AP Statistics · Topic 5.2 · Unit 5

AP Stats 5.2: Correlation Coefficient r

By Jude Wallis · Published July 28, 2026

Topic 5.2 in the Fall 2026 AP Statistics course is correlation. The correlation coefficient r summarizes the strength and direction of a linear association. It is unit-free and always between -1 and 1: near 0 means no linear association, near -1 or 1 means a strong one.

AP Statistics: Unit 5 (topics 5.2). Topic 5.2 (Correlation) is part of Unit 5 in the Fall 2026 CED. The exam expects you to interpret r from technology output; inference for a correlation or a regression slope is not part of the redesigned course.

What r measures

Topic 5.2 introduces the correlation coefficient, written $r$ , the single number that summarizes the strength and direction of a linear association between two quantitative variables. It is unit-free, so it does not depend on whether you measure in inches or centimeters, and it always falls between $-1$ and $1$ , inclusive.

The sign of $r$ matches the direction of the association. A negative value goes with a negative association, where the response tends to fall as the explanatory variable rises; a positive value goes with a positive association, where the response tends to rise. You can read that sign straight off the scatterplot: points that trend upward from left to right give a positive $r$ , and points that trend downward give a negative one.

Reading the strength

The strength of the linear association is set by how close $r$ is to $-1$ or $1$ . A value of $r = 0$ means there is no linear association at all, while $r = -1$ or $r = 1$ means a perfect linear association, with every point sitting exactly on a straight line.

Values in between are judged by distance from those extremes: $r = 0.9$ signals a strong linear relationship, $r = 0.5$ a moderate one, and $r = 0.1$ a weak one. On the AP exam a graphing calculator reports $r$ for you, so the skill being tested is reading and interpreting the value, not grinding through a formula. Even so, computing $r$ once on a tiny data set, as the worked example does, shows where the number comes from.

Two warnings the AP course stresses

Correlation carries two limits that the framework names directly.

First, a value of $r$ close to $-1$ or $1$ does not by itself mean a linear model is appropriate. Data that bends along a smooth curve can still produce a high $r$ , so you confirm linearity from the scatterplot and the residual plot, not from $r$ alone.

Second, correlation does not necessarily imply causation. A strong association between two variables does not prove that changes in one cause changes in the other, because a lurking variable or coincidence can drive the pattern. The correlation versus causation guide works through why ice cream sales and shark attacks rise together without either causing the other.

Compute the correlation for three points

A small data set has three points: (1, 2), (3, 3), and (5, 7), where x is the explanatory variable and y is the response. Compute the correlation coefficient $r$ from the z-scores, dividing the sum of the paired products by $n - 1$ , then interpret it.

Find the means. $\bar{x} = \frac{1+3+5}{3} = \frac{9}{3} = 3$ and $\bar{y} = \frac{2+3+7}{3} = \frac{12}{3} = 4$ .
Find the sample standard deviations (spread, dividing by $n-1$ ). For $x$ the deviations are $-2, 0, 2$ , so $s_x = \sqrt{\frac{4+0+4}{3-1}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2$ . For $y$ the deviations are $-2, -1, 3$ , so $s_y = \sqrt{\frac{4+1+9}{3-1}} = \sqrt{\frac{14}{2}} = \sqrt{7} \approx 2.6458$ .
Convert each value to a z-score (deviation divided by its standard deviation). For $x$ : $-1, 0, 1$ . For $y$ : $\frac{-2}{2.6458} \approx -0.7559$ , $\frac{-1}{2.6458} \approx -0.3780$ , $\frac{3}{2.6458} \approx 1.1339$ .
Multiply the paired z-scores. $(-1)(-0.7559) = 0.7559$ ; $(0)(-0.3780) = 0$ ; $(1)(1.1339) = 1.1339$ . Their sum is $0.7559 + 0 + 1.1339 = 1.8898$ .
Divide by $n - 1 = 2$ . $r = \frac{1.8898}{2} \approx 0.9449$ .

The correlation is $r \approx 0.945$ . It is positive and close to 1, so the three points show a strong positive linear association: larger x-values tend to come with larger y-values. On the exam you would read this value straight from calculator output.

Frequently asked questions

Can the correlation coefficient be greater than 1?

No. The correlation coefficient $r$ is always between $-1$ and $1$ , inclusive, so a value outside that range means an arithmetic error. A value of exactly $-1$ or $1$ means every point lies on a perfectly straight line.

Does a correlation of 0 mean the variables are unrelated?

Not necessarily. An $r$ of 0 means there is no linear association, but the variables could still have a strong non-linear relationship, such as a U-shaped curve. Always check the scatterplot before concluding the variables are unrelated.