Coefficient of Determination (R²) Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
For a simple linear regression, the coefficient of determination R² is the square of the Pearson correlation coefficient r, expressed as a percentage: R² = r² × 100. A correlation of 0.8 therefore gives an R² of 64%.
Explanation
The coefficient of determination R² answers a different question from the correlation coefficient: where r measures the strength and direction of a linear relationship between two variables (between −1 and 1), R² indicates what proportion of the total variation in the dependent variable is explained by this linear relationship with the explanatory variable — a reading directly as a percentage, often more meaningful for assessing a model's quality. For a simple linear regression (a single explanatory variable, like the one fitted by our simple linear regression calculator), this relationship R² = r² is exact and requires no further calculation: just square the correlation coefficient already known. Squaring r has a notable effect on interpretation: a moderate correlation of r=0.5, which seems to indicate a fairly marked relationship, gives only an R² of 25% — that is, three quarters of the observed variation remains unexplained by that single variable. This is why R² is generally considered a more rigorous (and stricter) measure of the quality of a linear fit than the raw correlation coefficient, notably in experimental science and econometrics, where an R² of 0.9 or more is often expected before considering a model truly predictive.
Example: correlation coefficient r=0.8
Inputs
Pearson correlation coefficient: 0.8.
Calculation
R² = 0.8² × 100 = 0.64 × 100 = 64%.
Result
In this simple linear regression, 64% of the variation in the dependent variable is explained by the explanatory variable.
Frequently asked questions
Why is R² always positive, unlike r?
Because squaring r systematically removes its sign: a negative correlation (r=−0.5) and a positive correlation of the same magnitude (r=0.5) give exactly the same R² (25%), since R² measures the strength of the explained relationship, not its direction. To know the direction of the relationship (positive or negative), you must go back to the sign of r itself, not R².
Is an R² of 64% a good result?
It depends heavily on the field of application: in social science or economics, where many external factors generally influence the studied phenomena, an R² of 60 to 70% can already be considered solid. In experimental physics or engineering, where the relationship between the two quantities is often more directly causal, an R² above 90%, or much closer to 100%, is frequently expected.
Does this relationship R² = r² also apply to a regression with several explanatory variables?
No: this simple identity is only valid for a simple linear regression, with a single explanatory variable. In a multiple regression (several explanatory variables), R² is calculated differently, from the total variance and the residual variance of the model, and can no longer be derived directly from the square of a single correlation coefficient.