A residual of zero, a point right on the line

Question

A model predicts a cyclist's average speed in km per hour from the training weeks completed:

$\widehat{\text{speed}} = 8 + 1.5 \cdot (\text{weeks}).$

After $4$ weeks the cyclist's measured average speed was exactly $14$ km per hour.

What is the residual, and what does it tell you about that data point?

Accepted Answer

Compute the prediction at 4 weeks: \widehat{	ext{speed}} = 8 + 1.5 	imes 4 = 8 + 6 = 14. The residual is observed minus predicted: 	ext{residual} = 14 - 14 = 0. A residual of zero means the point sits exactly on the fitted line: the model's guess and reality agreed perfectly for that observation. This is not the model being "right" everywhere, it just happened to nail this single point. In any real dataset a few points land on or very near the line while others sit above or below; the residuals are the collection of all these vertical gaps. Takeaway. A zero residual means the observation f…

A residual of zero, a point right on the line

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