Line of Best Fit
Draw a line of best fit on a scatter plot, use it to make predictions, and understand interpolation vs extrapolation.
Printable Worksheets
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Worksheet
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Q1 · If you had a scatter plot with lots of points going roughly upward, how would you draw a single line that "best" represents the trend?
Q2 · Can a line of best fit predict what would happen for values far outside your data? When might that prediction be risky?
Learning Intentions
Know
- A line of best fit summarises the linear trend in bivariate data. It can be drawn by eye or calculated using least squares.
Understand
- Why predictions within the data range (interpolation) are more reliable than predictions outside it (extrapolation).
Can Do
- Draw a line of best fit by eye, estimate its equation, and use it to make predictions.
Key Terms
Misconceptions to Fix
Wrong: Interpolation and extrapolation are equally reliable.
Right: Interpolation (within the data range) is more reliable because it uses observed patterns. Extrapolation assumes the trend continues, which may not be true.
Wrong: A trend will always continue in the same direction.
Right: Real-world trends often change direction due to external factors. Never assume a linear trend continues indefinitely.
Line of Best Fit
Work through the content, activities and worked examples below. Test your understanding with the questions in the Questions phase.
For each prediction, state whether it is interpolation or extrapolation and assess its reliability:
- Data: x = 1 to 10. Predict x = 5.
- Data: x = 1 to 10. Predict x = 15.
- Data: temperatures 20°C to 30°C. Predict at 25°C.
- Data: temperatures 20°C to 30°C. Predict at 50°C.
Worked Example
Step-by-step-
1Prediction at 10 km: 10 km is within the data range (2–20 km). This is interpolation and is relatively reliable.
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2Prediction at 30 km: 30 km is outside the data range. This is extrapolation and is less reliable.
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3Why less reliable: The trend may not continue. At 30 km, houses might be rural with different pricing factors (land size, infrastructure).
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4Conclusion: Interpolation at 10 km is trustworthy. Extrapolation at 30 km should be treated with caution.
Revisit Your Thinking
Look back at your Think First response. What new understanding do you have now?
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
Multiple Choice
Select the best answer for each question.
1 mark Interpolation is generally more reliable than extrapolation because:
1 mark A line of best fit drawn by eye should:
1 mark If a line of best fit has gradient 2 and y-intercept 5, its equation is:
1 mark A residual is:
1 mark Extrapolation should be used with caution because:
Short Answer
Show all working and justify your answers.
1. 4 marks A scatter plot shows the relationship between age (years) and running speed (km/h) for people aged 10 to 60.
(a) Predict the running speed for a 25-year-old. Is this interpolation or extrapolation?
(b) Predict the running speed for an 80-year-old. Is this interpolation or extrapolation? Explain why this prediction may be unreliable.
2. 3 marks A company uses sales data from 2015–2024 to predict sales in 2030. Explain why this prediction may be inaccurate, even if the historical trend is strong.
3. 2 marks Give two reasons why extrapolation is generally less reliable than interpolation.
Marking guidance: 1 mark each for MCQs. See mark allocations for each short answer question.