Construct and interpret a residual plot

Residual plots

A residual is the signed difference between an observed response and the response predicted by a fitted line. A residual plot uses explanatory values horizontally and residuals vertically to assess whether the departures have a clear pattern

1. Find the predicted response Substitute the explanatory value into the fitted line before calculating the residual \(\displaystyle \widehat{y}=12+5\times4=32\)
2. Subtract in the correct order Calculate the residual as the actual response minus the predicted response \(\displaystyle e=35-32=3\)
3. Keep the residual sign A point below the fitted line has a negative residual, while a point above it has a positive residual \(\displaystyle y<\widehat{y}\Rightarrow e<0,\quad y>\widehat{y}\Rightarrow e>0\)
4. Use the residual-plot axes Put the explanatory variable horizontally and the signed residual vertically, with zero as the reference line \(\displaystyle (x,e)=\left(4,3\right),\quad e=0\)

Worked example

A fitted model predicts an assessment score from preparation time. The fitted line and case values are shown below. Calculate the predicted score and residual, give the residual-plot coordinate, and state what irregular scatter around zero suggests

1. \(\displaystyle \widehat{y}=18+4\times6=42\)Substitute the preparation time into the fitted model
2. \(\displaystyle e=y-\widehat{y}=39-42=-3\)Subtract the predicted score from the actual score and keep the sign
3. \(\displaystyle (x,e)=\left(6,-3\right)\)Plot the explanatory value horizontally and the signed residual vertically
4. \(\displaystyle \text{irregular scatter about }e=0\Rightarrow\text{linear model supported}\)Irregular residual scatter around zero supports using a linear model for the data

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