What is residual sum of squares?

 The Residual Sum of Squares (RSS), also known as the Sum of Squared Errors (SSE), is a measure of the discrepancy between the actual data points and the values predicted by a regression model.

πŸ“Œ Definition:

RSS=∑(Yi−Y^i)2\text{RSS} = \sum (Y_i - \hat{Y}_i)^2

Where:

  • YiY_i = actual value of the dependent variable

  • Y^i\hat{Y}_i = predicted value from the regression model

  • Yi−Y^iY_i - \hat{Y}_i = residual or error term

🎯 What Does RSS Represent?

RSS quantifies the total amount of variation in the dependent variable that is not explained by the regression model.

  • A smaller RSS means the model's predictions are closer to actual values → better fit.

  • A larger RSS indicates poor model fit, with more prediction errors.

πŸ“Š Where It Fits in Total Variance

In regression or ANOVA:

Total Sum of Squares (SST)=Explained (SSR)+Residual (RSS or SSE)\text{Total Sum of Squares (SST)} = \text{Explained (SSR)} + \text{Residual (RSS or SSE)}
Type of Sum of SquaresFormulaMeaning
SST (Total)∑(Yi−YΛ‰)2\sum (Y_i - \bar{Y})^2
Total variance in data
SSR (Regression)∑(Y^i−YΛ‰)2\sum (\hat{Y}_i - \bar{Y})^2
Explained by model
RSS or SSE (Residual)∑(Yi−Y^i)2\sum (Y_i - \hat{Y}_i)^2
Not explained by model (errors)

🧠 Finance Example:

Suppose you're predicting monthly sales (₹ in '000s) based on advertising spend. You have the following data:

ObsActual Sales (Y)Predicted Sales (ΕΆ)Residual (Y - ΕΆ)Residual²
1220215525
2270265525
3310305525
RSS=25+25+25=75RSS = 25 + 25 + 25 = 75

So, the model left ₹75,000 (in squared units) of sales unexplained. The lower this number, the better your model.

πŸ” Summary:

TermMeaning
RSS / SSESum of squared deviations of actual values from predicted values
IndicatesError or unexplained portion in the regression model
Ideal ValueAs low as possible (but not zero unless overfitting)
Used InComputing R², Mean Squared Error (MSE), F-ratio

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