Combine uncertainties in a calculated result
Estimate a derived quantity's uncertainty by adding absolute or percentage uncertainties for simple sums, products and quotients.
AS Level · Cambridge International Physics 9702, 2025–2027 syllabus, Topic 1. This online sequence supports learning and practice; it does not replace supervised laboratory work, full timed assessment or human subject-teacher review.
Before you begin
- Read a number with its unit and use multiplication, division, powers and right-angled triangles.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For a sum of two measured lengths, which uncertainties are added in this simple rule?
What you will learn
- Propagate absolute uncertainty through addition or subtraction
- Propagate percentage uncertainty through multiplication or division
- Report a derived result with a consistent uncertainty
Choose the matching uncertainty form
For a sum or difference of measured quantities, add their absolute uncertainties for a simple maximum-uncertainty estimate. If a = 12.0 ± 0.2 cm and b = 5.0 ± 0.1 cm, then a + b = 17.0 cm with estimated absolute uncertainty 0.3 cm.
For a product or quotient, add percentage uncertainties. If A = xy and x and y have percentage uncertainties of 1% and 2%, the simple estimated percentage uncertainty in A is 3%. For a quotient, the same addition applies to the magnitudes of the percentage uncertainties.
Keep the estimate tied to the method
Convert absolute uncertainty to percentage using (absolute uncertainty ÷ measured value) × 100%. Convert back using (percentage uncertainty ÷ 100) × calculated value.
These are simple propagation rules for the independent measurements considered here. Keep guard digits during the calculation, then report the result and uncertainty at compatible precision. The estimate does not correct a systematic bias in the original measurements.
Pause and explain
For a product of independent measured quantities, which uncertainties are added?
Worked example
A rectangle has length 20.0 ± 0.2 cm and width 5.0 ± 0.1 cm. Estimate its area and percentage uncertainty.
Show the worked solution
- Area = 20.0 × 5.0 = 100 cm².
- Length percentage uncertainty = 0.2/20.0 × 100% = 1%. Width percentage uncertainty = 0.1/5.0 × 100% = 2%.
- For a product, add percentages: 1% + 2% = 3%.
Answer 100 cm² with estimated percentage uncertainty 3%, equivalent to about ±3 cm².
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Two rods have lengths 14.2 ± 0.1 cm and 8.5 ± 0.2 cm. Estimate their combined length and absolute uncertainty.
Give me a hint
This is a sum, so keep and add absolute uncertainties.
Compare my reasoning
- Add the lengths: 14.2 + 8.5.
- Add the absolute uncertainties: 0.1 + 0.2.
- Report the length with a compatible uncertainty.
22.7 ± 0.3 cm.
Look for these in your work
- Lengths added
- Absolute uncertainties added
- Unit and precision consistent
A cuboid has dimensions 2.00 ± 0.01 m, 1.50 ± 0.02 m and 0.80 ± 0.02 m. Estimate its volume and percentage uncertainty.
Give me a hint
Multiply the dimensions; first find each percentage uncertainty and then add them.
Compare my reasoning
- Volume = 2.00 × 1.50 × 0.80 = 2.40 m³.
- Percentage uncertainties are 0.5%, about 1.33% and 2.5%.
- Add them to obtain about 4.33%.
About 2.40 m³ with estimated percentage uncertainty 4.3%.
Look for these in your work
- Volume calculated
- Each percentage uncertainty calculated
- Percentages added for a product
A measured speed is calculated as distance/time. Distance is 120 ± 2 m and time is 8.0 ± 0.2 s. Estimate the speed and its percentage uncertainty.
Give me a hint
A quotient uses the sum of percentage uncertainties.
Compare my reasoning
- Speed = 120/8.0 = 15 m s⁻¹.
- Distance percentage uncertainty = 2/120 × 100% ≈ 1.67%.
- Time percentage uncertainty = 0.2/8.0 × 100% = 2.5%; add them.
15 m s⁻¹ with an estimated percentage uncertainty of about 4.2%, or approximately ±0.6 m s⁻¹.
Look for these in your work
- Quotient and units correct
- Both percentage uncertainties calculated
- Percentages combined
Common mistakes
- Adding absolute uncertainties for a product without converting to percentages
- Adding percentage uncertainties for a simple sum
- Reporting more precision than the uncertainty supports
What can you explain now?
For a sum of two measured lengths, which uncertainties are added in this simple rule?
Compare with the explanation
Absolute uncertainties
For sums and differences, add absolute uncertainties.
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Make it stick
When do you add absolute uncertainties, and when do you add percentage uncertainties?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.