Association in Two-Way Tables

A two-way table organizes counts for two categorical variables, one in rows and one in columns. To check for association, you cannot just stare at the raw counts, because groups are often different sizes. Instead you compute conditional percentages: within each row, what percent falls into each column category. If those percentages are roughly the same across rows, the variables are not associated, knowing the row tells you nothing new about the column. If the percentages differ noticeably, the variables are associated, the outcome depends on which group you're in.

Mechanically, pick one variable to condition on, usually the explanatory one, and turn each row into percentages that add to 100%. Then compare those percentages across rows directly. A ten or twenty point gap in exercise rates between morning people and night people is a real signal of association. A two or three point gap is probably just noise from sampling.

Seeing association this way is purely descriptive. It tells you the two variables move together in your data, but it says nothing about why. That gap in a sample could come from a real effect, from random chance, or from a lurking variable tangled up with both. This limitation is exactly why the course later builds the chi-square test for independence: it takes the same two-way table setup and asks whether the association you see is large enough that chance alone is an unlikely explanation, giving you a formal way to decide if the pattern is real.