Math 1 · Unit 10 · Lesson 3 · part 2 of 5

A residual: actual minus predicted

Every model is a little bit wrong.

A residual: actual minus predicted

Every model is a little bit wrong. The points are not exactly on the line, and for each point there is a gap between what really happened and what the model said. That gap has a name.

residual = actual − predicted

A residual is the actual y minus the predicted y, for one point. It is how far off the model is for that point. On the graph it is the vertical distance from the point to the line.

Positive residualThe actual value is bigger than the prediction. The point is above the line. The model guessed too low.

Negative residualThe actual value is smaller than the prediction. The point is below the line. The model guessed too high.

Zero residualThe point is exactly on the line. The model got that one right.

Always subtract in the same order, actual minus predicted, so that positive means above.

A state college's tuition from 1990 to 2000, with t = 0 for 1990. A suggested model is R(t) = 326t + 6440. Fill in a predicted column by putting each t into the model, then a residual column by subtracting.

year t	actual tuition ($)	predicted: 326t + 6440	residual: actual − predicted
0	6,500	6,440	+60
1	6,656	6,766	−110
2	7,187	7,092	+95
3	7,378	7,418	−40
4	7,874	7,744	+130
5	7,985	8,070	−85
6	8,376	8,396	−20
7	8,797	8,722	+75
8	8,928	9,048	−120
9	9,424	9,374	+50
10	9,665	9,700	−35

Check one: at t = 4 the model predicts 326(4) + 6440 = 7,744. The actual tuition was 7,874. Residual: 7,874 − 7,744 = +130. The point is above the line, and the model guessed $130 too low.

Where to go next

Next: The residual plot: random scatter or a pattern?One residual tells you about one point. All the partsResiduals, Checking the Fit