Correlation vs regression: how they differ

By Jude Wallis · Published July 28, 2026

Correlation gives a single number, r, that measures the strength and direction of a linear relationship; regression fits a line that predicts one variable from another, giving a slope and an intercept.

AP Statistics: Unit 5 (topics 5.2 Correlation, 5.3 Linear Regression Models, 5.5 Least-Squares Regression). In the Fall 2026 AP Statistics course, correlation is topic 5.2 and least-squares regression spans topics 5.3 through 5.5; the course treats regression descriptively, and inference for the slope of a regression model is not part of it.

What correlation is

Correlation measures the strength and direction of a linear relationship between two quantitative variables with a single number, the correlation coefficient $r$ . It ranges from $-1$ to $1$ : values near $1$ mean a strong positive linear pattern, values near $-1$ a strong negative one, and values near $0$ little linear association. Correlation has no units and does not change if you swap which variable is $x$ and which is $y$ . It answers how tightly the points cluster around a line, not what that line is.

What regression is

Regression fits an actual line to the data so you can predict one variable from another. The least-squares regression line is written $\hat{y} = a + bx$ (y-hat is the predicted response, $a$ the intercept, $b$ the slope), and it treats $x$ as the explanatory variable and $y$ as the response. The slope $b$ carries units, giving the predicted change in $y$ for each one-unit increase in $x$ . Unlike correlation, regression is directional: predicting $y$ from $x$ is a different line from predicting $x$ from $y$ .

Correlation vs regression at a glance

Correlation and regression describe the same relationship but answer different questions.

Feature	Correlation	Regression
Output	One number, $r$	A line, $\hat{y} = a + bx$
Question	How strong is the linear link?	What line best predicts $y$ ?
Units	None	Slope has units
Direction	Symmetric in $x$ and $y$	Depends on which is the response
Range	$-1 \le r \le 1$	Slope and intercept unbounded
Main use	Measure association	Make predictions

When to use which

Reach for correlation when your goal is to summarize how strong and in what direction two variables move together. Reach for regression when you want to model the relationship and describe or predict how $y$ responds to $x$ . The two are linked: the slope and correlation share a sign, and $b = r \cdot \frac{s_y}{s_x}$ connects them through the standard deviations. Squaring the correlation gives $r^2$ , the fraction of the variation in $y$ that the regression line accounts for.

The classic mix-up and how to avoid it

The common error is treating $r$ and the slope $b$ as the same thing; they are not, because $r$ is unitless while $b$ is in units of $y$ per unit of $x$ . They always share a sign, and they are equal only in the special case where $s_x = s_y$ . A second trap is reading either one as proof of causation, when both only describe association; see correlation vs causation. Keep them straight by remembering that correlation reports the tightness of the pattern, while regression reports the line itself.

From correlation to the regression line

A dataset has correlation $r = 0.8$ , with $\bar{x} = 10$ , $s_x = 2$ , $\bar{y} = 30$ , and $s_y = 5$ . Find the slope and intercept of the least-squares line, and interpret $r^2$ .

Slope from the correlation: $b = r \cdot \frac{s_y}{s_x} = 0.8 \times \frac{5}{2} = 0.8 \times 2.5 = 2.0$ .
Intercept, using that the line passes through $(\bar{x}, \bar{y})$ : $a = \bar{y} - b\bar{x} = 30 - 2.0 \times 10 = 30 - 20 = 10$ .
Regression line: $\hat{y} = 10 + 2x$ .
Coefficient of determination: $r^2 = 0.8^2 = 0.64$ , so about $64\%$ of the variation in $y$ is explained by the linear model.
Compare the two summaries: correlation is the single unitless number $0.8$ , while regression is the full line $\hat{y} = 10 + 2x$ whose slope $2.0$ is in units of $y$ per unit of $x$ .

The least-squares line is $\hat{y} = 10 + 2x$ with $r^2 = 0.64$ ; correlation compresses the relationship to $r = 0.8$ , while regression expresses it as a predictive line with a slope that carries units.

Frequently asked questions

Can you have a strong correlation but a small slope?

Yes. The slope depends on the spread of the two variables through $b = r \cdot \frac{s_y}{s_x}$ , so a high correlation paired with a small $s_y$ relative to $s_x$ produces a small slope. Correlation strength and slope size are separate ideas.

Does the AP exam test inference for the regression slope?

No. The Fall 2026 AP Statistics course covers regression descriptively, including the least-squares line, residuals, $r$ , and $r^2$ , but inference for the slope of a regression model is not part of this course. It remains standard content in a full college statistics course.