Statistics

Correlation Coefficient

Lesson

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It always sits between $-1$ and $1$ .

$r = 1$ : perfect positive — every point lies on one upward-sloping line.
$r = -1$ : perfect negative — every point lies on one downward-sloping line.
$r = 0$ : no linear relationship (the variables might still be related non-linearly).
$|r|$ close to 1 means a strong linear pattern; close to 0 means a weak one.

The full formula (you’ll usually use a calculator):

\begin{aligned} r &= \frac{n\,\sum xy - \sum x \sum y}{\sqrt{A B}} \\ A &= n\sum x^2 - (\sum x)^2 \\ B &= n\sum y^2 - (\sum y)^2 \end{aligned}

Squaring $r$ gives the coefficient of determination $r^2$ — the fraction of the variance in $y$ that is explained by the linear relationship with $x$ :

r^2 = r \cdot r

Example: if $r = 0.9$ , then $r^2 = 0.81$ — about 81% of the variation in $y$ is “explained” by $x$ . Note that $r^2$ is always non-negative, so two different $r$ values (positive and negative) give the same $r^2$ .

Worked example 1

Points (1, 2), (2, 4), (3, 6), (4, 8) — each lies exactly on $y = 2x$ . Perfect upward line, so:

r = 1

Worked example 2 — r²

A study reports $r = -0.6$ between hours of TV watched and exam score. What is $r^2$ ?

r^2 = (-0.6)^2 = 0.36

About 36% of the variation in exam scores is linearly associated with TV hours.

How to type your answer

A single number between -1 and 1 for $r$ , or between 0 and 1 for $r^2$ . Examples: 1, -0.8, 0.36, 0.9025.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

r \text{ for } (1,2),\ (2,4),\ (3,6),\ (4,8)

Problem 2

r \text{ for } (1,5),\ (2,3),\ (3,1),\ (4,-1)

Problem 3

r \text{ for } (1,2),\ (2,5),\ (3,8),\ (4,11)

Problem 4

r \text{ for } (2,10),\ (4,8),\ (6,6),\ (8,4)

Practice

Standard problems matching the lesson.

Problem 5

r \text{ for } (1,1),\ (2,3),\ (3,5),\ (4,7)

Problem 6

r \text{ for } (1,8),\ (2,6),\ (3,4),\ (4,2)

Problem 7

r \text{ for } (1,3),\ (2,5),\ (3,4),\ (4,6)

Problem 8

r \text{ for } (1,5),\ (2,3),\ (3,4),\ (4,2)

Problem 9

r \text{ for } (1,2),\ (2,3),\ (3,5),\ (4,4)

Problem 10

r^2 \text{ for } r = 0.5

Problem 11

r^2 \text{ for } r = -0.7

Problem 12

r^2 \text{ for } r = 1

Problem 13

r^2 \text{ for } r = -0.6

Problem 14

r^2 \text{ for } r = 0.9

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

r^2 \text{ for } r = -0.5

Problem 16

r^2 \text{ for } r = -0.4

Problem 17

r^2 \text{ for } r = 0.95

Problem 18

r^2 \text{ for } r = 0.7

Problem 19

r^2 \text{ for } r = -0.3

Problem 20

\text{Given } r^2 = 0.64,\ r > 0.\ r = ?

Problem 21

\text{Given } r^2 = 0.49,\ r < 0.\ r = ?

Problem 22

\text{Given } r^2 = 0.81,\ r < 0.\ r = ?

Practice

Standard problems matching the lesson.

Problem 23

Study hours and test grades have r = 0.9. Find r² (the coefficient of determination).

Problem 24

Ice cream sales and beach attendance have r = 0.8. What proportion of variance in sales is explained by attendance (find r²)?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Hours of TV watched and test grades have r = -0.7. Find r².

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