What does the residuals plot indicate when a least squares line is fitted to a scatterplot?

When analyzing a residuals plot after fitting a least squares line to a scatterplot, the primary purpose is to assess the appropriateness of the linear model used. A residuals plot displays the residuals—differences between observed and predicted values—on the vertical axis and the independent variable on the horizontal axis.

If the residuals are randomly dispersed around the horizontal axis (the line at zero), this indicates that the linear model is appropriate for the data. It suggests that there are no discernible patterns in the residuals, which would mean that the linear relationship captures the data well without systematic underestimations or overestimations.

In contrast, if the residuals show patterns such as curvature or clustering, it would suggest that a linear model might not be sufficient, indicating that the relationship could be better modeled with a non-linear approach. Therefore, observing randomly scattered residuals leads to the conclusion that the linear model is indeed suitable for the data at hand.