Correlation Coefficient Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Correlation Coefficient.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The correlation coefficient (Pearson's r) is a number between −1 and 1 that measures both the strength and direction of the linear relationship between two quantitative variables. A value of 1 indicates a perfect positive linear relationship, −1 a perfect negative linear relationship, and 0 no linear relationship at all.

r = 1 means perfect positive line, r = −1 means perfect negative line, r = 0 means no linear pattern.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Correlation Coefficient asks whether the same cases connect two variables or groups in a pattern that can be described carefully.

Common stuck point: Students often know a procedure related to correlation coefficient but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I studying a relationship between variables, and have I separated association from causation?

Worked Examples

Example 1

medium

Given

r=0.6

, compute

R^2

and interpret it.

Answer

R^2=0.36,\text{ 36\% of variance explained}

First step

R^2 = r^2 = 0.36

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Example 2

medium

Five points:

(1,2), (2,4), (3,6), (4,8), (5,10)

. Compute

r

without a formula.

Example 3

medium

Given

R^2 = 0.49

and a positive scatterplot slope, find

r

Example 4

medium

Two variables have

r = 0.9

. Approximate what percent of the variation in one is explained linearly by the other.

Example 5

hard

Five

(x,y)

pairs:

(1,3),(2,5),(3,7),(4,9),(5,11)

. Compute

r

Example 6

hard

r = 0.6

between hours of TV and exam score. After standardizing

x

but not

y

, what does

r

become?

Example 7

hard

Data:

x = 1,2,3,4,5

;

y = 5,4,3,2,1

. Find

r

Example 8

hard

Compute

r

for:

x=2,4,6

and

y=1,3,5

Example 9

challenge

For data

x = 1,2,3,4

y = 1,4,9,16

, is the correlation closer to

1

0

, or something between?

x = 1,2,3,4 and y = 1,4,9,16 — r ≈ 0.98 (very high but the trend is curved)

Example 10

challenge

In a sample of

n=4

\sum(x-\bar{x})(y-\bar{y}) = 8

\sum(x-\bar{x})^2 = 4

\sum(y-\bar{y})^2 = 25

. Compute

r

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

What is the range of possible values for the correlation coefficient

r

Example 2

easy

r = 1

describes what kind of relationship?

Example 3

easy

r = -1

describes what kind of relationship?

Example 4

easy

r = 0

indicates what about a linear relationship?

Example 5

easy

A scatter plot trends downward. Is

r

positive or negative?

Example 6

easy

Which is a stronger linear relationship:

r=0.3

r=-0.8

Example 7

easy

Does

r=0.95

between shoe size and reading ability prove one causes the other?

Example 8

easy

If a regression slope is positive, what is the sign of

r

Example 9

medium

r=0.6

between hours studied and score. Interpret strength and direction in words.

Hours studied vs. exam score — moderate positive association (r = 0.6)

Example 10

medium

Two variables have

r=0.85

. After adding one extreme outlier,

r

drops to 0.40. What does this illustrate?

Example 11

medium

Data follow a perfect parabola

y=x^2

symmetric about 0. Why might

r \approx 0

despite a strong relationship?

y = x² — strong curved relationship but r ≈ 0 (no linear pattern)

Example 12

medium

r=0.8

. Find

R^2

and state the percent of variance explained.

Example 13

medium

Why is a correlation of

r=0.5

NOT 'half as strong' in explained-variance terms as

r=1

Example 14

medium

r=-0.9

between price and quantity sold. Describe the relationship's strength and direction.

Price vs. quantity sold — strong negative association (r = −0.9)

Example 15

medium

A study finds

r=0.7

between number of firefighters at a blaze and damage done. Explain the likely lurking variable.

Example 16

medium

r=0.2

between two variables. Describe the strength of the linear relationship.

Example 17

medium

A correlation

r=0.65

is computed from only 3 data points. Why should you be cautious about it?

Example 18

challenge

A regression slope is

b=2

with

s_x=3

and

s_y=10

. Using

b=r\frac{s_y}{s_x}

, find

r

Example 19

challenge

Variable x has

r=0.6

with y. If every x value is multiplied by 3 (a positive linear transform), what is the new

r

Example 20

challenge

Data

(1,2),(2,4),(3,6)

lie exactly on the line

y=2x

. What is

r

, and why?

Points (1,2), (2,4), (3,6) on y = 2x — perfect linear fit, r = 1

Example 21

easy

A scatterplot shows points climbing steadily from lower-left to upper-right. Is

r

closer to

+1

-1

Points climbing from lower-left to upper-right — r closer to +1 or −1?

Example 22

easy

r = -0.95

, the scatterplot would look like what?

Tight downward linear scatter — what does r = −0.95 look like?

Example 23

easy

A perfect horizontal line of points (all same

y

) yields what value of

r

Example 24

easy

r = 0.05

between two variables. Which best describes the linear association?

Example 25

easy

For paired data with all points on a line

y = -2x + 5

, find

r

Example 26

medium

A dataset has

r = 0.3

. After removing two extreme outliers

r

jumps to

0.85

. What does this show?

Example 27

medium

r = 0.50

between

x

and

y

. If we transform

y

y' = 3y + 1

, what is the new correlation between

x

and

y'

Example 28

medium

Data form a perfect circle

x^2 + y^2 = 1

. Estimate

r

Example 29

medium

Why is

r

a 'dimensionless' number?

Example 30

hard

Researchers report

r = 0.20

in a sample of

20

and

r = 0.20

in a sample of

2000

. Which is more likely statistically significant?

Example 31

hard

Why might the correlation between ice cream sales and drownings be high in summer data, even though one doesn't cause the other?

Example 32

hard

Two scatterplots look nearly identical but one includes a single far-away point. The 'with outlier' version has

r=0.10

r=0.95

without. Which is more trustworthy without further study?

Example 33

hard

Two variables have

r = 0

. Can they still be strongly related?

r = 0 yet a strong U-shaped pattern exists — r only measures linear association

Example 34

challenge

Show that

r

and the regression-line slope

b

have the same sign.

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Related Concepts

Correlation Line of Best Fit Linear Regression

Background Knowledge

These ideas may be useful before you work through the harder examples.

correlation introline of best fit