---
title: "Correlation Coefficient — AP Stats Definition & Exam Guide"
description: "The correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables. Learn how AP Stats tests r, r², and the slope formula."
canonical: "https://fiveable.me/ap-stats/key-terms/correlation-coefficient"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 5"
---

# Correlation Coefficient — AP Stats Definition & Exam Guide

## Definition

The correlation coefficient, r, is a unit-free number between -1 and 1 that measures the direction and strength of the linear association between two quantitative variables, where r = 1 or r = -1 means a perfect linear relationship and r = 0 means no linear association (AP Stats Topic 2.5).

## What It Is

The correlation coefficient, r, puts a single number on what a [scatterplot](/ap-stats/key-terms/scatterplot "fv-autolink") shows you visually. It tells you two things about the linear relationship between two quantitative variables. First, direction. A positive r means the [variables](/ap-stats/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt "fv-autolink") tend to rise together, and a negative r means one tends to fall as the other rises. Second, strength. The closer r is to 1 or -1, the more tightly the points hug a straight line. An r of exactly 1 or -1 means every point sits perfectly on a line, and r = 0 means there is no linear association at all.

The formula is r = (1/(n-1)) × Σ[(xi - x̄)/sx][(yi - ȳ)/sy], which is really just averaging the products of the z-scores of x and y. In practice you'll almost always get r from your calculator. Because r is built from z-scores, it has no units, and it doesn't change if you rescale the data (multiply every value by 3 and r stays exactly the same). Two warnings the CED bakes in directly. A value of r close to 1 or -1 does not guarantee a [linear model](/ap-stats/unit-5/correlation/study-guide/LlS81pC6QricXgIKNuFM "fv-autolink") is appropriate (always check the scatterplot and residuals), and correlation does not imply causation.

## Why It Matters

Correlation lives in Topic 2.5 of [Unit 2](/ap-stats/unit-2 "fv-autolink") (Exploring Two-Variable Data) and supports learning objectives 2.5.A (determine the correlation for a [linear relationship](/ap-stats/key-terms/linear-relationship "fv-autolink")) and 2.5.B (interpret it). But r doesn't stay in its own topic. It's wired directly into least-squares regression in Topic 2.8, where the slope formula b = r(s_y/s_x) and the coefficient of determination r² both depend on it (2.8.A). It also feeds the big-picture question from Topic 2.1 about whether an apparent pattern in data is real or random, a question that comes back with full inference machinery in Unit 9 when you test whether a population slope is actually different from zero. If you can compute, interpret, and correctly limit the conclusions you draw from r, you've got one of the most-tested skills in the course.

## Connections

### [Scatterplot (Unit 2)](/ap-stats/key-terms/scatterplot)

A scatterplot is the picture and r is the number. Topic 2.4 has you describe form, [direction](/ap-stats/key-terms/direction "fv-autolink"), strength, and unusual features visually, and r quantifies the direction and strength parts. But r only measures LINEAR strength, so a curved pattern can have a high r and still be a bad fit for a line. Always look at the plot before trusting the number.

### Least Squares Method (Unit 2)

The slope of the [least-squares regression line](/ap-stats/key-terms/least-squares-regression-line "fv-autolink") is b = r(s_y/s_x), so the correlation literally lives inside the regression equation. Notice what this means. The sign of r and the sign of the slope always match, and a stronger correlation (holding the standard deviations fixed) means a steeper line.

### [Coefficient of Determination (Unit 2)](/ap-stats/key-terms/coefficient-of-determination)

Square the correlation and you get r², the coefficient of determination, which gives the percent of [variation](/ap-stats/unit-5 "fv-autolink") in y explained by the linear relationship with x. Going from r to r² is easy, but going backwards requires care because r² = 0.64 could come from r = 0.8 or r = -0.8. You need the scatterplot or slope to know the sign.

### Inference for Slopes (Unit 9)

Unit 2's correlation describes a sample. Unit 9 asks whether the linear relationship is real in the population by testing the slope. A sample r far from 0 is the descriptive hint, and the t-test for slope is the formal answer to Topic 2.1's question of whether the pattern could just be random noise.

