What do residuals represent in a regression analysis?

Residuals in a regression analysis are defined as the differences between the observed values and the predicted values of the dependent variable. Specifically, they measure how far each observed outcome deviates from the value that the model predicts based on its regression equation. This is crucial because residuals provide insight into the accuracy of the regression model; smaller residuals indicate a better fit of the model to the data.

Understanding residuals is essential, as they can also help identify patterns in the data that the model may not adequately capture. For example, if residuals display a random pattern, it suggests that a linear model is appropriate. However, if residuals show a structured pattern, it may indicate that the model needs refinement, such as inclusion of higher-order terms or transformations of variables.

The other options do not correctly capture the essence of what residuals represent in regression analysis. The total sum of squares refers to the total variability in the dependent variable, while the average of predicted values does not reflect the differences that residuals represent. Similarly, variability of independent variables relates to the spread of the predictor variables, which is not directly tied to the concept of residuals.