Calculate R², adjusted R², SS decomposition (SST/SSR/SSE), F-statistic, standard error, and residual analysis. Full ANOVA decomposition with interpretation guide.
R² — the coefficient of determination — is the single most reported metric for regression model quality. It answers one question: what fraction of the outcome's variation does the model explain? Our calculator computes R², adjusted R², the full ANOVA sum-of-squares decomposition, F-statistic, and standard error from your data.
Enter X and Y values and instantly see SS_Total decomposed into SS_Regression (explained) and SS_Residual (unexplained). The ANOVA identity — SS_Total = SS_Regression + SS_Residual — is shown explicitly so you can verify the mathematics. The residual table reveals where your model fits well and where it struggles.
The interpretation guide contextualizes your R² within different fields: 0.95 is unremarkable in physics but extraordinary in psychology. Understanding that R²=0.60 means "60% explained, 40% unexplained" is essential for honest statistical communication. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
R² is the lingua franca of regression analysis. Every research paper, business report, and data science project reports it. Understanding the decomposition SS_Total = SS_Explained + SS_Residual gives you deep insight into how models work.
This calculator makes the decomposition transparent. Instead of a single R² number, you see exactly how much variance the model captures vs leaves on the table. The residual analysis helps determine whether a better model form might reduce SS_Residual.
R² = 1 − SS_Res/SS_Tot = SS_Reg/SS_Tot. Adjusted R² = 1 − (1−R²)(n−1)/(n−p−1). F = (SS_Reg/p)/(SS_Res/(n−p−1)). SE = √(SS_Res/(n−p−1)).
Result: R² = 0.9988, Adjusted R² = 0.9986, SS_Tot = 165.90, SS_Reg = 165.70, SS_Res = 0.20
99.88% of Y's variation is explained by the linear model. Only 0.12% (SS_Res = 0.20) is unexplained, indicating excellent fit.
All regression analysis rests on one identity: SS_Total = SS_Regression + SS_Residual. SS_Total measures total variation in Y around its mean. SS_Regression measures how much of that variation the model captures. SS_Residual is what remains unexplained.
R² is simply the fraction SS_Reg/SS_Tot. When R² = 0.85, the model captures 85% of variation; the remaining 15% is due to factors not in the model or random noise.
ANOVA (Analysis of Variance) formally tests whether SS_Regression is "large enough" relative to SS_Residual. The F-statistic = (SS_Reg/p) / (SS_Res/(n-p-1)), where p is the number of predictors. Under the null hypothesis (model explains nothing), F follows an F-distribution. Large F → small p-value → model is significant.
1. "Higher R² = better model" — false. Overfitting gives high R² but poor predictions. 2. "R² shows causation" — false. It measures association only. 3. "R² < 0.5 means the model is useless" — false in social sciences where R² = 0.10 can represent an important finding. 4. "R² is always between 0 and 1" — false for adjusted R² and when using non-OLS methods.
R² always increases when you add more predictors, even useless ones. Adjusted R² penalizes for each additional predictor, decreasing if the predictor doesn't improve the model enough. Use adjusted R² when comparing models with different numbers of predictors.
Standard R² ranges 0 to 1. However, adjusted R² can be negative if the model fits worse than simply using the mean as a prediction. If adjusted R² is negative, your model is adding noise, not signal.
For simple linear regression (one predictor), R² = r² exactly. For multiple regression, R² is the square of the multiple correlation coefficient R, which differs from individual Pearson correlations.
F tests whether the model explains significantly more variance than expected by chance. Large F means the model is statistically significant. It's the ratio of per-predictor explained variance to per-observation unexplained variance.
It depends entirely on context. In physics: poor. In economics: decent. In social psychology: excellent. Always compare R² to typical values in your specific field, not to an abstract standard.
No — overfitting. Adding enough predictors always increases R² toward 1.0, but the model becomes worthless for prediction. Use adjusted R², AIC, BIC, or cross-validation to balance fit with simplicity.