← Statistics

Statistics

Correlation Coefficient

Lesson

The correlation coefficient rr measures the strength and direction of a linear relationship between two variables. It always sits between −1-1 and 11.

  • r=1r = 1: perfect positive — every point lies on one upward-sloping line.
  • r=−1r = -1: perfect negative — every point lies on one downward-sloping line.
  • r=0r = 0: no linear relationship (the variables might still be related non-linearly).
  • ∣r∣|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):

r=n ∑xy−∑x∑yABA=n∑x2−(∑x)2B=n∑y2−(∑y)2\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 rr gives the coefficient of determination r2r^2 — the fraction of the variance in yy that is explained by the linear relationship with xx:

r2=r⋅rr^2 = r \cdot r

Example: if r=0.9r = 0.9, then r2=0.81r^2 = 0.81 — about 81% of the variation in yy is “explained” by xx. Note that r2r^2 is always non-negative, so two different rr values (positive and negative) give the same r2r^2.

Worked example 1

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

r=1r = 1

Worked example 2 — r²

A study reports r=−0.6r = -0.6 between hours of TV watched and exam score. What is r2r^2?

r2=(−0.6)2=0.36r^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 rr, or between 0 and 1 for r2r^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 for (1,2), (2,4), (3,6), (4,8)r \text{ for } (1,2),\ (2,4),\ (3,6),\ (4,8)

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

r2 for r=0.5r^2 \text{ for } r = 0.5

Problem 11

r2 for r=−0.7r^2 \text{ for } r = -0.7

Problem 12

r2 for r=1r^2 \text{ for } r = 1

Problem 13

r2 for r=−0.6r^2 \text{ for } r = -0.6

Problem 14

r2 for r=0.9r^2 \text{ for } r = 0.9

Challenge

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

Problem 15

r2 for r=−0.5r^2 \text{ for } r = -0.5

Problem 16

r2 for r=−0.4r^2 \text{ for } r = -0.4

Problem 17

r2 for r=0.95r^2 \text{ for } r = 0.95

Problem 18

r2 for r=0.7r^2 \text{ for } r = 0.7

Problem 19

r2 for r=−0.3r^2 \text{ for } r = -0.3

Problem 20

Given r2=0.64, r>0. r=?\text{Given } r^2 = 0.64,\ r > 0.\ r = ?

Problem 21

Given r2=0.49, r<0. r=?\text{Given } r^2 = 0.49,\ r < 0.\ r = ?

Problem 22

Given r2=0.81, r<0. r=?\text{Given } r^2 = 0.81,\ r < 0.\ r = ?