once in one cell. Row totals, column totals and the grand total serve different denominators. For a within-region transport comparison, divide each bus count by its own region total. Check that the resulting within-row percentages total one across all choices.

Worked example

North has 13 bus and 9 car users; South has 7 bus and 15 car users. The bus rates are 13/22 and 7/22. Their difference describes an association in these observations. This calculation alone gives neither a population significance test nor evidence that region causes a transport choice.

Compare the alternatives

If another population table has A:(2 yes,3 no) and B:(4 yes,6 no), both conditional yes probabilities are 2/5. The overall probability is also 6/15=2/5. A uniformly chosen person therefore has independent group and response variables in this finite population, despite B having more yes observations.

Avoid a specific trap

Aggregates weight subgroup rates by subgroup sizes. A treatment can have a better rate in every difficulty group but a worse aggregate rate if it receives more difficult cases. This is not contradictory arithmetic. It means the aggregate comparison answers a different weighted question.

Transfer check with explanation

Transfer: with P(A)=1/2, P(B)=1/4 and overlap 1/20, P(A|B)=1/5 whereas P(B|A)=1/10. The overlap stays the same but the reference group changes. Explain which population is restricted before calculating either conditional probability.

WORKED EXAMPLE

Use the displayed two-way count table. Which conditional comparison addresses whether travel choice is associated with region?

Hold region fixed for each denominator, then compare the resulting choice distributions.
North bus rate 13/22 and South bus rate 7/22 differ.
Why an alternative fails: A single joint proportion does not compare travel-choice distributions between regions.

Answer: Compare the bus proportion within North with that within South

Compare the other answer choices

Compare North bus count with the grand total

A single joint proportion does not compare travel-choice distributions between regions.

Compare the total bus count with the total car count

Marginal choice totals erase the regional grouping needed for association.

Compare North bus count with North car count only

One region alone cannot establish whether choice distributions differ across regions.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

AP Statistics Course and Exam Description, effective Fall 2026