Two-Way Tables

Unit 2 · Exploring Two-Variable Data

What AP Stats asks here

Two-way tables hold counts for two categorical variables. The recurring trap is the denominator — marginal and joint use the grand total, conditional uses the row or column total of the category you condition on. Association is detected by comparing conditional distributions, and Simpson's paradox warns that pooled rates can mislead when groups have different compositions.

Three distributions, three denominators

Marginal

P(X = x) = \dfrac{\text{row or column total}}{\text{grand total}}

Joint

P(X = x \cap Y = y) = \dfrac{\text{cell count}}{\text{grand total}}

Conditional

P(Y = y \mid X = x) = \dfrac{\text{cell count}}{\text{row total of } x}

Association

Independent

P(Y \mid X_1) \approx P(Y \mid X_2) \text{ across all categories of } X

Associated

\text{conditional distributions of } Y \text{ differ across groups of } X

AP Tip: A segmented bar chart whose two bars have nearly identical slice heights is the picture of independence. Different slice heights are the picture of association.

Caution: Simpson's paradox: pooled rates can favor B even when A is better in every subgroup, because B is given to easier cases more often. Always check rates within homogeneous subgroups.

Type 1

Marginal, joint, conditional

Pick the right denominator for the question. 'Among walkers' = condition on walkers; 'overall' = grand total; 'both male and walks' = grand total in the denominator.

Example 1

A school surveyed 200 students: Bus Car Walk Total Male 40 30 20 90 Female 50 35 25 110 Total 90 65 45 200 Which of the following is the conditional probability

P(\text{male} \mid \text{walk})

? (A)

20/200 = 10\%

(B)

20/45 \approx 44.4\%

(C)

20/90 \approx 22.2\%

(D)

90/200 = 45\%

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Type 2

Detecting association

Compute conditional distributions across rows (or columns) and compare. If they differ substantially, the variables are associated.

Example 2

Among students who studied, 80 passed and 20 failed (n = 100). Among students who did not study, 30 passed and 70 failed (n = 100). Are 'study habit' and 'exam outcome' associated? (A) No — both groups have n = 100, so by symmetry there is no association. (B) Yes — pass rate is 80% if studied vs 30% if not; strong association. (C) Cannot tell without a formal hypothesis test. (D) Yes — but only because pass and fail counts are different in total.

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Type 3

Simpson's paradox

When subgroup composition differs across treatments, the pooled rate can reverse the within-subgroup story. Always inspect rates within homogeneous subgroups.

Example 3

A hospital reports overall survival rates of 80% for Treatment A and 85% for Treatment B. Among severe cases, A = 60% and B = 50%. Among mild cases, A = 95% and B = 90%. Which interpretation is correct? (A) Treatment B is better because the overall rate is higher. (B) Treatment A is better in both severity groups; Treatment B's higher pooled rate reflects that B was used more on mild (high-survival) cases. (C) The numbers are impossible — pooled and subgroup rates must agree. (D) The two treatments are equivalent.

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