Correlation Examples in Statistics

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 Statistics.

Concept Recap

Correlation is a statistical relationship between two variables where changes in one are associated with changes in the other. Positive correlation means both increase together; negative correlation means one increases as the other decreases; no correlation means no consistent pattern.

When one thing goes up and another tends to go up with it (like study time and test scores), that's positive correlation. When one goes up and the other goes down (like TV time and exercise), that's negative correlation. They 'move together' in some 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 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 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?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Correlation vs Causation Mistakes

Worked Examples

Example 1

hard

For

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

, compute

r

to two decimals.

Data: (1,2), (2,4), (3,5), (4,4), (5,5) — compute r.

Answer

r \approx 0.79

First step

\bar x = 3, \bar y = 4

. Deviations

x

-2,-1,0,1,2

;

y

-2,0,1,0,1

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Example 2

medium

A scatter plot shows that as hours of study increase, test scores tend to increase. Describe the correlation and state whether it implies causation.

Example 3

medium

Classify each as positive correlation, negative correlation, or no correlation: (a) Temperature and ice cream sales. (b) Shoe size and IQ. (c) Hours of TV watched and exercise done.

Practice Problems

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

Example 1

easy

As study time increases, test scores tend to increase. What type of correlation is this?

Example 2

easy

As outdoor temperature rises, hot-chocolate sales fall. What type of correlation is this?

Example 3

easy

Shoe size and favorite color show no consistent pattern. What type of correlation is this?

Example 4

easy

A correlation coefficient is

r=0.95

. Is the relationship strong or weak, and which direction?

Example 5

easy

A correlation coefficient is

r=-0.88

. Describe the relationship.

Example 6

easy

A correlation coefficient is

r=0.05

. Is this a strong or weak relationship?

Example 7

easy

Which correlation is stronger:

r=0.6

r=-0.8

Example 8

easy

The number of firefighters at a fire and the damage caused are positively correlated. Does sending fewer firefighters reduce damage?

Example 9

medium

Data points: as

x

goes

1,2,3,4

y

goes

2,4,6,8

. Describe the correlation and its strength.

x: 1, 2, 3, 4 → y: 2, 4, 6, 8

Example 10

medium

Data: as

x

goes

1,2,3,4

y

goes

8,6,4,2

. Describe the correlation.

x: 1, 2, 3, 4 → y: 8, 6, 4, 2

Example 11

medium

Two studies report

r=0.3

and

r=0.85

for the same kind of relationship. Which study's data shows points clustering more tightly around a line?

Example 12

medium

Hours of sleep and number of errors made are studied. More sleep, fewer errors. Name the direction and give a plausible

r

value sign.

Example 13

medium

A scatter shows points rising then falling in a clear arch (a curve). Is the linear correlation

r

likely near

+1

, near

-1

, or near

0

Example 14

medium

r=0

between two variables, can we conclude they are unrelated?

Example 15

medium

Variable

x

is in meters and

y

in kilograms with

r=0.7

. If

x

is reconverted to centimeters (multiply by 100), what happens to

r

Example 16

medium

Two variables have

r=0.4

. Roughly what fraction of the variation in

y

is explained by the linear relationship (

r^2

Example 17

medium

Two variables have

r=0.5

. What fraction of the variation in

y

is explained by the linear relationship?

Example 18

challenge

Five points:

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

. Without the last point,

r=1

. Explain qualitatively how the outlier affects

r

and whether it raises or lowers it below

1

Four perfectly collinear points, plus one outlier at (5, 100) — what happens to r?

Example 19

challenge

A dataset has

r=0.9

overall but splits into two subgroups each with

r\approx 0

. How can pooling create a strong correlation from groups with none?

Example 20

challenge

Suppose

z=x+y

where

x

and

y

are unrelated. Why would

z

be positively correlated with

x

even though

x

and

y

are not correlated?

Example 21

easy

As the number of hours worked increases, the paycheck total increases. Direction of correlation?

Example 22

easy

r = 0

for two variables. What does this say about their linear relationship?

Example 23

easy

Which is a stronger linear association:

r = 0.4

r = -0.7

Example 24

easy

Outdoor temperature vs. heating bill in winter usually shows what kind of correlation?

Example 25

easy

A car's age vs. its resale value. What sign of

r

would you expect?

Example 26

medium

A scatter plot shows a clear U-shape. What approximate value of

r

would you expect?

Scatter plot with a clear U-shape — estimate r.

Example 27

medium

Data:

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

. What is

r

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

Example 28

medium

A scatter plot of arm span vs. height shows points tightly along a line slanting upward. Estimate

r

Example 29

medium

A study finds

r = 0.92

between two variables. A reporter claims this proves causation. What is wrong with that claim?

Example 30

medium

For data

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

, what is

r

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

Example 31

medium

Two variables have

r=0.6

. A single new outlier is added that is far from the trend. Will

|r|

tend to increase or decrease?

Example 32

medium

A scatter plot has

r = 0.8

. What can you say about

r^2

as a percent?

Example 33

medium

A scatter plot of weight vs. price for diamonds shows tight upward trend except for one

\$10{,}000

diamond weighing

0.1

carat. Should this outlier raise concern?

Diamond weight (carats) vs. price ($) — should the outlier at 0.1 carat, $10,000 raise concern?

Example 34

hard

A study finds

r = 0.10

between coffee and longevity. The author claims coffee strongly affects longevity. Critique.

Example 35

hard

r

for points

(1,1), (2,4), (3,9), (4,16), (5,25)

— is it

+1

Points (1,1), (2,4), (3,9), (4,16), (5,25) — is r = +1?

Example 36

hard

Two variables have

r=0.5

on full data, but within each subgroup of a categorical variable,

r=0

. Why might this happen?

Example 37

hard

Two scatter plots have the same

r

, but one has tighter clustering. Is the correlation really the same?

Example 38

hard

A negative

r

becomes more negative when an extreme high-

x

, low-

y

point is added. Why?

Adding an extreme high-x, low-y point — why does r become more negative?

Example 39

challenge

Suppose

\bar x = \bar y = 0

\sum x_i^2 = \sum y_i^2 = 10

, and

\sum x_i y_i = 7

. Compute

r

Example 40

medium

A study finds a strong positive correlation between the number of firefighters at a fire and the damage caused. Does this mean sending more firefighters causes more damage? Explain.

Example 41

medium

A scatter plot shows that as outside temperature increases, hot chocolate sales decrease. Describe the correlation and explain why this pattern does not by itself prove temperature is the only cause.

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Related Concepts

Scatter Plot Data Collection Correlation Coefficient Correlation vs Causation

Background Knowledge

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

stat scatter plotdata collection