Explained Variation Calculator: R² Made Simple

This calculator works out the explained variation (R²) of a regression model from the sum of squares values you enter. You provide the regression sum of squares (SSR) and the total sum of squares (SST), and it returns the proportion of variance explained by the model, plus the residual sum of squares and the coefficient of determination as a percentage. The result tells you how well your independent variables account for the variation in your dependent variable.

Formula

The explained variation is the ratio of the regression sum of squares to the total sum of squares:

R² = SSR / SST

Where:

• SSR (regression sum of squares, also called SS_explained) – the variation in the dependent variable that the model accounts for.
• SST (total sum of squares) – the total variation in the dependent variable around its mean.
• SSE (residual sum of squares, also called SS_unexplained) – the variation left unexplained, computed as SST − SSR.

All three values are measured in the same squared units as the dependent variable. The ratio is unitless and always falls between 0 and 1 for ordinary least squares regression with an intercept.

Worked Example

Using the default values from the calculator (SSR = 80, SST = 100):

1. Compute R² = 80 / 100 = 0.8
2. Compute SSE = 100 − 80 = 20
3. Convert to percentage: 0.8 × 100 = 80%

The model explains 80% of the total variation, leaving 20% unexplained. The calculator shows R² = 0.8, SSE = 20, and 80%.

When to Use Explained Variation

Explained variation is the standard measure of fit for linear regression. Use it when you want to know how much of the variability in your outcome (e.g., house prices, test scores, crop yield) is captured by your predictor variables (e.g., square footage, study hours, rainfall). It is reported in nearly every regression output, often labeled R-squared or coefficient of determination.

It is also used in ANOVA to describe effect size (eta-squared), in finance to assess how much of a fund’s return is explained by a benchmark, and in quality control to evaluate measurement systems. Whenever you have a model that predicts a numeric outcome, explained variation gives you a quick, intuitive summary of its explanatory power.

How to Read the Result

R² values range from 0 to 1 (or 0% to 100%). A value of 1 means the model perfectly explains all variation; 0 means it explains none. In practice, values above 0.7 are often considered strong in social sciences, while physical sciences may expect 0.9 or higher. However, context matters – a good R² in one field may be poor in another.

Be careful: a high R² does not prove causation, and a low R² does not mean the model is useless. It only measures linear association for the sample you have. Also, adding more predictors will always increase R², even if they are irrelevant, so many analysts use adjusted R² instead when comparing models with different numbers of predictors.

Common Mistakes and Pitfalls

• Using the wrong sums of squares: Make sure SSR and SST come from the same model and same dataset. Mixing values from different models gives meaningless results.
• Confusing SSR with SSE: SSR is the explained variation, SSE is the unexplained (residual). They add up to SST, so if you have two of them, you can find the third.
• Ignoring the intercept: The standard R² formula assumes the model includes an intercept. Without one, R² can be negative or exceed 1, and the usual interpretation breaks down.
• Rounding too early: If you round SSR and SST before dividing, you can get noticeably different R². Keep full precision until the final step.
• Treating R² as a percentage in formulas: When doing further calculations, use the decimal form (0.8), not 80.

Limitations and Alternatives

Explained variation has well-known limitations. It does not tell you whether the model is correctly specified, whether the relationship is nonlinear, or whether the predictors are statistically significant. A high R² can occur with a badly misspecified model, and a low R² can occur with a useful but noisy model.

Common alternatives include:

• Adjusted R²: Penalizes adding extra predictors, useful for model comparison.
• RMSE (root mean square error): Measures prediction error in the original units, more interpretable for practical purposes.
• AIC/BIC: Information criteria for comparing models, balancing fit and complexity.
• Correlation coefficient (r): For simple linear regression, R² equals the square of the Pearson correlation, but r also gives direction.

These measures complement R² rather than replace it; a thorough analysis reports several of them.

Frequently Asked Questions

What is the difference between explained variation and R²?

None – they are the same thing. Explained variation is the conceptual name for the proportion of total variance that the model accounts for, and R² (the coefficient of determination) is the numeric measure. Both equal SSR/SST.

Can R² be negative?

Yes, but only in special cases. In ordinary least squares with an intercept, R² is always between 0 and 1. It can go negative when the model is forced through the origin (no intercept), when using a different fitting method, or when evaluating predictions on new data that are worse than simply using the mean.

What is a good R² value?

There is no universal cutoff. In fields like physics or chemistry, values above 0.9 are typical; in social sciences, 0.5 may be considered strong. The best way to judge is to compare with other models on the same outcome and consider the practical significance of the unexplained variation.

How do I calculate R² if I only have SSE and SST?

Use the relationship SST = SSR + SSE. Rearrange to get SSR = SST − SSE, then compute R² = (SST − SSE) / SST = 1 − (SSE/SST). This is a common shortcut when software reports residual sum of squares.

Does a high R² mean the model is good?

Not necessarily. A high R² can be achieved by overfitting, by including many irrelevant predictors, or by having too few data points. Always check residual plots, statistical significance, and validate the model on new data before trusting it.

What is adjusted R² and how is it different?

Adjusted R² modifies the standard R² to account for the number of predictors and sample size. It increases only if a new predictor improves the model more than expected by chance, making it more honest for model comparison. It is always lower than or equal to R².

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in ssr | Regression SS (SSR) | - | 80 | 0
in sst | Total SS (SST) | - | 100 | 0
out r2 | R² (explained variation) | - | ssr / sst | 4
out ssres | Residual SS (SSE) | - | sst - ssr | 4
out pct | Explained variation % | % | (ssr / sst) * 100 | 2