R-Squared & Adjusted R-Squared Calculator (Multiple Regression

Statistical Foundations of R-Squared (R^2) & Adjusted R-Squared

In regression modeling, the coefficient of determination (R^2) quantifies the proportion of variance in the dependent variable explained by independent predictors:

• The Flaw of Raw R^2: Standard R^2 has a fundamental mathematical bias: adding any explanatory variable to a regression model—even completely random white noise—will mathematically never decrease R^2 and will almost always increase it. This incentivizes overfitting.

• The Adjustment Penalty: Adjusted R^2 (derived by Mordecai Ezekiel and Henri Theil) adjusts for sample size n and predictor count p. It only increases if the newly added variable reduces residual variance enough to compensate for the loss of a degree of freedom (specifically, if |t| > 1.0).

• Negative Adjusted R^2: If a model contains many useless predictors and has an extremely low raw R^2, Adjusted R^2 can become negative, indicating that the model performs worse than simply predicting the sample mean.

How to Use This Calculator

Enter your Input Mode — choose from From Unadjusted R-Squared (R²) + n + Number of Predictors (p), From Sums of Squares (SS_res and SS_total), Unadjusted R-Squared (R²) (e.g. 0.785 (Between 0.00 and 1.00)), Residual Sum of Squares (SS_res / SSE - if mode is SS) (e.g. 215), Total Sum of Squares (SS_total / SST - if mode is SS) (e.g. 1000), Sample Size (n / Total Observations) (e.g. 45), and Number of Independent Predictors (p / k - Excluding Intercept) (e.g. 4). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Multiple Regression R^2 \& Adjusted R^2 (Theil-Ezekiel) formulas:

R^2 = 1 - SS_resSS_total = SS_regSS_total

R_adj^2 = 1 - [ ((1 - R^2)(n - 1)) / (n - p - 1) ] = 1 - SS_res / (n - p - 1)SS_total / (n - 1)

Degrees of Freedom: df_model = p, df_error = n - p - 1, df_total = n - 1

Overall Model ANOVA F-Statistic: F = (R^2 / p) / ((1 - R^2) / (n - p - 1))

Shrinkage Penalty: Δ R^2 = R^2 - R_adj^2

Worked Example

For example, evaluating a multiple regression model with \ observations and \ predictors that achieves an unadjusted Degrees of freedom for error are . The Adjusted R^2 is 1 - [ ((1 - 0.785)(44)) / (40) ] = 1 - [ (0.215 × 44) / (40) ] = 1 - 0.2365 = 0.7635 (76.35\%). The shrinkage penalty is . The overall F-statistic is \ (p < 0.0001).

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.