The idea, step by step
Locate the prediction in the data range
Interpolation predicts for an explanatory value inside the observed range. Extrapolation predicts outside it. Extrapolation is generally more uncertain because an apparent pattern may change beyond the data, but interpolation is not automatically accurate.
The range is the predictor range, not whether a predicted y happens to resemble previous responses. A prediction at a missing interior x-value can still be interpolation.
Check the situation as well as the arithmetic
After substitution, ask whether the answer has meaningful units, fits physical constraints, and uses the same population and conditions. A line may eventually predict negative fuel, impossible percentages or unlimited growth. Such results reveal a domain limitation.
Report a suitable rounded estimate and its scope. Do not present a regression equation as an exact law, especially when residual variation is substantial or sample data are sparse.
A common trap
If a prediction is interpolation, it must be correct.
Instead: Interpolation avoids leaving the x-range but still depends on the model, data and residual variation.
See the thinking
An interior prediction
A model fitted at ages 4–12 years is h=25+6a. Predict at age 8 and classify it.
- Eight is within the observed age range.
- h=25+6×8=73.
- The estimate is interpolation, subject to the model fit and data quality.
A mathematically valid but impossible result
Fuel data from 0–300 km produce F=45-0.09d litres. Evaluate at 600 km.
- F=45-54=-9 litres.
- Six hundred kilometres is outside the data range.
- A negative amount is impossible; the linear relation cannot be used indiscriminately.
Check your understanding
Observed masses range from 2 to 9 kg. Is a prediction at 10 kg interpolation?
Show answer
No; extrapolation.
Ten lies beyond the largest observed explanatory value.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- Data were collected for x from 5 to 20. Classify predictions at x=12 and x=25.3 marks
- The observed x-values are 1,3,5,7. Is a prediction at x=4 interpolation even though x=4 was not observed?3 marks
- A fitted line is ŷ=8+1.5x. Predict at x=10 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 x from 5 to 20. Classify predictions at x=12 and x=25.3 marks
Compare each input with the observed explanatory range.
12: interpolation; 25: extrapolation. - The observed x-values are 1,3,5,7. Is a prediction at x=4 interpolation even though x=4 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 ŷ=8+1.5x. Predict at x=10 and state why the equation alone is insufficient to judge reliability.3 marks
Substitute: 8+15=23.
Arithmetic does not reveal whether this is extrapolation or whether residual scatter is small.
23; the observed x-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 x-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 ↗