Coefficient of Determination (R²) Calculator

About the Coefficient of Determination (R²) Calculator

R², the coefficient of determination, summarizes how much of the outcome's variation is explained by a regression model. It is one of the most commonly reported fit statistics, but it is easiest to interpret when it is tied back to the underlying sum-of-squares breakdown.

This calculator shows that decomposition directly: total variation, explained variation, and residual variation. It also reports adjusted R², the F-statistic, and standard error so you can place the headline R² in a fuller regression context.

That makes the page useful both for quick model-fit checks and for understanding what an R² value actually means beyond a single decimal number.

When This Page Helps

R² is often quoted without context, which makes it easy to overstate what a model has accomplished. Showing the explained and unexplained pieces side by side makes the statistic more concrete and keeps the discussion anchored in what the model still misses.

How to Use the Inputs

1. Enter X and Y data points (comma-separated).
2. Or select a preset to see different fit qualities.
3. Review R², adjusted R², and correlation r.
4. Examine the ANOVA decomposition (SS_Total = SS_Reg + SS_Res).
5. Check F-statistic for overall model significance.
6. Analyze residuals for patterns indicating model inadequacy.
7. Use the interpretation guide to contextualizes your R² value.

Example Calculation

Tips & Best Practices

• R² = 1 doesn't mean the model is correct — it means the line passes through every data point (possible with 2 points).
• Always check residuals visually. A curved pattern in residuals means the linear model is wrong, regardless of R².
• Use adjusted R² when comparing models with different numbers of variables.
• A high F-statistic (> 10 or so) with p < 0.05 confirms the model explains real variance.
• Standard error is in Y-units — it directly tells you typical prediction accuracy.
• For time series, R² can be misleadingly high due to shared trends — always detrend first.

The SS Decomposition: Heart of Regression

All regression analysis rests on one identity: SS_Total = SS_Regression + SS_Residual. SS_Total measures total variation in Y around its mean. SS_Regression measures how much of that variation the model captures. SS_Residual is what remains unexplained.

R² is simply the fraction SS_Reg/SS_Tot. When R² = 0.85, the model captures 85% of variation; the remaining 15% is due to factors not in the model or random noise.

ANOVA and F-test

ANOVA (Analysis of Variance) formally tests whether SS_Regression is "large enough" relative to SS_Residual. The F-statistic = (SS_Reg/p) / (SS_Res/(n-p-1)), where p is the number of predictors. Under the null hypothesis (model explains nothing), F follows an F-distribution. Large F → small p-value → model is significant.

Common R² Misconceptions

1. "Higher R² = better model" — false. Overfitting gives high R² but poor predictions. 2. "R² shows causation" — false. It measures association only. 3. "R² < 0.5 means the model is useless" — false in social sciences where R² = 0.10 can represent an important finding. 4. "R² is always between 0 and 1" — false for adjusted R² and when using non-OLS methods.

Frequently Asked Questions

• Why use adjusted R² instead of R²?

  R² always increases when you add more predictors, even useless ones. Adjusted R² penalizes for each additional predictor, decreasing if the predictor doesn't improve the model enough. Use adjusted R² when comparing models with different numbers of predictors.

• Can R² be negative?

  Standard R² ranges 0 to 1. However, adjusted R² can be negative if the model fits worse than simply using the mean as a prediction. If adjusted R² is negative, your model is adding noise, not signal.

• What's the relationship between R² and r?

  For simple linear regression (one predictor), R² = r² exactly. For multiple regression, R² is the square of the multiple correlation coefficient R, which differs from individual Pearson correlations.

• What does the F-statistic tell me?

  F tests whether the model explains significantly more variance than expected by chance. Large F means the model is statistically significant. It's the ratio of per-predictor explained variance to per-observation unexplained variance.

• Is R² = 0.50 good or bad?

  It depends entirely on context. In physics: poor. In economics: decent. In social psychology: excellent. Always compare R² to typical values in your specific field, not to an abstract standard.

• Can I just maximize R²?

  No — overfitting. Adding enough predictors always increases R² toward 1.0, but the model becomes worthless for prediction. Use adjusted R², AIC, BIC, or cross-validation to balance fit with simplicity.