In the context of a linear regression, what does the term "residual" refer to?

The term "residual" in the context of linear regression is defined as the difference between observed values and predicted values. This concept is fundamental in regression analysis as it quantifies how far off the model's predictions are from the actual data points. For a given data point, the residual is calculated by subtracting the predicted value (obtained from the regression equation) from the actual observed value.

Thus, if the actual value of the dependent variable is higher than the predicted value, the residual will be positive, indicating that the model under-predicted; conversely, if the actual value is lower than the predicted value, the residual will be negative, showing that the model over-predicted that value. Analyzing the residuals helps assess the accuracy of the model and can indicate whether the linear model is appropriate for the data. It also plays a critical role in diagnosing potential problems with the model, such as non-linearity or outliers.

By understanding the role of residuals, one can enhance the model's effectiveness and identify areas where improvements may be needed, making this term a key element in regression analysis.