Coefficient of Determination (R²) Calculator

StatisticsLast updated: August 22, 2026

The coefficient of determination R² tells you what percentage of the total variability in the dependent variable is explained by the model. The total sum of squares SST splits into an explained (regression) part SSR and an unexplained (residual) part SSE; R² is the ratio SSR/SST and takes values between 0 and 1. A value such as 0.72 means that 72% of the variability in Y is explained by the predictors.

This calculator computes R² from your data, the adjusted R² that penalizes the number of predictors, and the standard error of the estimate (SEE); it lays out the sum-of-squares decomposition and the complete regression ANOVA table (SS, df, MS, F, p). For single-predictor models it also links R² numerically to the correlation coefficient r, so you can see how the two measures convert into one another.

The observed values of the variable you want to explain.
With a single predictor you get the R² of a simple regression (the square of r); with several lines you get the multiple coefficient of determination.
Sets the false positive risk. In the social sciences 5% is the common choice.

Coefficient of Determination Formulas

Total:         SST = Σ(yᵢ − ȳ)²
Regression:    SSR = Σ(ŷᵢ − ȳ)²
Residual:      SSE = Σ(yᵢ − ŷᵢ)²
Decomposition: SST = SSR + SSE
Determination: R² = SSR / SST = 1 − SSE / SST
Adjusted:      R²_adj = 1 − (1 − R²)·(n − 1)/(n − k − 1)
Std. error:    SEE = √[ SSE / (n − k − 1) ]
ANOVA:         F = (SSR / k) / (SSE / (n − k − 1)) ,  df = k , n − k − 1
One predictor: R² = r²  (r: the Pearson correlation coefficient)
Several:       R = √R² = corr(y, ŷ)

The identity SST = SSR + SSE holds only for least-squares models that include an intercept. In regression through the origin the decomposition breaks down and R² is defined differently; the no-intercept R² values reported by software are not comparable.

How to Calculate

  1. Paste the observed values of the dependent variable (Y) into the upper box.
  2. Write one predictor per line in the predictor box; a single line gives the R² of a simple regression, several lines give the multiple coefficient of determination.
  3. Choose the significance level; the decision of the F test is made against it.
  4. Study the SST = SSR + SSE decomposition and the pie chart to see the explained share.
  5. Compare R² with the adjusted R²: a wide gap means the model carries too many predictors for its number of observations.
  6. Use the F and p values in the ANOVA table to decide whether the model is significant as a whole, and report SEE as the practical size of the prediction error.

Worked Examples

One predictor: study hours and exam score

Of the SST of 1,049.67, the model explains 425.90 and leaves 623.77 as residual: R² = 0.4057 (40.6%), the adjusted R² is 0.3463, and SEE = 7.8979. With F(1, 10) = 6.828 and p = 0.026 the model is significant; because there is a single predictor, r = 0.6370 and the identity r² = R² holds.

Coefficient of determination (R²): 0.4057 (40.6%) · Adjusted R²: 0.3463 · Gap between R² and adjusted R²: 0.0594

Two predictors: R² rises, but adjusted R² is the real criterion

Adding absences as a second predictor lets the model explain 97.2% of the variability in Y: R² = 0.9717, adjusted R² = 0.9655, and the gap between them is only 0.0063 — since both rise together, the increase reflects a genuine contribution. The residual sum of squares falls to 29.66 and the standard error of the estimate to 1.8152; the model is significant with F(2, 9) = 154.780 at p < 0.001, and the multiple correlation is R = √R² = 0.9858.

Coefficient of determination (R²): 0.9717 (97.2%) · Adjusted R²: 0.9655 · Gap between R² and adjusted R²: 0.0063

Inflated R²: small sample, many predictors

When four predictors that have no real relationship with Y are fitted to 7 observations, R² = 0.5000 (50.0%) looks reasonable: 41.71 of the SST of 83.43 appears explained purely because of the number of predictors. The adjusted R², however, drops to -0.5001, and the gap of 1.0001 shows the size of the penalty. With only 2 residual degrees of freedom left, F(4, 2) = 0.500 and p = 0.750, so the model is not even significant: a high R² alone does not mean a good model.

Coefficient of determination (R²): 0.5000 (50.0%) · Adjusted R²: -0.5001 · Gap between R² and adjusted R²: 1.0001

Frequently Asked Questions

What counts as a good R²?
There is no universal threshold; the acceptable level depends on the field. In physics and engineering measurements anything below 0.90 may be considered low, while in studies of human behavior an R² of 0.20-0.30 is valuable. By Cohen's widely used classification, 0.02 is a small, 0.13 a medium, and 0.26 or more a large effect.
Why does R² inflate as predictors are added?
Least squares drives the residual sum of squares (SSE) to the smallest value it can reach. Adding a new column widens the solution space, so SSE can never increase; since R² = 1 − SSE/SST, R² can never decrease either. Even a completely random variable catches some small pattern by chance and lifts R² a little. In the extreme, k = n − 1 predictors give R² = 1 and the model passes through every point — while having learned nothing. That is why model comparison uses the adjusted R², which penalizes the number of predictors.
How is adjusted R² computed, and why can it be negative?
It is computed as R²_adj = 1 − (1 − R²)·(n − 1)/(n − k − 1); every extra predictor shrinks the denominator and raises the penalty. If the model explains almost nothing (R² very close to zero) the expression can turn negative. A negative adjusted R² means the predictors carry less information about Y than the mean does, and the whole model needs to be reconsidered.
How are R² and the correlation coefficient r related?
In a simple regression with one predictor, R² equals the square of the Pearson correlation coefficient: if r = 0.64 then R² = 0.41. In multiple regression the coefficient of determination is the square of the multiple correlation coefficient: R = corr(y, ŷ), the correlation between the observed values and the model's predictions. R is always positive, whereas r carries the direction of the relationship in its sign.
What is the standard error of the estimate (SEE) good for?
SEE = √[SSE/(n − k − 1)] is the average size of the prediction error, expressed in the same unit as Y. Because R² is a proportion, it can call a model 'good' while the prediction error remains unacceptably large in practice. Roughly 68% of the observations lie within ±1 SEE of the regression line and 95% within ±2 SEE.
Does a high R² show that the model is correct?
No. R² only measures how tightly the model fits; it says nothing about whether the functional form is right, whether the assumptions hold, or whether the relationship is causal. A linear model forced onto a curved relationship can still return a high R², and a handful of extreme observations can raise it artificially. Never interpret R² without inspecting the residual plot.
What does the F test in the ANOVA table examine?
It tests H₀: all slope coefficients are zero, that is, R² = 0. The larger the ratio F = (SSR/k)/(SSE/(n−k−1)), the greater the explained variance relative to the error variance and the smaller p becomes. In a single-predictor model this F test gives exactly the same result as the t test of the slope (F = t²).