What Is R² and How Does It Measure Model Fit?

The coefficient of determination, written as $R^2$ or sometimes $r^2$, measures how well a regression model accounts for variation in the dependent variable. A value closer to $1$ usually indicates a better fit to the evaluated data.

How Is It Calculated?

In machine learning, $R^2$ compares the model's prediction errors with the errors produced by a simple baseline that always predicts the mean:

$$R^2 = 1 - \frac{\sum_{i=1}^{n}(y_i-\hat{y}_i)^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}$$

Here, $y_i$ is an actual value, $\hat{y}_i$ is its predicted value, and $$ is the mean of the actual values. The numerator is the , while the denominator is the total variation around the mean.

For example, suppose the actual values are $2, 4, 6$, with mean $4$, and the predictions are $3, 4, 5$:

$$\sum_{i=1}^{3}(y_i-\hat{y}_i)^2 = 1+0+1=2$$ $$\sum_{i=1}^{3}(y_i-\bar{y})^2 = 4+0+4=8$$

Therefore, $R^2=1-\frac{2}{8}=0.75$. The model accounts for $$ of the variation relative to the mean-value baseline.

When comparing models, calculate $R^2$ on the same test dataset because scores from different datasets are not directly comparable. Training data measures how closely the model fits examples it has already seen; test data better indicates generalization. However, $R^2$ alone does not show the typical size or direction of prediction errors. Examine residuals and other evaluation metrics where appropriate.

A common misconception is that $R^2$ must lie between $0$ and $1$. It can be negative, especially when evaluating an unsuitable model on unseen test data. A high $R^2$ also does not prove causation or successful generalization; overfitting may produce a high training score but a lower test score.

IB Exam Technique

For A4.3, define $R^2$, interpret it relative to the mean baseline, and evaluate model fit using test data. Do not claim that a high value automatically means the model is reliable.