R² R-Squared Calculator
Calculate the coefficient of determination from observed and predicted values.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Calculate the coefficient of determination from observed and predicted values.
| Concept | Formula or rule |
|---|---|
| R-Squared Calculator | R² = 1 − Σ(y−ŷ)² / Σ(y−ȳ)² |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: R² compares model residual error with variation around the observed mean.
✓ Higher R² means more variation is explained, not necessarily a better causal model.
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Using R² alone to select models
Check residuals, validation performance, and scientific plausibility.
Check residuals, validation performance, and scientific plausibility.
⚠️
Assuming high R² proves accuracy for new data
Overfitting can produce high training R².
Overfitting can produce high training R².
⚠️
Interpreting R² as percent correct
It is a variance-based measure.
It is a variance-based measure.
Practice Questions
1. What does R²=0 mean relative to the mean predictor?
Answer: No improvement.
2. Can high R² imply causation?
Answer: No.
3. What error appears in the numerator?
Answer: Sum of squared residuals.
Frequently Asked Questions
Can R² be negative?▾
Yes for predictions from a model not fit with an intercept to the same data.
Is higher always better?▾
Not automatically.
What is R²=1?▾
Predictions match observations perfectly.