What is R² score in regression?

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🔹 Definition

• R² measures the proportion of variance in the dependent variable (Y) that is explained by the independent variables (X) in the regression model.

• It compares the fitted model against a simple baseline (the mean of Y).

🔹 Formula

$$R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$$

Where:

• $SS_{res} = \sum (y_i - \hat{y}_i)^2$ → Residual Sum of Squares (errors from predictions).

• $SS_{tot} = \sum (y_i - \bar{y})^2$ → Total Sum of Squares (variance of actual values).

• $y_i$ = actual values

• $\hat{y}_i$ = predicted values

• $\bar{y}$ = mean of actual values

🔹 Interpretation

• R² = 1 → Perfect fit (model explains 100% of variance).

• R² = 0 → Model is no better than predicting the mean.

• R² < 0 → Model performs worse than predicting the mean (poor fit).

🔹 Example

Suppose we predict house prices based on square footage:

• Actual prices: [200k, 220k, 250k]

• Predicted prices: [210k, 215k, 240k]

• Mean price: 223k

If $R² = 0.85$, that means 85% of the variation in house prices is explained by square footage in the model, and 15% is due to noise or missing factors.

🔹 Limitations of R²

• Doesn’t measure accuracy directly (just proportion of explained variance).
  Can increase with more features (even irrelevant ones). → That’s why Adjusted R² is used for multiple regression.
  Not useful for non-linear models in some cases.

✅ In short:
The R² score tells you how much of the variation in your target is explained by your regression model, ranging from 0 (useless) to 1 (perfect fit), but it must be used carefully alongside other metrics like MAE, RMSE, and Adjusted R².

Read more :

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