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Sum of Squared Residuals

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The sum of squared residuals, also known as the residual sum of squares (RSS), is a statistical measure used in regression analysis to quantify the goodness of fit of a model. It represents the total amount of variation in the dependent variable that is not explained by the independent variables in the regression model.

5 Must Know Facts For Your Next Test

  1. The sum of squared residuals is calculated by summing the squares of the differences between the observed and predicted values for the dependent variable.
  2. A lower sum of squared residuals indicates a better fit of the regression model to the data, as it means the model is able to explain more of the variation in the dependent variable.
  3. The sum of squared residuals is used in the calculation of the coefficient of determination (R-squared), which is a measure of the proportion of the variance in the dependent variable that is explained by the independent variables in the regression model.
  4. Minimizing the sum of squared residuals is the goal of the ordinary least squares (OLS) regression method, which is a common technique for fitting linear regression models.
  5. The sum of squared residuals can be used to compare the fit of different regression models, with the model having the lowest sum of squared residuals being the preferred model.

Review Questions

  • Explain how the sum of squared residuals is calculated and its relationship to the goodness of fit of a regression model.
    • The sum of squared residuals is calculated by summing the squares of the differences between the observed values of the dependent variable and the predicted values based on the regression model. A lower sum of squared residuals indicates a better fit of the model to the data, as it means the model is able to explain more of the variation in the dependent variable. The sum of squared residuals is a key component in the calculation of the coefficient of determination (R-squared), which is a measure of the proportion of the variance in the dependent variable that is explained by the independent variables in the regression model.
  • Describe the role of the sum of squared residuals in the ordinary least squares (OLS) regression method.
    • The ordinary least squares (OLS) regression method is a common technique for fitting linear regression models, and it aims to minimize the sum of squared residuals. This means that the OLS method seeks to find the regression coefficients that result in the smallest possible sum of squared differences between the observed and predicted values of the dependent variable. By minimizing the sum of squared residuals, the OLS method ensures that the regression model provides the best possible fit to the observed data.
  • Analyze how the sum of squared residuals can be used to compare the fit of different regression models and determine the most appropriate model for the data.
    • The sum of squared residuals can be used to compare the fit of different regression models, with the model having the lowest sum of squared residuals being the preferred model. This is because a lower sum of squared residuals indicates that the model is better able to explain the variation in the dependent variable. By comparing the sum of squared residuals across multiple regression models, researchers can determine which model provides the best fit for the data and is most appropriate for making inferences or predictions. This evaluation of model fit using the sum of squared residuals is a crucial step in the regression analysis process, as it helps ensure that the chosen model is reliable and valid for the research question at hand.

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