Use the Coefficient of Determination Calculator (R-squared) to compute R², the statistic that tells you how much of your data’s variation a model explains. Enter either R (correlation) or the sum of squares values and the calculator returns R² and checks for invalid inputs.
What the Coefficient of Determination (R-squared) Means
The Coefficient of Determination, written as R², measures the proportion of total variation in the outcome that your model explains. If R² is 0.80, your model explains about 80% of the variation relative to a baseline that ignores predictors.
R² is most common in linear regression, but the idea—explained versus unexplained variation—also appears in other modeling contexts.
Key Idea: Explained vs. Unexplained Variation
R² compares two quantities:
- Total Sum of Squares (SST): how much the observed values vary around their mean.
- Residual Sum of Squares (SSE): how much the model’s predictions miss the observed values.
With these, R² is defined as:
R² = 1 − (SSE / SST)
Common Equivalent Forms of R-squared
You may also compute R² from the correlation coefficient R in simple linear regression. In that case:
R² = (r)²
Where r is the Pearson correlation between the predictor and the outcome.
When to Use Each Method
- Use r-based input when you have a correlation coefficient from simple linear regression.
- Use SSE/SST input when you have regression output tables or you can compute sums of squares directly.
Variables and Units (What You Should Enter)
R² is unitless. That’s because it is based on ratios of sums of squares, and the units cancel out.
- r: a number between −1 and 1.
- SSE: a non-negative number (sum of squared residuals).
- SST: a non-negative number (total sum of squares). For R² to be defined, SST must be greater than 0.
If SST is 0, all observed values are identical, and R² is not meaningful because there is no variation to explain.
How to Interpret R-squared Correctly
R² answers: “How much better is my model than predicting the mean every time?” Higher values generally indicate a better fit, but interpretation depends on context.
Use these practical rules:
- R² close to 1: the model explains most variation.
- R² near 0: the model explains little beyond the mean.
- Negative R²: can occur in some setups (especially when SSE > SST), meaning the model fits worse than the baseline.
Also remember: R² does not automatically mean causation. A strong fit can still be driven by confounders, data leakage, or overfitting.
Limitations You Must Know
R² is useful, but it has limits:
- It can increase when you add predictors, even if they don’t improve real predictive power. That’s why adjusted R² is often used.
- It can be misleading with small datasets or complex models.
- It doesn’t measure predictive accuracy on new data by itself. For that, use validation metrics (like MAE, RMSE, or cross-validated R²).
Practical Example 1: Quick Regression Fit Check
Suppose you ran a simple linear regression and got a correlation coefficient r = 0.92. In simple linear regression, R² is r².
- R² = (0.92)² = 0.8464
- Interpretation: about 84.6% of the outcome variation is explained by the predictor.
If your R² is high but residual plots show patterns, the model may still be misspecified (for example, nonlinearity or heteroscedasticity).
Practical Example 2: Using SSE and SST from Regression Output
Imagine your regression output shows:
- SSE = 120
- SST = 500
Then:
- R² = 1 − (120/500) = 1 − 0.24 = 0.76
- Interpretation: the model explains 76% of the variation relative to using the mean alone.
This is especially common when you have ANOVA-style tables or regression summaries.
How the Coefficient of Determination Calculator Works
This calculator computes R² using the method you choose:
- If you input r, it squares r to get R².
- If you input SSE and SST, it applies R² = 1 − (SSE/SST).
It also validates inputs to avoid impossible values (like negative sums of squares or SST = 0) and returns a clear error message when needed.
Frequently Asked Questions
What does an R-squared value of 0.75 mean?
An R-squared of 0.75 means your model explains about 75% of the variation in the outcome around its mean. The remaining 25% is unexplained by the model and appears in the residuals. This comparison is always relative to the mean-only baseline.
Can R-squared be negative?
Yes. While many regression settings yield R² between 0 and 1, some formulations can produce negative values when SSE is larger than SST. That means the model fits worse than predicting the outcome’s mean. Negative R² signals a poor model fit.
Is R-squared the same as correlation?
No. Correlation (r) measures the strength and direction of a linear relationship between two variables. R-squared is r² in simple linear regression, so it measures explained variance. Correlation can be negative; R-squared is always nonnegative in that case.
Does a higher R-squared always mean a better model?
Not always. R-squared often increases as you add predictors, even if they don’t improve real predictive performance. That’s why you should consider adjusted R-squared and validate on new data. Use cross-validation or test-set metrics to confirm improvement.
What’s the difference between R-squared and adjusted R-squared?
Adjusted R-squared corrects for the number of predictors used in the model. Regular R-squared can rise simply by adding variables. Adjusted R-squared only increases when the new predictors improve the fit enough to outweigh the added complexity.
Next Steps: Turn R-squared Into Action
After you compute R², check model quality beyond the single number:
- Plot residuals to look for patterns.
- Validate on a holdout set or using cross-validation.
- Consider adjusted R² when comparing models with different numbers of predictors.
When used this way, R² becomes a reliable indicator of fit quality—not a final verdict.