The Linear Correlation Coefficient

Introduces Pearson's linear correlation coefficient $r$ : its interpretation as a measure of the strength and direction of a linear relationship, its computational formula, key properties (symmetry, invariance under linear transformations), and how to compute $r$ from paired data.

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Tutorial

Introduction

The linear correlation coefficient (also called Pearson's correlation coefficient), denoted $r$ , is a numerical measure of the strength and direction of the linear relationship between two quantitative variables in a paired dataset $(x_i, y_i)$ .

The value of $r$ always satisfies

-1 \leq r \leq 1.

The sign of $r$ indicates the direction of the linear relationship:

$r > 0$ : as $x$ increases, $y$ tends to increase (positive association).
$r < 0$ : as $x$ increases, $y$ tends to decrease (negative association).
$r = 0$ : no linear association.

The magnitude $|r|$ indicates the strength of the linear relationship:

$r = \pm 1$ : a perfect linear relationship — all data points lie exactly on a straight line.
$|r|$ close to $1$ : strong linear association.
$|r|$ close to $0$ : weak linear association.

A common rule of thumb for describing strength:

$|r| \geq 0.8$ : strong
$0.5 \leq |r| < 0.8$ : moderate
$|r| < 0.5$ : weak

Strength and direction are independent properties. A correlation of $r = -0.9$ is just as strong as $r = 0.9$ ; they differ only in direction.