5.8 Using Residual Plots to Assess a Linear Model

What residual plots show

A residual plot graphs residuals (x on the x-axis, residuals on y-axis) to check if a linear model is appropriate. If the linear model fits well, points should show no pattern — a random, uniform scatter around the horizontal line at zero. A curved pattern in the residual plot signals that a linear model is not appropriate, even if the scatterplot looked fairly linear.

Checking for equal variability

Besides checking for curvature, look at the spread of residuals across all x-values. If the residuals fan out or narrow as x increases (unequal spread), this suggests the variability of y around the line changes — violating the assumption of constant variance, which makes predictions less reliable in some regions of x.

Why residual plots matter

Residual plots amplify deviations from linearity that are hard to see in a regular scatterplot. Always examine both randomness of scatter (no curve) and consistency of spread (no fanning) before trusting a linear model's predictions. A good linear fit shows residuals randomly scattered with roughly constant spread around zero.

Key terms

Residual plot

A scatterplot of x-values (or predicted values) against residuals, used to assess if a linear model fits the data.

Residual

Observed y-value minus predicted y-value (residual = actual − predicted).

Curved pattern in residual plot

Indicates the linear model is not appropriate; a non-linear model may fit better.

Random scatter in residual plot

Indicates the linear model is a good fit for the data.

Fanning pattern in residuals

Residuals with increasing or decreasing spread across x-values, indicating non-constant variance.

Try an AP-style question

A residual plot shows random scatter with no pattern. What does that suggest?

1. A curved model is needed
2. The data have no relationship
3. The correlation is zero
4. A linear model is appropriate