Understand the idea.
Locate the prediction in the data range
Compare the requested predictor value with the observed -range before substituting. Interpolation uses an input inside that range; extrapolation uses one outside it. An unmeasured input between two observed inputs still counts as interpolation.
Extrapolation relies on a pattern continuing beyond the evidence and is generally more uncertain. Interpolation still depends on a suitable model, representative data and residual variation. Being inside the range does not guarantee that a particular individual will match the prediction.
Check the situation as well as the arithmetic
Check whether the calculated response has appropriate units and respects the situation. Negative fuel, impossible proportions or forecasts for a different population can reveal limits that correct substitution alone does not address. The applicable domain may be smaller than the algebraic domain.
Report a prediction as an estimate and state its scope. Consider the data range, gaps in observations and whether conditions have changed. A model that fitted one period or group need not keep working under new conditions.
See the method in action.
An interior prediction
A model fitted at ages 4–12 years is h=25+6a. Predict at age 8 and classify it.
Worked solution
- 1
Compare the requested age with the data range.
Eight years lies inside the observed predictor interval, so the prediction is interpolation.
- 2
Substitute the age into the model.
The predictor is age, so the input is 8 rather than a previous response value.
- 3
State the prediction and its scope.
The model predicts 73 units at age 8. This is an interpolated estimate, whose reliability still depends on the fit and the population represented by the data.
73 units; interpolation.
A mathematically valid but impossible result
Fuel data from 0–300 km produce F=45-0.09d litres. Evaluate at 600 km.
Worked solution
- 1
Locate the requested distance.
The requested input is outside the observed 0–300 km range, making it extrapolation.
- 2
Evaluate the equation at that distance.
Substitution gives the algebraic output of the fitted line.
- 3
Check the physical constraint.
Fuel remaining cannot be negative. The result of −9 L is not a valid practical prediction; it shows that continuing this line to 600 km is unsuitable.
-9 L is the algebraic result, not a valid practical prediction.
Try it yourself.
Check your understanding
Observed masses range from 2 to 9 kg. Is a prediction at 10 kg interpolation?
Show answer & reasoning
No; extrapolation.
The observed predictor range ends at , while . The prediction is extrapolation because its input lies outside that range, regardless of the response the equation produces.
Keep practising
6 worksheet questions · 18 marks
Read the questions online
- Data were collected for from 5 to 20. Classify predictions at and .3 marks
- The observed -values are 1,3,5,7. Is a prediction at interpolation even though was not observed?3 marks
- A fitted line is . Predict at and state why the equation alone is insufficient to judge reliability.3 marks
- A model predicts 112% attendance. Explain the appropriate response.3 marks
- A model from one machine type is applied to a newer machine. What assumption needs checking?3 marks
- Why might a prediction inside a range containing only two distant observations be unreliable?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Data were collected for from 5 to 20. Classify predictions at and .3 marks
Compare each input with the observed explanatory range.
12: interpolation; 25: extrapolation. - The observed -values are 1,3,5,7. Is a prediction at interpolation even though was not observed?3 marks
Four is inside the range 1–7.
Interpolation does not require an existing observation at that exact input.
Yes. - A fitted line is . Predict at and state why the equation alone is insufficient to judge reliability.3 marks
Substitute: .
Arithmetic does not reveal whether this is extrapolation or whether residual scatter is small.
23; the observed -range and fit information are also needed. - A model predicts 112% attendance. Explain the appropriate response.3 marks
Do not report the impossible value uncritically.
Attendance cannot exceed 100%; check the model, data and prediction domain. - A model from one machine type is applied to a newer machine. What assumption needs checking?3 marks
Being inside the numerical -range does not guarantee the same population/model.
The relationship and operating conditions must transfer to the new machine type. - Why might a prediction inside a range containing only two distant observations be unreliable?3 marks
Numerical interpolation classification does not measure evidence density.
The unobserved interior form is poorly supported; the relationship may curve or vary substantially.
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 ↗