Build the method
Check random collection, appropriate independence and sufficiently large expected cell counts. The revised AP framework describes expected counts greater than five for these procedures. An observed cell count alone is not the relevant expected-count check. Each person should contribute to one cell, not be counted repeatedly as independent units.
Worked example
In a two-by-two table with rows (22,18) and (18,22), each row and column totals forty and the grand total is eighty. Thus each null expected count is forty times forty divided by eighty, or twenty. Each cell contributes 4/20=0.2, so the full statistic is 0.8. Adding signed deviations first would give zero and conceal the departures that squaring is intended to measure.
Compare the alternatives
A two-by-three table has (2−1)(3−1)=2 degrees of freedom. Row and column totals impose constraints, so the six cells are not six freely varying deviations. The single-variable categories-minus-one formula belongs to a different testing situation and is not used here.
Avoid a specific trap
A small p-value supports an association or unequal categorical distributions under the null model. It does not give association strength, identify which cells differ without further analysis, or prove causation for observational data. Sample size can make modest departures statistically detectable.
Transfer check with explanation
Transfer: independently sampling residents from three cities and comparing their transport choices is a homogeneity design. Sampling once from a combined city population and recording city plus transport is an independence design. The numerical table can look similar, but the sampling story changes the parameter description and conclusion.
A chi-square independence test uses a 2-row by 3-column table, with no structural zero cells. What are its degrees of freedom?
- Fixed row and column totals constrain the free cell counts in both directions.
- (2−1)(3−1)=2.
- Why an alternative fails: Count cells rather than the independent deviations after marginal constraints.
Compare the other answer choices
Add the reduced dimensions instead of multiplying them.
Use the categories-minus-one rule for a different single-variable procedure.
Count cells rather than the independent deviations after marginal constraints.
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.
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