- use the positive difference
Worked example
A length is measured as 8.4 cm, but the actual length is 8.1 cm.
Find the absolute error.
Step by step
- Find the positive difference.
Final answer
0.3 cm
Absolute error is a size: It tells you the size of the mistake, not whether the measurement was above or below the actual value.
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- actual value
- use the true/reference value in the denominator
Worked example
Use absolute error 0.3 and actual value 8.1 to find the relative error.
Step by step
- Substitute into the formula.
Final answer
Relative error ≈ 0.0370
Denominator trap: Relative error compares the error with the actual value, not the measured value.
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Why relative error matters: A 0.5 cm error might be tiny for a 5 m object but large for a 1 cm object.
Relative error tells you how important the error is compared with the size of the measurement.
| Situation | Absolute error | Relative effect |
|---|---|---|
| Length about 100 cm | 0.5 cm | small |
| Length about 1 cm | 0.5 cm | very large |
Quick interpretation
Which measurement is more accurate if both have absolute error 0.2, but one actual value is 2 and the other is 20?
Step by step
- Relative error for 2 is 0.2/2 = 0.1.
- Relative error for 20 is 0.2/20 = 0.01.
Final answer
The measurement with actual value 20 is more accurate because its relative error is smaller.
Turning it into a percentage: If you multiply the relative error by 100, you get percentage error.
That full topic comes next.
Worked example — relative to percentage
A relative error is 0.04.
What percentage is this?
Step by step
- Multiply by 100%.
Final answer
4%
Decimal or percent?: Read the question carefully.
Some questions want relative error as a decimal, others want it as a percentage.
🎯 IB-style worked example
Skills you'll use here:
- Plug a value into a quadratic model to estimate a quantity.
- Compute percentage error with the formula |estimate − actual| ÷ actual × 100.
- Solve a quadratic equation using the GDC (PolySmlt) — keep only the realistic root.
- Solve an exponential equation by isolating the power, then using logs.
Scenario: A delivery drone has its braking distance d (in metres) modelled in terms of speed s (in m s⁻¹) by:
d = 1.15s² − 4.8s, for s ≥ 0.
(In a later part of the question, a revised stopping model D = 120 + 35(1.08)ᵗ is used, where t is time in seconds.)
Part (a) — estimate braking distance at 22 m s⁻¹
Using d = 1.15s² − 4.8s:
(a) Calculate the estimated braking distance when s = 22 m s⁻¹. [2 marks]
Step by step
- Substitute s = 22 into the model:
- Evaluate each term:
- Subtract:
Final answer
d = 451 m
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (b) — find percentage error
The actual measured braking distance at 22 m s⁻¹ is 450 m.
(b) Calculate the percentage error in your estimate from part (a). [2 marks]
Step by step
- Write the formula (use absolute value so the answer is positive):
- Substitute estimate = 451 and actual = 450:
- Evaluate and round to 3 s.f.:
Final answer
Percentage error ≈ 0.222%
Part (c) — solve the quadratic for s
Using d = 1.15s² − 4.8s:
(c) Find the speed s when d = 300 m. Give your answer in m s⁻¹. [2 marks]
Step by step
- Set d = 300 and rearrange to standard form (all on one side):
- Solve on the GDC using PolySmlt (degree 2) or the quadratic formula. Two roots come out:
- Reject the negative root — speed can't be negative:
Final answer
s ≈ 18.4 m s⁻¹
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (d) — solve the exponential for t
A revised stopping model is D = 120 + 35(1.08)ᵗ, where t is time in seconds.
(d) Find t when D = 420. [2 marks]
Step by step
- Substitute D = 420 and isolate the exponential. Subtract 120, then divide by 35:
- Take logs of both sides and use the power rule:
- Solve for t and evaluate on the GDC:
Final answer
t ≈ 27.9 s
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Exam tips for mixed-model questions:
- Write the percentage-error formula before substituting — examiners reward seeing the method, not just the number.
- Use absolute value in the numerator so the answer is positive (a 'percentage error' is never negative).
- When you get two roots from a quadratic, throw out the unrealistic one (negative speed, negative time, negative distance, etc.).
- For exponentials, isolate the power first, then take logs — never take logs while the constant is still attached.