Compounding percentage uncertainties

The rules

For independent measurements:

• Multiply or divide: add percentage uncertainties.
• Power n: n x the percentage uncertainty (a square root is n = 1/2).
• Add or subtract: add absolute uncertainties.
• Constants (π, g, 4) carry no uncertainty.

For example, for ρ = RA/l with A = πd²/4: %U in ρ = %U in R + %U in l + 2 x %U in d.

Adding absolute uncertainties for a product or quotient is an invalid method and is not credited, even if it gives a similar number.

Formulae with sums or differences inside

Deal with the sum or difference first, in absolute terms, then convert to a percentage. Examples:

• the cross-section of a tube, A = π(D² - d²)/4;
• the density difference (ρ_s - ρ_l) in the viscosity equation;
• a temperature rise (θ_2 - θ_1).

A small difference between two large numbers can have a large percentage uncertainty.

Good practice

• Find each percentage uncertainty first and list them.
• Do not round until the end.
• Give a compounded percentage uncertainty to at least 2 s.f.: recent WPH16 mark schemes ask for a minimum of 2 s.f. here (a half-range percentage uncertainty may be 1 or 2 s.f.).
• Identify the largest contribution (after multiplying by its power): that is where an improvement matters.

Worked example

A tube has outer diameter 25.0 mm and inner diameter 21.0 mm, each ± 0.05 mm. Find the percentage uncertainty in D² - d².

1. D² = 625 mm², uncertainty 2 x (0.05 / 25.0) x 625 = 2.5 mm²
2. d² = 441 mm², uncertainty 2 x (0.05 / 21.0) x 441 = 2.1 mm²
3. D² - d² = 184 mm², uncertainty 2.5 + 2.1 = 4.6 mm²
4. Percentage = 4.6 / 184 x 100 = 2.5%

Answer: About 2.5%, much larger than the 0.2% in each diameter

Check yourself

1. In core practical 7, the percentage uncertainties are: R 1.5%, l 0.4%, d 0.8%. Find the percentage uncertainty in ρ = RA/l, where A = πd²/4.
   Show answer

   3.5 %. 1.5 + 0.4 + 2 x 0.8 = 3.5%.

2. v = sqrt(2gh). The percentage uncertainty in h is 1.2%. What is the percentage uncertainty in v?
   Show answer

   0.6 %. Square root: 1/2 x 1.2% = 0.6%.

3. In the viscosity equation, ρ_s = 7800 ± 50 kg m^-3 and ρ_l = 1260 ± 10 kg m^-3. Find the percentage uncertainty in (ρ_s - ρ_l), to 2 s.f.
   Show answer

   0.92 %. Difference = 6540 kg m^-3 with uncertainty 50 + 10 = 60 kg m^-3; 60 / 6540 x 100 = 0.92%.

4. E = 4FL / (πd² Δx). The percentage uncertainties are F 0.5%, L 0.1%, d 1.3% and Δx 2.0%. Find the percentage uncertainty in E.
   Show answer

   5.2 %. 0.5 + 0.1 + 2 x 1.3 + 2.0 = 5.2%.

5. A sphere's diameter has a percentage uncertainty of 0.4%. Find the percentage uncertainty in its volume.
   Show answer

   1.2 %. V depends on d³: 3 x 0.4% = 1.2%.

6. Kinetic energy is (1/2)mv². The mass has a percentage uncertainty of 0.2% and the speed 1.5%. Find the percentage uncertainty in the kinetic energy.
   Show answer

   3.2 %. 0.2 + 2 x 1.5 = 3.2%.

7. A temperature rises from 21.5 °C to 34.0 °C, each reading ± 0.5 °C. Find the percentage uncertainty in the rise, to 2 s.f.
   Show answer

   8.0 %. Rise = 12.5 °C, uncertainty 1.0 °C; 1.0 / 12.5 x 100 = 8.0%.

8. A student finds the uncertainty in a density by adding the absolute uncertainties of mass and volume. Why is this not credited?
   • Absolute uncertainties cannot be added in any case
   • Density has no uncertainty
   • For division, percentage uncertainties must be added, not absolute ones
   • Mass and volume have the same units
   Show answer

   For division, percentage uncertainties must be added, not absolute ones. Absolute uncertainties of different quantities cannot be added for a quotient; examiners call it an invalid method.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel A level Physics: Appendix 10, Uncertainties and practical work
• Pearson examiners' report: IAL Physics WPH16/01, June 2024
• Pearson examiners' report: IAL Physics WPH16/01, October 2024

Last updated 8 November 2026. Spotted a mistake? Tell us.