- true/reference value
Worked example
Measured value 49, actual value 50.
Find the percentage error.
Step by step
- Absolute error = |49 - 50| = 1.
- Divide by actual and multiply by 100%.
Final answer
2%
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Smaller is better: A smaller percentage error means the measurement or estimate is closer to the true value relative to the size of the quantity.
| Percentage error | Interpretation |
|---|---|
| 1% | very accurate |
| 5% | moderate error |
| 20% | large error |
Context matters: A 5% error might be acceptable in one context and poor in another.
Always link your answer to the situation.
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Worked example
An object is measured as 12 cm, actual length 12.5 cm.
Find the percentage error.
Step by step
- Absolute error = 0.5 cm.
- Percentage error = 0.5 / 12.5 × 100%.
Final answer
4%
Use the actual value: Percentage error is based on the true/reference value, not the measured one.
Worked context example
A student estimates a journey as 18 km, but the actual distance is 20 km.
Interpret the percentage error.
Step by step
- Absolute error = 2 km.
- Percentage error = 2/20 × 100% = 10%.
- Interpretation: the estimate is 10% away from the actual distance.
Final answer
The estimate has a 10% error, so it is moderately inaccurate.
Interpret, do not just calculate: If the question asks you to comment, say what the percentage means for the quality of the estimate.
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Percentage error and ranges: A percentage error means the real value could be a little bigger or smaller than the estimate.
Use it to create a possible range of answers.
Quick warm-up
An estimate is 80 and the percentage error can be 10%.
Find the possible range.
Step by step
- 10% of 80 is 8.
- Smallest possible value: 80 − 8 = 72.
- Largest possible value: 80 + 8 = 88.
Final answer
72 to 88
Quick method: To find the minimum value, decrease the number by the percentage error.
To find the maximum value, increase the number by the percentage error.
Example: with a 6% error, use ×0.94 for the maximum and ×1.06 for the minimum.
🎯 IB-style worked example
Scenario: The approximate wind chill index W on a cold day is modelled by:
W = -34.1 - 7.33 ln(v)
where v is the wind speed in km/h.
Part (a) — calculate the wind chill index
Find W when v = 13 km/h. [2 marks]
Step by step
- Substitute v = 13 into the formula.
- On the calculator, type the whole expression in one line.
- The calculator gives:
- Round to 3 significant figures.
Final answer
W ≈ -52.9
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (b) — use 6% error to find a range
The percentage error in the approximate value from part (a) can be as high as 6%.
Predict the maximum and minimum possible values of W. [3 marks]
Step by step
- Use the unrounded value from part (a) for better accuracy.
- For a 6% error, multiply by 0.94 for the maximum and 1.06 for the minimum.
- Calculate the maximum value.
- Calculate the minimum value.
- With negative numbers, the value closer to zero is larger.
Final answer
Maximum ≈ -49.9, Minimum ≈ -56.3
Negative numbers trap: For negative values, the number closer to zero is larger.
So -49.9 is greater than -56.3.
Part (c) — solve for wind speed
Find v when W = -60. [2 marks]
Step by step
- Substitute W = -60.
- Add 34.1 to both sides.
- Divide by -7.33.
- Undo ln using ex.
- Evaluate.
Final answer
v ≈ 34.2 km/h
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Calculator display warning: If your GDC shows something like 3.4241E1, the E1 means ×10¹.
So 3.4241E1 = 34.241, not 3.4241.
Exam tips for this type of question:
- Keep the full calculator value from part (a) before using percentage error.
- For percentage error, use ×0.94 and ×1.06 when the error is 6%.
- With negative answers, maximum means least negative and minimum means most negative.
- When solving logs, isolate ln(v) first before using ex.