Calculation history
Adjusted R-squared formulas
Model with an intercept
Model without an intercept
R² is the coefficient of determination. n is the observation count. p is the predictor count.
The adjustment penalizes predictors that add little explanatory value. It can decrease when unnecessary variables enter the model. Negative values can occur for weak models.
How to use this calculator
- Select summary mode when R-squared is already known.
- Select dataset mode to derive R-squared from prediction pairs.
- Enter the number of independent predictors.
- Confirm whether the regression includes an intercept.
- Choose the desired decimal precision.
- Calculate and inspect adjusted fit, penalties, and warnings.
- Use model comparison to rank alternative specifications.
- Export the results as CSV or PDF.
Example regression inputs
| Model | Observations | Predictors | R-squared | Adjusted R-squared |
|---|---|---|---|---|
| Baseline | 100 | 2 | 0.6800 | 0.6734 |
| Expanded | 100 | 4 | 0.7200 | 0.7082 |
| Complex | 100 | 6 | 0.7350 | 0.7179 |
The complex model has the strongest adjusted score here. The improvement remains small relative to added complexity. Cross-validation should guide the final choice.
Limitations and assumptions
Adjusted R-squared measures in-sample explanatory fit. It does not measure causal validity or calibration. It cannot replace residual diagnostics.
Compare models fitted to identical response observations. Different datasets can make rankings misleading. Use cross-validation for predictive selection.
Check linearity, independence, homoscedasticity, and residual behavior. Investigate multicollinearity before interpreting coefficients. Domain relevance should guide predictor choices.
Frequently asked questions
What is adjusted R-squared?
It estimates explained variation while penalizing model complexity. The penalty depends on observations and predictors. It helps compare nested regression models.
Why is adjusted R-squared lower than R-squared?
Ordinary R-squared never decreases after adding predictors. Adjusted R-squared applies a complexity penalty. Weak predictors can therefore lower it.
Can adjusted R-squared be negative?
Yes, weak models can produce negative values. This indicates poor adjusted explanatory performance. Reconsider predictors and model form.
Should I always choose the highest value?
Not automatically. Consider validation performance, interpretability, assumptions, and cost. Small differences may not justify complexity.
Does adjusted R-squared detect overfitting?
It provides a useful complexity signal. It cannot fully detect out-of-sample overfitting. Cross-validation remains more reliable.
What counts as a predictor?
Count estimated independent-variable terms in the model. Include dummy variables and transformations separately. Do not count the intercept.
Can I use actual and predicted values?
Yes, enter paired lists or upload CSV data. The calculator derives SSE, SST, and R-squared. Both lists need equal lengths.
Why does the intercept option matter?
Centered and uncentered models use different baselines. Their adjustment formulas also differ. Compare values only under compatible specifications.
Is a high adjusted R-squared sufficient?
No, high fit can coexist with biased estimates. Inspect residuals and validation results. Confirm assumptions and practical usefulness.