The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. It ranges from −1 to +1, where −1 is a perfect negative correlation, 0 is no correlation, and +1 is a perfect positive correlation. It is one of the most used (and misused) statistics in science, business, and everyday reasoning.
The Correlation Coefficient
r = +1.0: Perfect positive (both increase together) r = +0.7 to +0.9: Strong positive r = +0.4 to +0.7: Moderate positive r = +0.1 to +0.4: Weak positive r = 0.0: No linear relationship Negative values: same scale, opposite direction The sign indicates direction; the magnitude indicates strength.
r-squared: The Explained Variance
Squaring r gives r², the coefficient of determination. This tells you what percentage of variation in one variable is explained by the other. If r = 0.80, then r² = 0.64, meaning 64% of the variation is explained by the relationship. The remaining 36% is due to other factors.
| r value | r² | Interpretation |
|---|---|---|
| ±0.90 | 0.81 | 81% of variance explained — very strong |
| ±0.70 | 0.49 | 49% of variance explained — strong |
| ±0.50 | 0.25 | 25% of variance explained — moderate |
| ±0.30 | 0.09 | 9% of variance explained — weak |
Calculate correlation for your own data with the Correlation Calculator.
Correlation Does Not Imply Causation
This is the most important principle in statistics. A strong correlation between two variables does not mean one causes the other. Ice cream sales and drowning deaths are positively correlated — not because ice cream causes drowning, but because both increase in summer (a confounding variable). Always look for confounders, reverse causation, and coincidence before drawing causal conclusions.
Key Takeaways
- r ranges from −1 to +1 measuring strength and direction of linear relationship.
- r² tells you the percentage of variation explained by the relationship.
- Correlation does not imply causation — always consider confounders.
- r only measures linear relationships — it misses curved or complex patterns.
Frequently Asked Questions
What is a strong correlation?
In most fields, r above 0.70 (or below -0.70) is considered strong. In social sciences, where human behavior is noisy, r above 0.50 may be considered strong. In physics and engineering, r below 0.95 may be considered weak. Context matters.
Can correlation be negative?
Yes. A negative correlation means as one variable increases, the other decreases. For example, hours of exercise per week and body fat percentage have a negative correlation. The strength is still measured by the magnitude (closer to -1 is stronger).
What is the difference between correlation and regression?
Correlation measures the strength of a relationship (r-value). Regression builds a predictive equation (y = mx + b) that lets you predict one variable from the other. Regression uses correlation as part of its calculation but provides more practical predictive power.