## On the AP Exam

Multiple-choice questions love r's properties. Expect stems asking what happens to r when you rescale the data (answer: nothing, since r is unit-free), what r equals when all points lie exactly on a line with positive slope (r = 1), or how to interpret a value like r = -0.85 (a strong negative linear association). You may also compute r from summary statistics like Σx, Σy, and Σxy. On FRQs, correlation usually appears inside a regression question. The 2017 FRQ on gray wolf length and weight and the 2018 FRQ on checkout times both started from a scatterplot and computer regression output. Common tasks include interpreting r or r² in context, connecting r to the slope, and explaining why a strong correlation doesn't prove causation. The graders want context (the actual variables named) and the word "linear" in your interpretation. "Strong negative association" loses credit; "strong negative linear association between length and weight" earns it.

## Correlation Coefficient vs Coefficient of Determination (r²)

The correlation r measures the direction and strength of a linear relationship and runs from -1 to 1. The coefficient of determination r² is its square, runs from 0 to 1, and answers a different question: what percent of the variation in the response variable is explained by the linear model? They have different interpretations and you can't swap them. An r of -0.9 means a strong negative linear association, while the matching r² of 0.81 means 81% of the variation in y is explained by the line. Also, r² alone can't tell you the direction, since squaring erases the sign.

## Key Takeaways

- The correlation coefficient r is always between -1 and 1, where the sign gives the direction of the linear association and the distance from 0 gives its strength.
- r is unit-free and doesn't change when you multiply or rescale the data, because it's calculated from z-scores.
- An r close to 1 or -1 does not prove a linear model is appropriate; you still need to check the scatterplot and residual plot.
- Correlation does not imply causation, so a strong r between two variables never lets you conclude that one variable causes the other.
- The correlation connects directly to regression through the slope formula b = r(s_y/s_x) and through r², the coefficient of determination.
- When interpreting r on the exam, always state direction, strength, the word 'linear,' and the variables in context.

## FAQs

### What is the correlation coefficient in AP Stats?

It's the number r, between -1 and 1, that measures the direction and strength of the linear relationship between two quantitative variables. It's covered in Topic 2.5 of Unit 2 and is usually computed with technology rather than by hand.

### Does a correlation of 0.95 mean a linear model is appropriate?

No. The CED states explicitly that an r close to 1 or -1 does not necessarily mean a linear model fits. A strongly curved pattern can still produce a high r, so you have to check the scatterplot and residual plot before fitting a line.

### Does a strong correlation mean one variable causes the other?

No, and AP Stats tests this constantly. Correlation does not imply causation, because lurking variables or coincidence can create an association. Only a well-designed randomized experiment supports a causal conclusion.

### What's the difference between r and r squared?

r is the correlation, running from -1 to 1, and describes the direction and strength of the linear association. r² is the coefficient of determination, running from 0 to 1, and gives the proportion of variation in y explained by the linear model. An r of -0.8 gives an r² of 0.64, but r² alone can't tell you the relationship is negative.

### What happens to r if you multiply all the data values by a constant?

Nothing. Because r is built from standardized values, rescaling or shifting the data leaves it unchanged. If r = 0.4 and you multiply every value by 3, r is still 0.4, which is a classic multiple-choice question.

## Related Study Guides

- [5.1 Graphical Representations Between Two Quantitative Variables](/ap-stats/unit-5/representing-relationship-between-two-quantitative-variables/study-guide/3rWWsKXcnbYlqY64hQ1j)
- [Unit 2 Overview: Probability, Random Variables, and Probability Distributions](/ap-stats/unit-2/review/study-guide/JeaowG56kC80Eu94VWs7)
- [Legacy AP Statistics Skills Guide: Inference Procedure Selection](/ap-stats/unit-CL5B675bCTuba5g2/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ)
- [5.5 Least-Squares Regression](/ap-stats/unit-5/least-squares-regression/study-guide/cRc4EhpHno3A4KvWrqyj)
- [AP Statistics Exam Guide](/ap-stats/study-tools/2024-ap-statistics-exam-guide/study-guide/udcPq20FAYBVIqHJVa8Y)

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