Coefficient of Determination (R²)
Proportion of variance in the target explained by the model.
Formula
Variables
Example
R² = 0.85 means 85% of variance is explained
How it works
R² compares the squared prediction error of a model to the total squared variation you would see if you always predicted the mean of the target. If the model fits better than the mean-only baseline, the ratio of leftover error is small and R² moves toward 1; if it fits no better than the mean, R² is 0. It is built from sums of squared deviations because squaring penalizes large misses and makes positive and negative errors count equally.
When to use it
Use R² when evaluating regression models on continuous targets, such as predicting house prices, sales, or temperatures. It is especially useful for comparing how much better a model is than a simple average-based baseline on the same dataset.
Worked example, step by step
Actual values: y = [10, 12, 9, 15] Predicted values: ŷ = [11, 11, 10, 14] Step 1: Compute the mean of actual values ȳ = (10 + 12 + 9 + 15) / 4 = 46 / 4 = 11.5 Step 2: Compute the residual sum of squares Σ(y_i - ŷ_i)² = (10-11)² + (12-11)² + (9-10)² + (15-14)² = 1 + 1 + 1 + 1 = 4 Step 3: Compute the total sum of squares Σ(y_i - ȳ)² = (10-11.5)² + (12-11.5)² + (9-11.5)² + (15-11.5)² = 2.25 + 0.25 + 6.25 + 12.25 = 21 Step 4: Plug into the formula R² = 1 - 4/21 = 1 - 0.190476... R² ≈ 0.81 Final answer: the model explains about 81% of the variance in the target values.
Common mistakes to avoid
- Thinking R² is the same as accuracy or a percent of correct predictions
- Assuming R² can only be between 0 and 1, even though it can be negative for poor models
- Using R² to compare models across different datasets without considering that the target variance may differ
Did You Know?
R² can be negative if the model is worse than predicting the mean.
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More in Data Science & ML
View allMean Squared Error (MSE)
BasicAverage of squared differences between predicted and actual values.
Root Mean Squared Error (RMSE)
BasicSquare root of MSE, in the same units as the target variable.
Mean Absolute Error (MAE)
BasicAverage of absolute differences between predictions and actual values.
Classification Accuracy
BasicFraction of predictions that are correct.
Precision
IntermediateOf all positive predictions, the fraction that were correct.
Recall (Sensitivity)
IntermediateOf all actual positives, the fraction correctly identified.