Practice Correlation Coefficient in Statistics

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

medium

r = 0.50

between

x

and

y

. If we transform

y

y' = -2y + 5

, what is the new correlation?

Example 3

medium

A dataset has

r = 0.3

. After removing two extreme outliers

r

jumps to

0.85

. What does this show?

Example 4

easy

For paired data with all points on a line

y = -2x + 5

, find

r

Example 5

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 6

easy

Does the correlation coefficient change if both variables are measured in different units (e.g., meters vs feet)?

Example 7

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 8

medium

Given

r=0.6

, compute

R^2

and interpret it.

Example 9

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 10

easy

r = 1

describes what kind of relationship?

Example 11

medium

Five points:

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

. Compute

r

without a formula.

Example 12

easy

A perfect horizontal line of points (all same

y

) yields what value of

r

Example 13

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 14

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 15

easy

Which is a stronger linear relationship:

r=0.3

r=-0.8

Example 16

medium

Two variables have

r = 0.9

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

Example 17

hard

Compute

r

for:

x=2,4,6

and

y=1,3,5

Example 18

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

Example 19

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 20

easy

r = -0.95

, the scatterplot would look like what?

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

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Correlation Line of Best Fit Linear Regression