Association

Association is the claim that two variables are not independent: the distribution of one shifts depending on the value of the other. Nothing in that requires a straight line, and nothing in it requires numbers. For two quantitative variables you describe an association by its form, direction, strength and unusual features. For two categorical variables you have no form or direction to report, so you compare conditional distributions inside a two-way table instead.

Take 200 students, 150 who walk to school and 50 who drive, and record who was late at least once last term: 45 of the walkers and 20 of the drivers. Work within each group. Among walkers, $45/150 = 30$ percent were late; among drivers, $20/50 = 40$ percent. The two conditional percentages differ, so knowing how a student travels changes what you expect about lateness, and that is an association.

The misreading is almost automatic: "45 walkers were late and only 20 drivers were, so walking is associated with being late." The counts run the opposite way to the rates, because there are three times as many walkers to begin with. A raw count mixes the effect you are looking for with the size of the group it came from, which is exactly what dividing by the group total removes.

No association at all means every conditional distribution is the same. Had 30 percent of the drivers been late too, 15 of the 50, both rows would sit exactly on the overall rate, $60/200 = 30$ percent, and you would report no association. Note also that an association can be strong and curved, in which case the correlation $r$ can understate it badly or miss it entirely, so "no correlation" and "no association" are different verdicts. The correlation vs association page holds that contrast.

In the Fall 2026 AP Statistics course, association between two categorical variables is topics 2.1 and 2.2, and association between two quantitative variables is topic 5.1. Neither one licenses a causal claim on its own.