Calculator
Choose an input mode, paste your data, and compute R squared, error metrics, and residual diagnostics.
Example data table
| Row | Observed (y) | Predicted (ŷ) | Residual |
|---|---|---|---|
| 1 | 3.10 | 3.00 | 0.10 |
| 2 | 4.90 | 5.10 | -0.20 |
| 3 | 6.80 | 6.60 | 0.20 |
| 4 | 8.20 | 8.40 | -0.20 |
| 5 | 9.90 | 10.00 | -0.10 |
You can paste two columns for y and ŷ, two columns for x and y, or a multi-column CSV with headers for multiple regression.
Formula used
- SST = Σ(yᵢ − ȳ)² total variance in y
- SSE = Σ(yᵢ − ŷᵢ)² unexplained error
- SSR = Σ(ŷᵢ − ȳ)² explained variance
- R² = 1 − (SSE / SST)
- Adj R² = 1 − (1 − R²)·(n−1)/(n−k−1)
- k is predictor count (not intercept).
Error metrics: MSE = SSE / n, RMSE = √MSE, and MAE = mean(|y − ŷ|).
How to use this calculator
- Select a mode based on the data you have.
- Paste your dataset using commas or tabs.
- For multiple regression, set the target header name.
- Press Calculate to see results above the form.
- Download a CSV or PDF report when needed.
Professional notes
Interpreting R Squared in practice
R squared measures the share of variance in y explained by the fitted values ŷ. If SST is 250 and SSE is 50, then R² = 1 − 50/250 = 0.80, meaning 80% of variability is captured. High values are common in smooth physical systems, but can be rare in noisy behavioral data. Always review the scale of RMSE to understand practical error. Note that R² can be negative when SSE exceeds SST, signaling worse than predicting ȳ.
Adjusted R Squared for fair comparisons
Adding predictors never increases SSE, so raw R² can rise even when new variables add little signal. Adjusted R² corrects this using n and k: Adj R² = 1 − (1 − R²)·(n−1)/(n−k−1). With n=40, k=6, and R²=0.72, adjusted R² drops to about 0.67, warning that complexity may be excessive. Use it to compare models with different k, especially when k approaches n.
Residual patterns that change decisions
Residuals should look random around zero. Systematic curvature, widening spread, or clusters often imply nonlinearity, heteroscedasticity, or missing features. Track residual mean, standard deviation, and extremes to spot drift. Durbin–Watson near 2 suggests low autocorrelation, while values below 1.5 can indicate time-order dependence that inflates apparent fit. Investigate large residuals for outliers, leverage points, or data entry errors.
Why input format and cleaning matter
This calculator supports observed vs predicted pairs, simple x–y regression, and multi-column models. Missing values, duplicated rows, and unit mismatches can distort SST and SSE, moving R² dramatically. In multiple regression, collinearity can make the matrix singular and invalidate coefficients. Prefer standardized columns, consistent units, and at least k+2 complete rows. When variables differ by orders of magnitude, scaling can improve stability.
Reporting results with context
Use R² alongside RMSE, MAE, and a small row preview to communicate fit and error. When sharing, include the model type, intercept choice, predictor list, and sample size. Export the CSV for audits and the PDF for stakeholders. If R² is near 1.0, confirm you are not evaluating on training data only. For performance claims, report validation or test-set R² and note preprocessing steps.
FAQs
1) What does R squared tell me?
R squared is the proportion of variation in the target explained by the model’s predictions. It is calculated from sums of squares and ranges widely by domain, noise level, and data scale.
2) Can R squared be negative?
Yes. If the model’s SSE is larger than SST, then R² becomes negative. That means the model performs worse than predicting the mean of y for every row.
3) Why is adjusted R squared sometimes lower?
Adjusted R squared penalizes extra predictors using n and k. If a new feature does not reduce SSE enough, adjusted R squared falls, warning that added complexity is not justified.
4) Which mode should I choose?
Use Observed & Predicted when you already have ŷ values. Use x & y for a quick simple regression fit. Use multiple regression when you have a headered CSV with several predictors.
5) How much data do I need for multiple regression?
You need more complete rows than predictors. A practical minimum is k+2 valid rows, but stable estimates usually require far more, especially when predictors are correlated or noisy.
6) Should I rely on R squared alone?
No. Pair R squared with RMSE, MAE, and residual checks. A high R squared can still hide bias, leakage, or autocorrelation. When possible, report performance on a separate validation or test set.