Step by Step
r
The correlation coefficient
r measures both the direction and strength of a LINEAR relationship between two variables, ranging from −1 to +1. r = 1 or −1 means a perfect linear relationship; r = 0 means no linear relationship (though a strong nonlinear relationship could still exist).
r²
The coefficient of determination
r² (r-squared) represents the proportion of variation in the response variable (y) that is explained by the explanatory variable (x), through the linear model.
%
Always interpret r² as a percentage
When reporting r², always phrase it as a percentage of variation explained — for example, r² = 0.81 should be interpreted as "81% of the variation in y is explained by the linear relationship with x."
Note
r only captures linear relationships
A low r value doesn't necessarily mean there's no relationship at all — it only means there's no strong LINEAR relationship. A clear curved (nonlinear) pattern could still exist even when r is close to 0.
Applied Walkthrough
1
A study finds r = 0.9 between hours studied and exam score. This indicates a strong, positive linear relationship.
2
Calculate r²: 0.9² = 0.81.
3
Interpret r² as a percentage: 81% of the variation in exam scores is explained by the linear relationship with hours studied.
4
The remaining 19% of variation in exam scores is due to other factors not captured by this linear model (such as test anxiety, prior knowledge, or sleep quality) — r² doesn't explain everything, just the portion attributable to this specific linear relationship.
Exam Application
Exams test whether you can correctly interpret r (direction and strength of a linear relationship) versus r² (percentage of variation explained), and specifically whether you phrase r² as a percentage in your written interpretation.
⚠ Common Trap
The most common trap is failing to phrase r² as a percentage of variation explained, or confusing r and r² — remember r describes direction and strength, while r² gives the percentage of variation in y accounted for by the linear relationship with x.
✓ Quick Self-Check
1. What range does the correlation coefficient r fall within?
−1 to +1.
Tap to reveal / hide
2. What does r measure?
The direction and strength of a linear relationship between two variables.
Tap to reveal / hide
3. What does r² represent?
The proportion (expressed as a percentage) of variation in the response variable explained by the explanatory variable.
Tap to reveal / hide
4. If r = 0.9, what is r², and how should it be interpreted?
r² = 0.81; 81% of the variation in y is explained by the linear relationship with x.
Tap to reveal / hide
5. Does a low r value always mean there's no relationship between two variables?
No — it only means there's no strong linear relationship; a nonlinear (curved) relationship could still exist.
Tap to reveal / hide