The idea, step by step
Observed minus predicted
A residual is e=y-ŷ. A positive residual means the observed response is above the model prediction, so the model underestimates that observation. A negative residual means the model overestimates. Residual units are the same as response units.
Use the fitted response at the same x as the observation. A residual is not y divided by ŷ, nor the difference between two adjacent responses.
Look for structure in a residual plot
Plot residuals vertically against the explanatory variable or fitted values, with a horizontal zero line. A useful linear model tends to leave a patternless cloud around zero. A curved pattern suggests that the model has missed curvature; a widening spread suggests unequal variability.
Residual checks assess model suitability. For a least-squares line with an intercept fitted to those observations, residuals sum to zero before rounding, but individual residuals are usually not zero.
A common trap
A positive residual means the model prediction was too high.
Instead: A positive observed-minus-predicted residual means the model predicted too low.
See the thinking
A signed error
For the model ŷ=5+2x, an observation is (4,16). Find and interpret its residual.
- At x=4, ŷ=5+8=13.
- e=16-13=3.
- The observed response is 3 units above the prediction.
A missed curve
Residuals at x=1,2,3,4,5 are 2,-1,-2,-1,2. What pattern is present?
| x | Residual |
|---|---|
| 1 | 2 |
| 2 | -1 |
| 3 | -2 |
| 4 | -1 |
| 5 | 2 |
- The residuals are positive at both ends and negative in the middle.
- They form a U-shaped pattern rather than random scatter.
- A curved model should be investigated.
Check your understanding
Observed y=12 and predicted y=15. Find the residual.
Show answer
-3.
Subtract predicted from observed: 12-15=-3.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- For ŷ=3+4x and the observation (2,13), find the residual.3 marks
- For ŷ=20-0.5x and the observation (6,15), find the residual.3 marks
- A residual is -4 kg. Interpret its sign and unit.3 marks
- Residuals are -0.4,0.6,-0.2,0.1,-0.1. Find their sum. Does a zero sum alone prove a good fit?3 marks
- Residual spread grows as x increases. What feature needs investigation?3 marks
- A line was chosen by eye, not fitted by least squares. Must its residuals sum to zero?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For ŷ=3+4x and the observation (2,13), find the residual.3 marks
Predicted response is 3+8=11.
Residual=13-11=2.
2. - For ŷ=20-0.5x and the observation (6,15), find the residual.3 marks
Predicted response is 17.
Residual=15-17=-2.
-2. - A residual is -4 kg. Interpret its sign and unit.3 marks
Residuals retain response units.
The observation is 4 kg below the prediction, so the model overestimates mass by 4 kg. - Residuals are -0.4,0.6,-0.2,0.1,-0.1. Find their sum. Does a zero sum alone prove a good fit?3 marks
Add the signed residuals.
A zero sum can coexist with curvature or other unsuitable patterns.
Sum=0; no. - Residual spread grows as x increases. What feature needs investigation?3 marks
The fit may have non-constant error spread.
A single typical prediction error may not describe the whole range.
The variability of errors may increase with x. - A line was chosen by eye, not fitted by least squares. Must its residuals sum to zero?3 marks
A freely chosen line is not constrained by that fitting condition.
No. The zero-sum property applies to the least-squares intercept fit to the same data.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.
Official curriculum source ↗