Calculator inputs
Multiple-model comparison
| Model | R² | Adjusted R² | SSE | RMSE | MAE | Predictors | Best | Action |
|---|
Comparison entries are stored in this browser only.
Formula used
Standard R-squared
R² = 1 − [Σ(yᵢ − ŷᵢ)² ÷ Σ(yᵢ − ȳ)²]
Adjusted R-squared
Adjusted R² = 1 − (1 − R²) × [(n − 1) ÷ (n − p − 1)]
Here, yᵢ is actual, ŷᵢ is predicted, and ȳ is the actual mean. The value n represents observations, while p represents predictors. Weighted mode multiplies each squared term by its sample weight.
How to use
- Select standard, weighted, or summary-statistics mode.
- Enter actual and predicted values with matching lengths.
- Add sample weights when weighted mode is selected.
- Enter the predictor count for adjusted R-squared.
- Choose precision, then submit the calculator form.
- Review metrics, residuals, charts, and interpretation.
- Export results or add models for comparison.
Example data
| Observation | Actual | Predicted | Weight |
|---|---|---|---|
| 1 | 3 | 2.8 | 1 |
| 2 | 5 | 5.4 | 1 |
| 3 | 7 | 6.6 | 1 |
| 4 | 9 | 9.3 | 1 |
| 5 | 11 | 10.7 | 1 |
| 6 | 13 | 13.2 | 1 |
Interpretation guide
| R-squared range | General interpretation |
|---|---|
| Below 0 | Worse than predicting the actual-value mean. |
| 0 | No improvement over the mean baseline. |
| Above 0 to below 0.50 | Limited explanatory power. |
| 0.50 to below 0.75 | Moderate explanatory power. |
| 0.75 to below 0.90 | Strong explanatory power. |
| 0.90 and above | Very strong explanatory power. |
These ranges are only broad guidance. Acceptable scores vary across domains and datasets. Always inspect errors, bias, leakage, and validation quality.
Important limitations
R-squared measures explained variance, not practical usefulness. A high score can hide biased predictions. It also cannot prove a causal relationship.
Adding predictors never lowers ordinary training R-squared. Adjusted R-squared applies a complexity penalty. Test-set evaluation gives a more realistic estimate.
Outliers may strongly influence squared-error calculations. Nonlinear patterns can also mislead simple interpretations. Compare R-squared with MAE, RMSE, and residual plots.
Frequently asked questions
1. What does R-squared measure?
R-squared estimates the proportion of target variance explained by predictions. It compares model error against mean-baseline error. Higher values usually indicate closer overall fit.
2. Can R-squared be negative?
Yes, negative values are possible on evaluated data. They mean predictions perform worse than the mean baseline. This often signals poor generalisation or unsuitable modelling.
3. Is an R-squared of one perfect?
An R-squared of one means zero squared residual error. Training data may still contain leakage or overfitting. Validate performance on unseen data before trusting it.
4. Why use adjusted R-squared?
Adjusted R-squared penalises unnecessary predictors in a model. It can decrease when weak features are added. This supports fairer comparison across model complexities.
5. What predictor count should I enter?
Enter the number of independent input features used. Do not count the response variable itself. Treat engineered columns as predictors when the model uses them.
6. When should weighted R-squared be used?
Use weights when observations have unequal importance or reliability. Each weight scales its squared residual and deviation. Zero weights exclude observations from weighted sums.
7. Why is R-squared undefined sometimes?
R-squared needs variation among actual target values. Identical actual values make total variance equal zero. The standard ratio is then mathematically undefined.
8. Should I prefer R-squared over RMSE?
Neither metric is universally better for evaluation. R-squared describes relative variance explained. RMSE describes error magnitude in target units.
9. Can I compare different datasets?
Direct comparison across different datasets can be misleading. Target variance and noise may differ greatly. Compare models under consistent data and validation conditions.