R-Squared (R²) Calculators

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R-squared (R²), the coefficient of determination, measures the proportion of variance in the dependent variable that is explained by the independent variable(s) in a regression model. It ranges from 0 to 1 — R² = 0 means the model explains none of the variance; R² = 1 means it explains all of it. R² is one of the most widely reported regression statistics, but it is also frequently misinterpreted — a high R² does not guarantee a good model, does not imply causation, and increases automatically with additional predictors in multiple regression.

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R² Formula

R² = 1 − SS_res / SS_tot

Where:

  • SS_res (residual sum of squares) = Σ(yᵢ − ŷᵢ)² — variation not explained by the model
  • SS_tot (total sum of squares) = Σ(yᵢ − ȳ)² — total variation in y
  • ŷᵢ = predicted value; ȳ = mean of y

Equivalently: R² = SS_reg / SS_tot = proportion of total variance explained by the regression.

Interpreting R²

  • R² = 0.85: Model explains 85% of the variance in y; 15% is unexplained
  • R² near 1: Very high fit — model predicts y closely
  • R² near 0: Model performs barely better than the mean of y as a predictor

Context matters: R² = 0.3 may be excellent in psychology or ecology (noisy biological data); R² = 0.95 may be expected in physics or analytical chemistry.

R² vs. Pearson's r

  • r (Pearson correlation coefficient): Measures strength and direction of linear relationship: −1 to +1
  • R² = r² for simple linear regression only: R² = 0.81 means r = ±0.90
  • For multiple regression, R² is the proportion of variance explained by all predictors combined; it cannot be converted back to r

Limitations of R²

  • R² always increases when predictors are added (even uninformative ones) — use adjusted R² to penalize for extra predictors
  • High R² ≠ good model (check residual plots for pattern)
  • Does not indicate whether the relationship is causal
  • Cannot compare R² values between models with different dependent variables
  • R² is uninformative for non-linear relationships assessed with linear regression

Glossary

R² (Coefficient of Determination)
The proportion of variance in the dependent variable explained by the regression model: R² = 1 − SS_residual/SS_total. Ranges 0–1. Does not imply causation and increases with additional predictors.
Adjusted R²
A modified R² that penalizes for adding predictors that don't improve model fit: 1 − [(1−R²)(n−1)/(n−p−1)]. Use instead of R² when comparing models with different numbers of predictors.
Residual Sum of Squares (SS_res)
Σ(observed − predicted)². The unexplained variation in y after fitting the regression model. SS_res = SS_total × (1 − R²). Minimized by ordinary least squares regression.

Frequently Asked Questions

R² (coefficient of determination) = 1 − SS_residual/SS_total. It measures what proportion of the variance in the dependent variable (y) is explained by the regression model. R² = 0.75 means the model accounts for 75% of y's variance; 25% is unexplained residual variation. R² ranges 0–1; values closer to 1 indicate a better fit.

There is no universal standard — it depends on the field. In physics or analytical chemistry, R² > 0.99 is expected for controlled experiments. In ecology, biology, or psychology, R² = 0.3–0.6 may be considered reasonable given natural variability. In econometrics, R² varies widely. Always interpret R² relative to the complexity of the phenomenon, the number of predictors, and whether residual patterns indicate model adequacy, not by a single threshold.

Adjusted R² = 1 − [(1 − R²)(n − 1)/(n − p − 1)], where n = sample size and p = number of predictors. Unlike R², adjusted R² penalizes for adding unhelpful predictors — it decreases when an added predictor explains less variance than expected by chance. Use adjusted R² (instead of R²) when comparing regression models with different numbers of predictors or evaluating whether adding variables genuinely improves model fit.

No — R² measures statistical association, not causation. A high R² tells you the model accounts for much of the variance in y, but does not indicate whether x causes y. Classic example: ice cream sales and drowning rates both correlate with summer temperature — high R² for each, but neither causes the other. Causal inference requires experimental design, randomization, or directed causal reasoning beyond regression statistics.