Correlation

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.