R Squared Calculator

Calculator

Choose your input style, then submit to compute R².

Mode x and y pairs (fit regression) actual and predicted (evaluate model)

Pick regression fitting or evaluation-only.

Input source Paste data Editable table

Paste is best for long datasets.

Decimals 2 3 4 5 6 7 8 9 10 11 12

Controls display rounding only.

Intercept Include intercept (y = a + b x) Force through origin (y = b x)

Affects the fitted line and interpretation.

Adjusted R² Show when applicable Hide

Shown only with intercept-based fitting.

Load example data Clear inputs

Paste data (two columns per line)

Supported separators: comma, semicolon, tab, or spaces. One pair per line.

Editable table

Add rows as needed. Blank rows are ignored.

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Example data table

Use this dataset to test the calculator quickly.

Formula used

R² is computed as:

Notes: SSE is the sum of squared errors and SST is total sum of squares. Adjusted R² is shown only when an intercept is included.

What R² Represents

R² describes the share of total outcome variance explained by the fitted line. When SST is positive, R² = 1 − SSE/SST, so lower squared error yields higher fit. For example, R² = 0.80 suggests the model explains about 80% of the variability around the mean. In linear regression with an intercept, R² equals the squared Pearson correlation between x and y. Negative values can appear when predictions are worse than using the mean as a baseline.

Intercept Choice Matters

With an intercept (y = a + b x), the fit is centered at x̄ and ȳ, making R² comparable across many datasets. Forcing the line through the origin (y = b x) can be valid for proportional processes, but it changes residual structure and can distort R² if measurements include offsets or units do not start at zero. Use domain knowledge to justify the constraint, not only a higher score.

Residuals and RMSE

R² alone can hide large typical errors. The residual table highlights row‑level differences: residual = actual − predicted. The calculator also reports RMSE = sqrt(SSE/n), which stays in y units and is easier to interpret operationally. A high R² with a high RMSE usually means the model tracks direction well but misses the required accuracy threshold. Look for patterns such as increasing spread with x, which can signal non‑constant variance.

Data Quality Checks

Stable R² needs enough spread and enough points. With fewer than 10 observations, one outlier can shift slope and R² noticeably. If all x values are identical, the slope cannot be estimated; if all y values are identical, SST becomes zero and R² is undefined. Scan the table for duplicates, entry errors, and extreme leverage points before trusting the summary.

Reporting and Comparison

Use the export options to keep results reproducible. The CSV stores inputs, predictions, and residuals for audits, peer review, and quick charts. The PDF captures a compact snapshot of R², SSE, SST, RMSE, and key parameters. When comparing models, compute R² on the same hold‑out data and confirm improvements are consistent across residual patterns. If you include an intercept, the calculator can also show adjusted R² to reflect model complexity.

FAQs

1) What is considered a “good” R²?

It depends on the field and noise level. In controlled processes, 0.90+ may be expected, while in human behavior data, 0.20–0.50 can still be useful. Always compare against a baseline and check residuals.

2) Can R² be negative?

Yes. If SSE exceeds SST, the model performs worse than simply predicting the mean of y for every point. This often signals a poor functional form, incorrect data pairing, or inappropriate constraints.

3) Why does the calculator sometimes say R² is undefined?

R² uses SST in the denominator. If all y values are identical, SST becomes zero, so explained variance cannot be computed. Add variation in y or choose a dataset where outcomes actually change.

4) When should I force the intercept to zero?

Only when theory and measurement support a true zero point, such as direct proportionality with reliable origin calibration. For many datasets, forcing zero creates biased residuals and misleading fit statistics.

5) How do RMSE and R² work together?

R² summarizes relative fit, while RMSE measures absolute error size in y units. Two models can have similar R² but different RMSE. Use RMSE to judge whether errors are acceptable for your use case.

6) Does a higher R² guarantee better predictions?

No. R² can improve by overfitting or by fitting trends that do not generalize. Validate on hold‑out data, inspect residual patterns, and confirm that improvements persist across new samples before adopting a model.