Direct comparison
Correlation Coefficient vs Determination: r vs R²
Correlation coefficient vs coefficient of determination: compare r and R², formulas, direction of association, variance explained, and regression reporting.
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How do Correlation Coefficient (r) and Coefficient of Determination (R²) compare side by side?
The table below compares Correlation Coefficient (r) and Coefficient of Determination (R²) across 11 dimensions, from primary statistical focus to when to choose.
Side-by-side comparison
| Dimension | Correlation Coefficient (r) | Coefficient of Determination (R²) |
|---|---|---|
| Primary Statistical Focus | Measures the strength and direction of a linear association between two continuous variables. | Measures the proportion of total variance in a dependent variable accounted for by one or more independent predictors in a regression model. |
| Mathematical Formula | r = Σ((x - x̄)(y - ȳ)) / [sqrt(Σ(x - x̄)²) * sqrt(Σ(y - ȳ)²)], defined as the sample covariance divided by the product of standard deviations. | R² = 1 - (SS_res / SS_tot) = SS_reg / SS_tot, where SS_res is the residual sum of squares and SS_tot is the total sum of squares. |
| Numerical Range & Bounds | Strictly bounded between -1.0 and +1.0 (-1 ≤ r ≤ +1). | Bounded between 0.0 and 1.0 (0% to 100%) in standard OLS regression with an intercept (can be negative if fitted without an intercept on out-of-sample data). |
| Directional Information (Sign) | Preserves direction: positive values (r > 0) indicate direct relationships; negative values (r < 0) indicate inverse relationships. | Non-directional: squaring eliminates algebraic signs, so R² cannot indicate whether the underlying relationship is positive or negative. |
| Scale & Unit of Interpretation | Dimensionless standardized index of co-variation; cannot be interpreted directly as a proportion or percentage of variance. | Dimensionless ratio directly interpretable as the percentage (proportion) of response variance explained by the model. |
| Variable Roles & Symmetry | Symmetric: corr(X, Y) = corr(Y, X); requires no distinction between predictor (independent) and outcome (dependent) variables. | Asymmetric: dependent on model specification; regressing Y on X yields different residual sums of squares than regressing X on Y. |
| Multiple Predictors Context | Strictly bivariate: calculates pairwise association between two individual variables at a time (matrix of pairwise correlations). | Multivariable: extends naturally to multiple regression, quantifying the collective variance explained by p predictor variables simultaneously. |
| Mathematical Relationship in Simple Regression | In simple linear regression (single predictor with intercept), r equals the signed square root of R²: r = sign(b₁) * sqrt(R²). | In simple linear regression (single predictor with intercept), R² equals the square of Pearson's r: R² = r². |
| Sensitivity to Outliers & Non-Linearity | Highly sensitive to bivariate outliers; non-linear curves can produce r near 0 despite strong underlying deterministic relationships. | Highly sensitive to influential leverage points; per NIST PMD 4.4.4, high R² alone does not confirm that a linear model is adequate without residual analysis. |
| Software Implementation (Python & R) | Python: scipy.stats.pearsonr(x, y) or pandas.Series.corr(); R: cor(x, y, method='pearson'). | Python: statsmodels.api.OLS(y, X).fit().rsquared or sklearn.metrics.r2_score(y, y_pred); R: summary(lm(y ~ x))$r.squared. |
| When to Choose | Choose for exploratory bivariate data screening, hypothesis testing of association, construct validation, and assessing feature collinearity. | Choose for regression modeling, assessing explanatory power, model comparison, predictive goodness-of-fit, and reporting variance explained. |
Sources
- SciPy pearsonr documentation (docs.scipy.org)
- APA Publication Manual 7th ed. § 6.44 (read 2026-11)
- NIST/SEMATECH e-Handbook § 4.4.4 (itl.nist.gov)
- Statsmodels RegressionResults API (statsmodels.org) (read 2026-11)
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Common questions
Common questions about Correlation Coefficient (r) vs Coefficient of Determination (R²)
What is the primary difference between the correlation coefficient and the coefficient of determination?
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The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables on a scale from -1.0 to +1.0. In contrast, the coefficient of determination (R²) measures the proportion of variance in a dependent variable explained by one or more independent variables in a regression model on a scale from 0.0 to 1.0 (0% to 100%). While r indicates whether variables move together positively or negatively, R² quantifies explanatory power and model fit without indicating direction.
Why does R² equal r² in simple linear regression but not in multiple regression?
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In simple linear regression with one predictor and an intercept, ordinary least squares mathematics ensures that the squared Pearson correlation coefficient (r²) exactly equals the ratio of regression sum of squares to total sum of squares (R²). In multiple regression with two or more predictors, R² measures the combined variance explained by the entire set of predictors, accounting for inter-correlations and collinearity among predictors. Because predictors often overlap in the variance they explain, multiple R² is not simply the sum or square of individual pairwise r values.
Can the coefficient of determination (R²) ever be negative?
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In standard Ordinary Least Squares (OLS) regression fitted with an intercept on the training data, R² is mathematically constrained to lie between 0.0 and 1.0 because the residual sum of squares cannot exceed the total sum of squares. However, R² can become negative in two specific situations: when a linear model is fitted without an intercept term (forcing the regression line through the origin), or when evaluating model predictions on an independent test dataset where the model performs worse than simply predicting the mean of the dependent variable.
How do you report r and R² in APA 7th edition style?
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Under APA 7th edition style (Section 6.44), statistics that cannot exceed 1.0 are reported without a leading zero before the decimal point. Both symbols are italicized: italic r for the correlation coefficient (e.g., r(48) = -.80, p < .001) and italic R with a superscript 2 for the coefficient of determination (e.g., R² = .64, F(1, 48) = 85.33, p < .001). When reporting multiple regression, also report adjusted R² (R²_adj = .62) to account for model complexity.
Going deeper
