In the context of regression, what is a residual?

A residual in the context of regression analysis represents the difference between the actual observed value of the dependent variable and the value predicted by the regression model. It quantifies the error in the prediction, providing insight into how well the model fits the data. By examining these differences across all observations, one can evaluate the model's performance, identify patterns that the model may not be capturing, and potentially improve the model by addressing those discrepancies.

Residuals are crucial for diagnosing model adequacy and checking for assumptions, such as homoscedasticity and normality. They are fundamental in determining how much of the variance in the dependent variable is explained by the independent variables in the regression model.

The other options describe concepts related to regression but do not accurately define a residual. For instance, the difference between independent and dependent variables does not capture the error involved in predictions. The average distance between the data points and the regression line describes a concept related to fit but is not a residual itself. Although the total sum of residuals should approximate zero, this characteristic doesn't define what a residual is but reflects a property of the least squares method used in regression analysis.