Coefficient of Determination Calculator (R²)

This calculator works out the coefficient of determination (R²) from two paired lists of values: your observed (actual) data and the values predicted by a model. You paste or type the two lists, and the calculator returns R² as a number between 0 and 1, which tells you how much of the variance in the observed data the model explains.

Formula

The coefficient of determination is defined as:

R² = 1 – (SS_res / SS_tot)

• SS_res (residual sum of squares) = Σ(y_i – Å·_i)², where y_i is each observed value and Å·_i is the corresponding predicted value.
• SS_tot (total sum of squares) = Σ(y_i – ȳ)², where ȳ is the mean of the observed values.

Both sums are in squared units of the dependent variable. The ratio SS_res/SS_tot is the fraction of variance left unexplained, so subtracting it from 1 gives the fraction explained.

Worked Example

Using the default lists: observed = [4, 5, 6, 7, 8], predicted = [3.8, 5.1, 5.9, 7.2, 7.9].

1. Mean of observed: (4+5+6+7+8)/5 = 6.0
2. Total sum of squares: (4–6)² + (5–6)² + (6–6)² + (7–6)² + (8–6)² = 4 + 1 + 0 + 1 + 4 = 10
3. Residuals: (4–3.8)=0.2, (5–5.1)=–0.1, (6–5.9)=0.1, (7–7.2)=–0.2, (8–7.9)=0.1
4. Residual sum of squares: 0.04 + 0.01 + 0.01 + 0.04 + 0.01 = 0.11
5. R² = 1 – (0.11 / 10) = 1 – 0.011 = 0.989

The calculator shows R² ≈ 0.989, meaning the predicted values explain about 98.9% of the variance in the observed data.

When to Use R²

Use R² when you have a model (regression line, machine learning prediction, physical law fit) and you want a single number describing how well it tracks the actual outcomes. Common applications include:

• Evaluating a linear regression fit in statistics coursework.
• Comparing two forecasting models to see which explains more variance.
• Checking the fit of a theoretical curve to experimental measurements.

R² is scale‑free, so it works whether your data is in meters, dollars, or degrees Celsius. It does not, however, tell you whether the model is appropriate – only how well it fits the given points.

How to Read the Result

R² ranges from 0 to 1 in most practical cases (it can be negative if the model is worse than simply using the mean).

• R² = 1: perfect fit – all predicted values equal observed values.
• R² = 0.8–0.99: strong explanatory power, common for well‑fitting regressions.
• R² = 0.5: the model explains half the variance; the other half is noise or missing variables.
• R² = 0: the model is no better than predicting the mean every time.
• R² < 0: the model fits worse than the mean – a sign of a badly specified model or wrong predictions.

A high R² does not prove causation. It also does not indicate whether the coefficients are statistically significant – that requires additional tests like p‑values.

Common Mistakes and Limitations

• Mismatched list lengths: the calculator will fail or give nonsense if the observed and predicted lists have different numbers of entries. Always check that each observed value has exactly one prediction.
• Ignoring outliers: a single large residual (e.g., a data entry error) can dramatically lower R². Plot the residuals to spot them.
• Using R² for non‑linear models blindly: R² is still computable, but it may not be comparable across different model forms. For non‑linear fits, consider adjusted R² or AIC.
• Confusing R² with correlation r: R² is the square of the Pearson correlation coefficient for a simple linear regression. For a linear model, r = √R², but the sign of r is lost – R² is always non‑negative.
• Overfitting: adding more predictors always increases R², even if they are irrelevant. Use adjusted R² when comparing models with different numbers of terms.

R² vs. Adjusted R²

Adjusted R² modifies the standard R² to penalize adding extra predictors. The formula is:

Adjusted R² = 1 – (1 – R²) × (n – 1) / (n – k – 1)

where n is the number of observations and k is the number of predictors. When you add a useless variable, R² goes up slightly, but adjusted R² may go down. Use adjusted R² when comparing models with different numbers of predictors, especially in multiple regression.

Frequently Asked Questions

What does R² = 0.75 mean in plain words?

It means that 75% of the variation in your observed values is explained by the model. The remaining 25% is due to other factors, measurement error, or randomness. It does not mean the predictions are 75% accurate – accuracy is measured by error metrics like RMSE.

Can R² be negative?

Yes. This happens when the model’s predictions are worse than simply using the mean of the observed data. For example, if you fit a line to data that is strongly curved, the residuals can be so large that SS_res exceeds SS_tot, making R² negative. Many software packages report negative R² as 0, but the true value is negative.

What is the difference between R² and correlation coefficient r?

The correlation coefficient r measures the strength and direction of a linear relationship between two variables, ranging from –1 to +1. R² is the square of r (for simple linear regression) and represents the proportion of shared variance. R² is always between 0 and 1, and it loses the direction information – you can’t tell if the relationship is positive or negative from R² alone.

How do I calculate R² for a non‑linear model?

The same formula works: R² = 1 – (SS_res / SS_tot). You just need the observed values and the model’s predictions. However, R² for non‑linear models is not directly comparable to R² from linear models, and it can be misleading. For non‑linear fits, consider using the standard error of the estimate or information criteria like AIC.

Why does my calculator show a different R² than Excel?

Excel’s RSQ function and most statistical software compute R² the same way, but rounding differences or using the wrong list order can cause small discrepancies. Also, Excel’s chart trendline R² for a linear fit is identical to the manual formula, but for polynomial fits it uses a different calculation. Double‑check that you have not swapped observed and predicted values – R² is symmetric for linear regression, but not for non‑linear models.

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list obs | Observed values | 4, 5, 6, 7, 8
list pred | Predicted values | 3.8, 5.1, 5.9, 7.2, 7.9
out r2 | Coefficient of determination (R²) | - | 1 - (sum((obs - pred)^2) / sum((obs - mean(obs))^2)) | 4
out r | Correlation coefficient (r) | - | sqrt(max(0, r2)) | 4