R-squared (R²)
Also known as: R², coefficient of determination, r2 score
The coefficient of determination, denoted R², measures the proportion of variance in the dependent variable explained by the independent variables in a regression model. Introduced by Karl Pearson in the late 19th century, R² is one of the most widely used metrics for assessing how well a model fits observed data.
Key highlights
- Highly interpretable: directly shows the proportion of variance explained
- Scale-independent, allowing comparison across different datasets
- Simple to calculate and widely understood across disciplines
- Provides immediate feedback on model quality
Intuition
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How it works
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When to use it
R² is essential for evaluating linear regression models and other predictive models. Use it when you want a single summary statistic of overall model fit. However, be aware that R² can be misleading: it always increases with more predictors (even irrelevant ones), and high R² does not guarantee good predictions outside the training data. It assumes linear relationships and is sensitive to outliers.
Strengths & limitations
- Highly interpretable: directly shows the proportion of variance explained
- Scale-independent, allowing comparison across different datasets
- Simple to calculate and widely understood across disciplines
- Provides immediate feedback on model quality
- Increases mechanically with additional predictors, even if they have no real effect
- Assumes linear relationships; inappropriate for nonlinear models
- Can be misleading on small datasets or with extreme outliers
- Does not directly measure prediction accuracy on future data
Common pitfalls
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Applications
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Frequently asked
Can R² be negative?
Yes. A negative R² occurs when your model performs worse than simply predicting the mean for every observation. This typically indicates a severely misspecified model (e.g., fitting a linear regression to highly nonlinear data) or incorrect calculation.
Why does R² increase when I add more variables, even useless ones?
This is called overfitting. Adding any variable (even random noise) reduces residual sum of squares slightly by random chance. Use adjusted R² or information criteria (AIC, BIC) to penalize extra parameters.
Is a high R² guarantee that my model will predict well on new data?
No. High training R² often signals overfitting. Always validate on held-out test data. Use cross-validation or compute prediction intervals on independent data.
How does R² differ from correlation coefficient r?
R² is the square of the Pearson correlation coefficient for simple linear regression. However, for multiple regression or nonlinear models, R² is defined differently and cannot be interpreted as a squared correlation.
Should I use R² to compare models fit to different datasets?
With caution. R² depends on the variance structure of the target variable. If one dataset has much higher variance than another, R² values are not directly comparable. Use cross-validation or other standardized metrics instead.
Sources
- 1.Pearson, K. (1896). Mathematical contributions to the theory of evolution. Philosophical Transactions of the Royal Society A, 187, 253-318.
- 2.Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 559-572.
- 3.Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society A, 222, 309-368.
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Cite this page
ScholarGate. (2026, June 3). R-squared. ScholarGate. https://scholargate.app/model-evaluation/r-squared