A contingency table (or two-way table) displays frequency counts for two categorical variables at once, with one variable's categories as rows and the other's as columns, so you can compare distributions and test for association with a chi-square test.
A contingency table is a grid of counts for two categorical variables. One variable's categories run down the rows, the other's run across the columns, and each cell holds the number of individuals who fall into both categories at once. For example, a table might cross smoking status (Smoker, Former Smoker, Never Smoker) with lung disease (Present, Absent), so each of the six cells counts people with that exact combination. Add up a row or column and you get the totals for one variable by itself, which are called marginal distributions.
The word "contingency" is the clue to what the table is for. It shows whether one variable's distribution is contingent on (depends on) the other. If the proportion with lung disease looks roughly the same across all three smoking groups, the variables look independent. If the proportions shift noticeably from group to group, that's evidence of an association. In AP Stats you read these tables descriptively in Unit 1 and then test them formally with a chi-square test in Unit 8.
Contingency tables show up at both ends of the course. In Unit 1, Topic 1.4 (learning objectives 1.4.A, 1.4.B, and 1.4.C), they're the raw material for comparing categorical data sets. You turn the table into side-by-side or segmented bar graphs and describe what the proportions suggest in context. Then in Unit 8, Topic 8.6 (learning objectives 8.6.A through 8.6.D), the same table becomes the input for a chi-square test for homogeneity or independence. You compute expected counts from the row and column totals, calculate χ² = Σ(Observed − Expected)²/Expected, and use degrees of freedom equal to (rows − 1)(columns − 1). The table also matters in Topics 7.10 and 9.6, where the skill is choosing the right inference procedure. Spotting that your data live in a two-way table of counts is exactly how you know to reach for chi-square instead of a t-test or a slope test.
Keep studying AP Statistics Unit 8
Chi-Square Test for Homogeneity or Independence (Unit 8)
The contingency table is the entire data set for these tests. Observed counts come straight from the cells, expected counts come from (row total × column total) / grand total, and degrees of freedom come from the table's dimensions. A 2×5 table gives (2−1)(5−1) = 4 degrees of freedom.
Marginal Distribution (Unit 1)
The row and column totals along the edges of a contingency table are the marginal distributions. They tell you about each variable on its own, while the inside cells (and conditional distributions built from them) tell you whether the two variables are related.
Bar Graph (Unit 1)
A segmented or side-by-side bar graph is essentially a contingency table drawn as a picture. If the bars look nearly identical across groups, the table's conditional distributions match and the variables look independent. Visibly different bars hint at association.
Selecting an Inference Procedure (Units 7 and 9)
Topics 7.10 and 9.6 test whether you can match data to the right procedure. The mental shortcut is simple. Two categorical variables in a table of counts means chi-square, quantitative data and means means t-procedures, and bivariate quantitative data means inference for slopes.
Contingency tables are tested two ways. In Unit 1-style questions, you read the table, compute conditional or marginal proportions, and describe whether an association seems plausible. In Unit 8 questions, you carry out the full chi-square machinery. Multiple-choice stems regularly give you a table's dimensions and ask for degrees of freedom (a 5×4 table gives 12, a 3×2 table gives 2), give you row and column totals and ask for an expected count, or give you a χ² value and p-value and ask for the correct conclusion. For example, with χ² = 9.2 from a 2×5 table and p = 0.056, you'd fail to reject the null at α = 0.05 and interpret the p-value as the probability of getting a statistic at least that extreme if the null is true. On the FRQ side, chi-square questions typically hand you a contingency table and expect a complete test, naming the procedure, checking conditions (expected counts), computing the statistic and p-value, and writing a conclusion in context linked back to the populations sampled.
A frequency table summarizes ONE categorical variable, listing each category with its count. A contingency table crosses TWO categorical variables, so every cell is a count of individuals in a combination of categories. This matters for inference. One-way tables of counts go with a chi-square goodness-of-fit test, while two-way contingency tables go with a chi-square test for homogeneity or independence.
A contingency table displays counts for two categorical variables, with one variable's categories as rows and the other's as columns.
Row and column totals give the marginal distributions, while proportions computed within a single row or column give conditional distributions used to judge association.
For a chi-square test, expected counts come from (row total × column total) divided by the grand total, and degrees of freedom equal (rows − 1)(columns − 1).
If the conditional distributions are roughly the same across groups, the variables look independent; large differences suggest an association worth testing.
Seeing two categorical variables in a table of counts is your signal to choose a chi-square test for homogeneity or independence, not a t-test or slope test.
The same table can also be displayed as a segmented or side-by-side bar graph when a question asks for a graphical comparison.
It's a two-way table of counts for two categorical variables, where rows are the categories of one variable and columns are the categories of the other. Each cell counts individuals falling into both categories, like the number of smokers with lung disease.
Yes. "Contingency table," "two-way table," and "crosstabulation" all describe the same thing. The AP exam most often says "two-way table," so treat the terms as interchangeable.
Multiply (number of rows − 1) by (number of columns − 1). A 2×5 table gives 1 × 4 = 4 degrees of freedom, and a 5×4 table gives 4 × 3 = 12.
A frequency table summarizes one categorical variable; a contingency table crosses two. That difference decides your test, since one-way tables call for chi-square goodness-of-fit and two-way tables call for chi-square homogeneity or independence.
It gives evidence, but the table alone doesn't settle it. Compare conditional distributions to describe a possible association, then run a chi-square test for independence or homogeneity to decide whether the differences are statistically significant or could just be sampling variability.