The idea, step by step
Distinguish an outlier from an influential point
A point can be far from the general response pattern, or far from the other explanatory values. A point that substantially changes a fitted equation is influential. A large residual does not automatically imply large influence; its position on the predictor axis matters too.
First check whether the value was recorded or paired incorrectly. A genuine unusual observation is still information about the population.
Compare, explain and retain an audit trail
Fit with and without a questioned point as a sensitivity analysis. Compare slope, intercept, r and predictions. If a verified error can be corrected, use the corrected value and explain the correction. If a valid point is excluded because it belongs to a different population, state the criterion and report its effect.
Never delete a point merely to increase r² or obtain a preferred answer. When the cause is unclear, report both analyses and the uncertainty.
A common trap
Delete every point that lowers r².
Instead: Selection based on improving the fit can distort the evidence. Investigate validity and report sensitivity.
See the thinking
Measure an effect
Fit the supplied five points, then compare with the first four.
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 20 |
- All five give ŷ=-4+4x.
- The first four follow y=2x exactly.
- The fifth observation changes the slope; investigate whether it is a valid unusual case or a recording error.
Correct a verified error
A table records a 170 cm person as 1700 cm. The original measurement sheet confirms 170 cm. What should the analyst do?
- Correct the transcription to 170 cm and document the source of the correction.
- Recalculate the fitted model from the corrected paired data.
- Do not describe this as removing an inconvenient genuine person.
Check your understanding
A valid point is unusual but its cause is unknown. What is a responsible report?
Show answer
Show the fit’s sensitivity with and without it, explain the uncertainty, and avoid silently deleting it.
Validity and relevance, not a desired correlation, govern treatment.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- For (1,2),(2,4),(3,6),(4,8),(5,20), calculate the fitted slope using all observations and compare with the first four.3 marks
Paired measurements x y 1 2 2 4 3 6 4 8 5 20 - An unusually high score is confirmed as correct. Is this alone a reason to delete it?3 marks
- An isolated point has a very large x-value but lies close to a regression line. Could it be influential?3 marks
- Two reports use the same data, but one silently omits three low outcomes. State the problem.3 marks
- An all-data model predicts 40 and the model without one point predicts 55 for the same x. What does this show?3 marks
- Suggest a justified reason to exclude a correctly recorded point from a particular analysis.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For (1,2),(2,4),(3,6),(4,8),(5,20), calculate the fitted slope using all observations and compare with the first four.3 marks
Paired measurements x y 1 2 2 4 3 6 4 8 5 20 For all five, Sxy/Sxx gives the fitted slope.
The first four are exactly y=2x.
All-data slope 4; first-four slope 2. - An unusually high score is confirmed as correct. Is this alone a reason to delete it?3 marks
Investigate whether it belongs to the target population.
Report justified sensitivity rather than silently improving the fit.
No. A genuine score should not be excluded merely because it is unusual. - An isolated point has a very large x-value but lies close to a regression line. Could it be influential?3 marks
Influence is about changing the fit, not just distance vertically from it.
Yes. Its extreme predictor location can strongly affect the slope even with a small residual. - Two reports use the same data, but one silently omits three low outcomes. State the problem.3 marks
A report must disclose exclusions and give a defensible criterion.
Selective omission can bias the model and mislead readers. - An all-data model predicts 40 and the model without one point predicts 55 for the same x. What does this show?3 marks
The difference is 15 response units.
Do not hide the sensitivity behind a single prediction.
The prediction is sensitive to that observation; further investigation and both results should be reported. - Suggest a justified reason to exclude a correctly recorded point from a particular analysis.3 marks
The population criterion should be scientifically relevant and disclosed, not chosen to maximise correlation.
It may be from outside the explicitly defined target population, such as a different machine type.
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 ↗