Formula Used
SSE = Σ(yᵢ − ŷᵢ)²
SST = Σ(yᵢ − ȳ)²
SSR = SST − SSE
n = sample size, p = predictors
For simple linear regression with an intercept, R² equals r². This equivalence does not always hold for forced-origin or more complex models.
How to Use This Calculator
- Select the calculation mode matching your available data.
- Choose a data entry method for regression calculations.
- Enter numeric values with matching series lengths.
- Select regression degree, precision, and predictor settings.
- Press Calculate R Squared to generate results.
- Review residuals, charts, warnings, and step details.
- Copy, print, or export the results when finished.
Example Data
| X | Observed Y | Possible predicted Y |
|---|---|---|
| 1 | 2.1 | 2.03 |
| 2 | 4.0 | 4.02 |
| 3 | 5.9 | 6.01 |
| 4 | 8.2 | 8.00 |
| 5 | 9.8 | 9.99 |
| 6 | 12.1 | 11.98 |
Understanding R Squared
R squared measures explained variation within a fitted statistical model. It compares model errors with variation around the observed mean. Higher values usually indicate closer agreement within the analyzed sample.
The value does not prove that a model is correct. A high result can still hide biased residuals or overfitting. Always inspect plots, assumptions, and subject knowledge before making decisions.
Adjusted R squared adds a penalty for unnecessary predictors. It can decrease when extra variables add little useful information. This makes model comparisons more balanced when predictor counts differ.
Negative R squared values are possible with poor predictions. They mean the model underperforms the simple observed-mean benchmark. Forced-origin regression can also produce unusual coefficient-of-determination behavior.
Residual statistics provide another important model-quality view. RMSE emphasizes larger errors, while MAE treats errors proportionally. Their useful scale depends on the original response variable.
Correlation and R squared describe related but different ideas. Correlation carries direction, while R squared reports explained proportion. Their simple square relationship requires specific regression conditions.
Polynomial models may increase ordinary R squared automatically. Extra flexibility can fit noise instead of meaningful structure. Compare adjusted values and residual patterns before selecting higher degrees.
No universal R squared threshold defines an acceptable model. Scientific fields, data noise, and practical costs differ widely. Interpret results using context, uncertainty, and validation evidence together.
Frequently Asked Questions
What does R squared measure?
It measures the proportion of observed variation explained by a model. The value usually ranges from zero to one. Negative values can appear when predictions perform especially poorly.
Is a higher R squared always better?
No, higher values may result from added complexity or overfitting. Residual patterns and validation performance remain important. Model purpose should guide the final evaluation.
Why can R squared be negative?
A negative value means predictions are worse than using the mean. This often occurs with weak external predictions. It may also appear under unusual fitting constraints.
What is adjusted R squared?
Adjusted R squared penalizes models for adding more predictors. It increases only when added variables improve fit sufficiently. This supports fairer comparisons between differently sized models.
Does R squared equal correlation squared?
Yes, for simple linear regression containing an intercept. The relationship may fail for forced-origin or nonlinear models. Use the correct definition for your modeling approach.
Can this calculator fit polynomial regression?
Yes, it supports polynomial degrees from one through five. Higher degrees require enough distinct data points. Review residuals carefully before accepting added complexity.
What are SSE and SST?
SSE totals squared prediction errors across observations. SST totals squared deviations from the observed mean. Their ratio determines the coefficient of determination.
How are possible outliers highlighted?
The detailed table compares residual size with twice RMSE. Large residual rows receive a visible highlight. This is a screening aid, not definitive proof.
Can I export the results?
Yes, results can be copied, printed, or downloaded. CSV exports preserve tables and summary values. PDF export creates a compact report for records.