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 by 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 or −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
Example 8
medium
Given r=0.6, compute R2 and interpret it.
Example 9
medium
r=0.50 between x and y. If we transform y by 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
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 or 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, ∑(x−xˉ)(y−yˉ)=8, ∑(x−xˉ)2=4, ∑(y−yˉ)2=25. Compute r.
Example 19
challenge
A regression slope is b=2 with sx=3 and sy=10. Using b=rsxsy, find r.
Example 20
easy
If r=−0.95, the scatterplot would look like what?Tight downward linear scatter — what does r