What is R-squared (R²)?

R-squared (R²) measures how much better a regression model explains the target values compared with a simple baseline that always predicts the mean target value.

It is also called the coefficient of determination.

Formula

R² = 1 - (SS_res / SS_tot)

where:

• SS_res is the sum of squared residuals from the model
• SS_tot is the total squared deviation of the actual values from their mean

Intuition

Suppose the baseline predicts the average target for every observation.

R² compares the model's squared error with the error from that mean baseline.

If the model reduces the error substantially, R² becomes larger.

How to interpret R²

R² = 1.0

means the model predicts the evaluated data perfectly.

R² = 0

means the model performs no better, in squared-error terms, than simply predicting the mean target value.

For example:

R² = 0.80

is commonly interpreted as the model accounting for about 80% of the variation in the target relative to the mean baseline.

Can R² be negative?

Yes.

A negative R² means the model performs worse than the mean-prediction baseline on the evaluated data.

This can happen especially on validation or test data when a model generalizes poorly.

Is a higher R² always better?

A higher R² generally indicates a better fit on the same dataset, but it should not be used alone.

A model can have a high R² and still:

• overfit the training data
• make errors that are too large for the business problem
• show systematic patterns in its residuals
• rely on unnecessary features

R² does not show error size

R² is unitless.

It does not directly tell you whether predictions are off by ₹100, ₹10,000, or another practical amount.

That is why it is often used together with metrics such as MAE or RMSE.

Be careful when comparing models

R² is most meaningful when models are evaluated on the same target and the same dataset.

Adding more predictors to an ordinary regression model can also increase training R² even when the new features are not useful, which is one reason adjusted R² is sometimes used.

In short, R² measures how much a regression model improves over predicting the target mean in terms of squared error. Higher values usually indicate a better fit, but R² should be interpreted together with error metrics and validation performance.