Lines of good fit and predictions
A by-eye line summarises an approximately linear cloud. Predictions depend on the fit and the data; they are not guaranteed observations
- Placement Follow the whole main pattern with small, reasonably balanced vertical departures. Exact equal counts alone are insufficient
- Unusual observations Investigate them and justify treatment. Do not automatically delete them
- Interpolation Predict within the observed explanatory-variable range. At a range endpoint you are still within the observed range
- Extrapolation Predict outside that range. The pattern may not continue, so use extra caution
- Two points on a line Read from the line or use two points on that fitted line to calculate its equation. These need not be original observations
Worked example
A fitted line for length in cm and mass in g passes through (2,20) and (6,40). Observed lengths range from 1 to 7 cm. Estimate mass at length 5 cm and classify the estimate
- \(\displaystyle m=\frac{40-20}{6-2}=5\)The fitted mass increases by 5 g for each additional centimetre
- \(\displaystyle 20=5\times2+b,\quad b=10\)Use a point on the fitted line. The equation is mass = 10 + 5 × length
- \(\displaystyle \widehat{y}=10+5\times5=35\)The fitted prediction is 35 g, not an exact promise for an individual sample
- \(\displaystyle 1<5<7\)The explanatory value is inside the observed range, so this is interpolation
Common mistakes
Difficulty levels
Questions about this topic
Must the line pass through an observed point?+
No. It summarises the cloud. Points chosen to calculate its equation should be on the fitted line, and need not be observations
Is a by-eye fit unique?+
No. Several nearby lines may be reasonable. Explain the placement and use sensible precision for predictions
Should a separated point always be removed?+
No. Investigate measurement and context. If modelling a stated main group, explain that decision and how the unusual point could affect conclusions
Can interpolation still be unreliable?+
Yes. Weak association, poor measurements, biased sampling or an unsuitable model can make it unreliable even inside the observed range
Which range determines interpolation?+
Use the observed explanatory-variable range, not the visible axis limits or the length of the drawn line
Does a fitted line prove causation?+
No. A predictive association does not by itself establish a causal effect
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 5: Statistics