5.7 Assessing a Regression Model Using Residuals

Residuals reveal fit

A residual = observed y − predicted y. If residuals are small and show no pattern when plotted against x (a random scatter around zero), the linear model fits well. A residual plot with a curved pattern signals that a linear model is not appropriate — a nonlinear relationship exists.

Checking for equal spread

Besides shape, check the spread of residuals across x-values. If the vertical spread of points changes noticeably (e.g., a fan or funnel shape), this indicates non-constant variability, meaning predictions are less reliable in some regions of x than others.

Outliers and influential points

Points with large residuals are outliers in the y-direction — poorly predicted by the model. Points with extreme x-values that strongly pull the regression line are influential points; removing them can noticeably change the slope or intercept, even if their residual isn't large.

Key terms

Residual

Observed y-value minus predicted y-value for a data point.

Residual plot

A scatterplot of residuals (y-axis) vs. x-values, used to check the appropriateness of a linear model.

Random scatter in residual plot

Indicates the linear model is a good fit — no systematic pattern remains.

Curved pattern in residual plot

Signals the linear model is not appropriate; relationship may be nonlinear.

Influential point

A point, often with an extreme x-value, whose removal significantly changes the regression line's slope or intercept.

Try an AP-style question

Actual y = 18, predicted ŷ = 21. What is the residual?

1. 3
2. 39
3. −3
4. 0.86