Correlation vs Association

Correlation is a single number, written $r$, that reports how tightly the points of a scatterplot cluster around a straight line and which way that line tilts. It is the average product of standardized values, $r = \frac{1}{n-1}\sum \left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)$, where $\bar{x}$ (read x-bar) and $s_x$ are the mean and standard deviation of the explanatory variable. Both variables have to be quantitative, and it is the standardizing that strips the units out.

Five afternoons at a drinks stall pair the high temperature in degrees Celsius, 20, 22, 24, 26 and 28, with cups sold, 15, 21, 18, 27 and 30. Those data give $r = 0.9150$. Rewrite the temperatures in Fahrenheit as 68, 71.6, 75.2, 78.8 and 82.4 and $r$ is still 0.9150, while the fitted slope drops from 1.8 cups per degree to 1.0. Swap the roles of the two variables so that cups predicts temperature and $r$ comes out at 0.9150 again, because the formula treats $x$ and $y$ symmetrically. The slope does not.

Here is the sentence to stop writing: "$r = 0.90$ is twice as strong as $r = 0.45$." Correlation is not on a ratio scale, so the arithmetic of everyday comparison does not apply to it. Squaring turns each value into a share of variation explained, and $0.90^2 = 0.81$ against $0.45^2 = 0.2025$, four times as much rather than twice. The same mistake produces "$r = 0.90$ means 90 percent of the points lie on the line", which describes nothing $r$ measures.

Two limits matter more than any threshold for what counts as strong. $r$ sees straight-line patterns only, so a tight curve can still return a value near 0. And $r$ is not resistant: one point far from the rest moves it on its own, which is why you read the scatterplot before you trust the number.

Correlation is topic 5.2 in Unit 5 of the Fall 2026 AP Statistics course. Association is the wider idea that $r$ is one measure of, and the two are set against each other on the correlation vs association page.

Correlation is the entire mechanism behind diversification. Combining assets reduces risk only when their returns are less than perfectly correlated, so the value of $r$ between two holdings decides whether owning both helps at all: diversification.

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.