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 βˆ’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, βˆ’1 a perfect negative linear relationship, and 0 no linear relationship at all.

r = 1 means perfect positive line, r = βˆ’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=0.6r=0.6 with y. If every x value is multiplied by 3 (a positive linear transform), what is the new rr?

Example 2

medium
r=0.50r = 0.50 between xx and yy. If we transform yy by yβ€²=βˆ’2y+5y' = -2y + 5, what is the new correlation?

Example 3

medium
A dataset has r=0.3r = 0.3. After removing two extreme outliers rr jumps to 0.850.85. What does this show?

Example 4

easy
For paired data with all points on a line y=βˆ’2x+5y = -2x + 5, find rr.

Example 5

easy
A scatterplot shows points climbing steadily from lower-left to upper-right. Is rr closer to +1+1 or βˆ’1-1?

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)(1,2),(2,4),(3,6) lie exactly on the line y=2xy=2x. What is rr, and why?

Example 8

medium
Given r=0.6r=0.6, compute R2R^2 and interpret it.

Example 9

medium
r=0.50r = 0.50 between xx and yy. If we transform yy by yβ€²=3y+1y' = 3y + 1, what is the new correlation between xx and yβ€²y'?

Example 10

easy
r=1r = 1 describes what kind of relationship?

Example 11

medium
Five points: (1,2),(2,4),(3,6),(4,8),(5,10)(1,2), (2,4), (3,6), (4,8), (5,10). Compute rr without a formula.

Example 12

easy
A perfect horizontal line of points (all same yy) yields what value of rr?

Example 13

hard
Two variables have r=0r = 0. Can they still be strongly related?

Example 14

hard
Researchers report r=0.20r = 0.20 in a sample of 2020 and r=0.20r = 0.20 in a sample of 20002000. Which is more likely statistically significant?

Example 15

easy
Which is a stronger linear relationship: r=0.3r=0.3 or r=βˆ’0.8r=-0.8?

Example 16

medium
Two variables have r=0.9r = 0.9. Approximate what percent of the variation in one is explained linearly by the other.

Example 17

hard
Compute rr for: x=2,4,6x=2,4,6 and y=1,3,5y=1,3,5.

Example 18

challenge
In a sample of n=4n=4, βˆ‘(xβˆ’xΛ‰)(yβˆ’yΛ‰)=8\sum(x-\bar{x})(y-\bar{y}) = 8, βˆ‘(xβˆ’xΛ‰)2=4\sum(x-\bar{x})^2 = 4, βˆ‘(yβˆ’yΛ‰)2=25\sum(y-\bar{y})^2 = 25. Compute rr.

Example 19

challenge
A regression slope is b=2b=2 with sx=3s_x=3 and sy=10s_y=10. Using b=rsysxb=r\frac{s_y}{s_x}, find rr.

Example 20

easy
If r=βˆ’0.95r = -0.95, the scatterplot would look like what?
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