Correlation Examples in Math

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

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

Concept Recap

Correlation measures the strength and direction of the linear relationship between two quantitative variables, ranging from $-1$ to $+1$ .

Do two things go up and down together? $r = +1$ means perfectly together, $r = -1$ means perfectly opposite.

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 describes how two variables move together; it does not by itself prove cause.

Common stuck point: The procedure for correlation is the easy part; the trap is saying correlation proves causation. Asking "Do the dots show a direction and tightness of pattern?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the dots show a direction and tightness of pattern?

Worked Examples

Example 1

medium

Given five data points

(1,2), (2,4), (3,5), (4,4), (5,5)

, compute the Pearson correlation coefficient

r

Answer

r \approx 0.775

First step

Compute means:

\bar{x} = 3

\bar{y} = 4

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Example 2

easy

A study finds

r = -0.85

between hours of TV watched per day and exam scores. Interpret this value.

Example 3

medium

Scatterplot shows data tightly clustered around a flat horizontal trend. Estimate

r

Example 4

hard

Calculate

r

for the data

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

Example 5

challenge

Anscombe's quartet shows four datasets with identical mean, SD, and

r = 0.816

but radically different shapes. What lesson does this teach about

r

Practice Problems

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

Example 1

medium

r = 0.92

, what is the coefficient of determination

r^2

, and what does it mean?

Example 2

easy

A study finds a correlation coefficient of

r = 0.88

between height and arm span. Describe the direction and strength of the linear relationship.

Example 3

easy

A correlation coefficient is

r=+1

. What does this mean?

Example 4

easy

What does a correlation of

r=-1

indicate?

Example 5

easy

A correlation of

r=0

means what about the linear relationship?

Example 6

easy

Which is stronger: a correlation of

r=-0.9

r=+0.6

Example 7

easy

As study hours increase, test scores tend to increase. What sign is the correlation?

Example 8

easy

As a car's age increases, its resale value tends to decrease. What sign is the correlation?

Example 9

easy

Can you compute a correlation between eye color and favorite number?

Example 10

easy

Ice cream sales and drowning incidents both rise in summer, giving

r=+0.8

. Does ice cream cause drowning?

Example 11

medium

A scatterplot shows points tightly clustered along an upward line. Estimate whether

|r|

is closer to

0

0.5

, or

1

Example 12

medium

Two variables have

r=0.3

. Is this a strong relationship? Explain.

Example 13

medium

A data set's

x

and

y

values are perfectly linearly related by

y=2x+1

. What is

r

Example 14

medium

If every

y

value is multiplied by

3

(a positive constant), how does

r

between

x

and

y

change?

Example 15

medium

A relationship is described by

y=-0.5x+4

exactly. What is

r

Example 16

medium

A study finds towns with more firefighters have more fire damage (

r=+0.7

). Why is concluding 'firefighters cause damage' wrong?

Example 17

medium

Data points are

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

. What is the correlation

r

Example 18

medium

Two variables show a strong U-shaped (curved) relationship. Why might

r

be near

0

Example 19

medium

Data points are

(2,10),(4,20),(6,30)

. What is the correlation

r

Example 20

challenge

Points are

(1,5),(2,4),(3,3),(4,2),(5,1)

. Without full computation, state

r

and justify.

Example 21

challenge

Correlation between height and weight is

r=0.8

in adults. A researcher claims height 'explains' weight. What two cautions apply?

Example 22

challenge

Variable

A

correlates with

B

r=0.9

, and

B

correlates with

C

r=0.9

. Must

A

and

C

be strongly correlated?

Example 23

easy

r = -0.95

, describe the direction and strength of the linear relationship.

Example 24

easy

Hours studied and exam scores have

r = 0.65

. Is the relationship positive or negative? Strong, moderate, or weak?

Example 25

medium

r = 0.7

, compute the coefficient of determination and interpret it.

Example 26

medium

A linear model

\hat{y} = 3 + 2x

fits data perfectly. What are

r

and

r^2

Example 27

medium

Data shows

r = 0.9

between shoe size and reading ability among children aged

5

–

12

. Why might this be misleading as evidence of causation?

Example 28

medium

Data points are

(1, 10), (2, 7), (3, 4), (4, 1)

. What is

r

Example 29

medium

A study reports

r = 0.7

between exercise frequency and happiness. Can you conclude exercising more will make a specific person happier?

Example 30

medium

Why can

r

be near

0

even for a tight non-linear (e.g., parabolic) relationship?

Example 31

hard

Compute

r

for the data

(1, 1), (2, 2), (3, 3), (4, 4), (5, 100)

Example 32

hard

A researcher reports

r = 0.95

from a sample of size

4

. Why is this fragile evidence of a real relationship?

Example 33

hard

Data

(0,0), (1,1), (2,4), (3,9), (4,16)

y = x^2

. Is

r

near

0

or near

1

Example 34

hard

r_{XY} = 0.6

and

r_{YZ} = 0.6

, what is the minimum possible value of

r_{XZ}

Example 35

challenge

A study finds

r = 0.85

between math scores and music scores in

1000

students. Does this prove studying music improves math?

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Related Concepts

Mean Standard Deviation Inference for Regression Scatter Plot

Background Knowledge

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

meanstandard deviation