Proportion of Variation Calculator: R² Explained

This calculator works out the proportion of variation explained by a model, commonly known as R-squared (R²). You enter observed values and predicted values (or the model’s errors), and it returns R² as a decimal or percentage, plus the total sum of squares and residual sum of squares. It is a standard tool for evaluating regression models in statistics, data science, and quality control.

Formula

The proportion of variation explained, R², is defined as:

R² = 1 – (SS_res / SS_tot)

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

Both sums are in squared units of the dependent variable. R² itself is unitless and ranges from –∞ to 1 (though negative values are rare in ordinary regression with a constant term).

Worked Example

Using the default values: observed = 3,5,7,9,11; predicted = 2.8,5.2,6.9,9.1,11.2.

1. Compute the mean of observed: (3+5+7+9+11)/5 = 35/5 = 7.
2. Compute SS_tot: (3-7)² + (5-7)² + (7-7)² + (9-7)² + (11-7)² = 16+4+0+4+16 = 40.
3. Compute residuals: 3-2.8=0.2, 5-5.2=-0.2, 7-6.9=0.1, 9-9.1=-0.1, 11-11.2=-0.2.
4. Square each residual: 0.04, 0.04, 0.01, 0.01, 0.04.
5. Sum them: SS_res = 0.04+0.04+0.01+0.01+0.04 = 0.14.
6. R² = 1 – (0.14/40) = 1 – 0.0035 = 0.9965.

The calculator shows R² = 0.9965, meaning the predicted values explain 99.65% of the variation in the observed values.

How to Read the Result

R² is the fraction of the total variance in the observed data that is accounted for by the model. An R² of 0.80 means 80% of the variation is explained; the remaining 20% is unexplained. It is often multiplied by 100 and reported as a percentage.

Context matters: in physical sciences, R² above 0.99 is common; in social sciences, 0.30 might be considered strong. A high R² does not prove causation, nor does it guarantee the model is good – a model with many predictors can artificially inflate R² (see Adjusted R² below).

When to Use This Calculator

• After fitting any regression line or curve to compare fitted predictions against actual data.
• When comparing two models on the same data set – the one with higher R² explains more variance.
• In quality control or forecasting to quantify how well a model tracks observed outcomes.
• For educational purposes to verify hand calculations or spreadsheet results.

You need both the observed values and the predicted values (or the residuals). If you only have residuals and the observed mean, you can still compute SS_res and SS_tot from the data.

Common Mistakes and Misconceptions

• Mixing up which values go where: Observed values are the actual measurements; predicted values come from the model. Swapping them changes the result.
• Using mismatched data lengths: The calculator requires the same number of observed and predicted values. If one list is longer, it will produce an error.
• Interpreting R² as accuracy: R² is not the percentage of correct predictions; it is the proportion of variance explained. A model can have high R² and still be biased.
• Ignoring negative R²: If predictions are worse than simply using the mean, R² can be negative. This indicates a very poor fit.
• Assuming R² is always between 0 and 1: With non-linear models or models without an intercept, R² can exceed 1 or go negative.

Limitations and Alternatives

R² has known limitations. It always increases when you add more predictors, even if they are irrelevant. Adjusted R² corrects for the number of predictors and is better for comparing models with different sizes. Other metrics include:

• Adjusted R² – penalizes extra predictors.
• RMSE (Root Mean Square Error) – gives error in the same units as the dependent variable.
• MAE (Mean Absolute Error) – less sensitive to outliers than RMSE.
• AIC/BIC – for model comparison with a penalty for complexity.

For nonlinear models, R² is still computable but should be interpreted cautiously – the total sum of squares uses the mean, which may not be a meaningful baseline for nonlinear relationships.

Frequently Asked Questions

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 simply r² for a simple linear regression – it represents the proportion of shared variance. For multiple regression, R² is the square of the multiple correlation coefficient.

Can R² be greater than 1?

In ordinary least squares regression with an intercept, R² cannot exceed 1. However, in some nonlinear models or when using a different definition (e.g., for models without a constant term), R² can be negative or exceed 1. Always check the model type before interpreting.

What does an R² of 0 mean?

It means the model explains none of the variation in the observed data – the predictions are no better than just using the mean of the observed values. In practice, a low R² does not necessarily mean the model is useless; it might still have significant predictors but high noise.

How do I calculate R² in Excel?

Excel’s RSQ function returns R² for two data ranges: =RSQ(known_y’s, known_x’s). Alternatively, add a trendline to a scatter plot and check “Display R-squared value on chart.” For manual calculation, use the formula 1 – (SS_res/SS_tot) with the SUMSQ function.

Is a higher R² always better?

Not necessarily. A higher R² can come from overfitting – adding too many predictors that capture noise rather than true patterns. Always consider adjusted R², the model’s purpose, and out-of-sample performance. A parsimonious model with slightly lower R² may be more reliable.

What is a good R² value?

There is no universal threshold. In physics or chemistry, values above 0.95 are expected; in social sciences, 0.20 might be considered good. The context, the field, and the data quality all matter. Compare your R² to benchmarks from similar studies or models.

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in obs | Observed values (comma-separated) | - | 3,5,7,9,11 | 0
in pred | Predicted values (comma-separated) | - | 2.8,5.2,6.9,9.1,11.2 | 0
out r2 | R-squared (proportion) | - | 1 - (sum((obs-pred)^2) / sum((obs-mean(obs))^2)) | 4
out ss_res | Residual sum of squares | - | sum((obs-pred)^2) | 2
out ss_tot | Total sum of squares | - | sum((obs-mean(obs))^2) | 2