Uncertainty in a result: what Unit 3 expects

What Unit 3 expects

Specification 3.5 asks you to find the percentage uncertainty of a single reading from half the resolution and of repeat readings from the half range. It also states that Unit 3 students are not expected to compound percentage uncertainties (adding percentages for a product or quotient, or multiplying by a power). That is a Unit 6 skill, taught in the Unit 6 lesson on compounding percentage uncertainties.

Two readings added or subtracted

When a quantity is found from two readings, such as a length from two positions on a rule or a temperature rise, add the two absolute uncertainties: (85.0 ± 0.1) cm - (62.5 ± 0.1) cm = 22.5 ± 0.2 cm. The percentage uncertainty of the difference is 0.2 / 22.5 x 100 = 0.89%. A small difference between two readings has a large percentage uncertainty.

Dividing by a counted number

Dividing a measurement by an exact number (20 oscillations, 500 sheets) divides the absolute uncertainty by the same number, so the percentage uncertainty is unchanged. Halving a diameter to get a radius works the same way. Numbers such as 20, π and 4 carry no uncertainty.

Using a percentage uncertainty you are given

Questions often give you the percentage uncertainty of a result. IAL mark schemes accept either of two methods:

• Limits: value ± (percentage / 100 x value); state whether the accepted value lies inside the limits.
• Percentage difference against percentage uncertainty: %D = |measured - accepted| / accepted x 100%; if %D is smaller than the percentage uncertainty, the result is consistent with the accepted value.

Either way, write the comparison out with both numbers and a conclusion.

Worked example

A student finds the density of a metal nut to be 8.81 g cm^-3 with a percentage uncertainty of 2.0%. Could the nut be copper (density 8.96 g cm^-3)?

1. Limits: 8.81 x 0.98 = 8.63 and 8.81 x 1.02 = 8.99 g cm^-3; 8.96 lies between them.
2. Or: %D = |8.81 - 8.96| / 8.96 x 100 = 1.7%, which is less than 2.0%.

Answer: Yes: by either method the result is consistent with copper

Check yourself

1. Two lengths are 85.0 ± 0.1 cm and 62.5 ± 0.1 cm. What is the absolute uncertainty in their difference, in cm?
   Show answer

   0.2 cm. Subtracting: add absolute uncertainties, 0.1 + 0.1 = 0.2 cm.

2. The difference of those lengths is 22.5 ± 0.2 cm. Find its percentage uncertainty, to 2 s.f.
   Show answer

   0.89 %. 0.2 / 22.5 x 100 = 0.89%.

3. A student times 20 oscillations as 31.6 ± 0.2 s. What is the absolute uncertainty in the period, in s?
   Show answer

   0.01 s. Dividing by the exact number 20 divides the uncertainty by 20: 0.2 / 20 = 0.01 s.

4. A density is 7800 kg m^-3 with a percentage uncertainty of 1.3%. Find the absolute uncertainty in kg m^-3, to 2 s.f.
   Show answer

   100 kg m^-3. 1.3 / 100 x 7800 = 101.4 kg m^-3, so about 100 kg m^-3.

5. A measured density is 2.62 g cm^-3; aluminium has density 2.70 g cm^-3. Find the percentage difference, to 2 s.f.
   Show answer

   3.0 %. |2.62 - 2.70| / 2.70 x 100 = 3.0%.

6. The measured density 2.62 g cm^-3 has a percentage uncertainty of 2.5%, and its percentage difference from aluminium (2.70 g cm^-3) is 3.0%. What should the student conclude?
   • Consistent with aluminium, because the values are close
   • Not consistent with aluminium, because the percentage difference (3.0%) is larger than the percentage uncertainty (2.5%)
   • Consistent with aluminium, because 2.5% is small
   • No conclusion is possible without a graph
   Show answer

   Not consistent with aluminium, because the percentage difference (3.0%) is larger than the percentage uncertainty (2.5%). %D is larger than %U. The limits method agrees: the upper limit is 2.62 x 1.025 = 2.69 g cm^-3, below 2.70.

7. The percentage uncertainties in a student's measurements are: length 0.2%, diameter 1.3%, mass 0.05%. Which measurement should be improved first?
   • The length
   • The mass
   • All equally
   • The diameter
   Show answer

   The diameter. The largest percentage uncertainty is where an improvement matters most.

8. Which uncertainty skill does the specification say Unit 3 students are NOT expected to use?
   • Half the resolution for a single reading
   • Half the range for repeat readings
   • Compounding percentage uncertainties, such as doubling the percentage for a squared diameter
   • Comparing a result with an accepted value
   Show answer

   Compounding percentage uncertainties, such as doubling the percentage for a squared diameter. Specification 3.5: 'students are not expected to compound percentage uncertainties'. Compounding is assessed in Unit 6.

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 WPH13/01, January 2024
• Pearson examiners' report: IAL Physics WPH13/01, October 2024

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