R-Squared

Statistics · Axiom Academy

Understanding the Coefficient of Determination R² (R-squared), also called the coefficient of determination , measures the proportion of variance in the dependent variable that is predictable from the independent variable(s). 2 R² = r² for Simple Regression For simple linear regression (one predictor variable), R² equals the square of the correlation coefficient r. This connects our understanding of correlation to model fit. R² is interpreted as the percentage of variation in y that is explained by x . It ranges from 0 to 1, where higher values indicate better model fit. R² = 0.25 → 25% of variation explained (weak model) R² = 0.50 → 50% of variation explained (moderate model) R² = 0.75 → 75% of variation explained (strong model) R² = 1.00 → 100% of variation explained (perfect fit) R² is always bounded between 0 and 1. Understanding these extreme cases helps us interpret intermediate values. R² is a key measure of model quality, but it must be interpreted carefully. Higher R² generally means better fit, but context matters. Context matters: A "good" R² varies by field (0.5 may be excellent in social sciences, poor in physics) Not the only metric: Check residual plots, significance tests, and practical relevance Adjusted R²: For multiple regression, use adjusted R² to account for number of predictors Causation vs. correlation: High R² doesn't imply causation, just predictive relationship

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