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Combining uncertainties

Choose the uncertainty rule, then calculate the uncertainty in a sum, quotient or expression with powers.

Practise

Dividing measurements

Use the approximate maximum-uncertainty method. Treat the quoted input quantities as independent.

For a resistor, V = (9.8 ± 0.1) V and I = (0.22 ± 0.01) A. Calculate the percentage uncertainty in the resistance R = V / I. Give the uncertainty magnitude to 2 significant figures.

linear-maximum-uncertainty

Use decimals or e notation, e.g. 4.2e-3. Enter the unit separately. Use % for a percentage.

Hint

Convert each quoted absolute uncertainty to a percentage. For either multiplication or division, add those percentages.

Worked solution
  1. Rule: Add percentage uncertainties.
  2. R = V / I.
  3. Percentage uncertainties: voltage = (0.1 / 9.8) × 100%; current = (0.01 / 0.22) × 100%.
  4. Add those percentages without rounding the intermediate values.
  5. Answer: 5.6 % (2 significant figures).
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Question type: Mixed practice
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Uncertainty in a product or quotient

For R = V/I, let V = (6.0 ± 0.1) V and I = (0.30 ± 0.01) A. The school-level maximum percentage uncertainties are 1.67% and 3.33%. Add them for R: 5.0%. R = 20 Ω, so its absolute uncertainty is 1.0 Ω.

For a difference, add absolute uncertainties even though the measured values are subtracted. For a squared quantity, double its percentage uncertainty. These are the stated maximum-uncertainty rules, not a statistical root-sum-of-squares calculation.

Apply the rules to resistivity

Choose the rule from the expression

Approximate maximum-uncertainty rules
OperationUncertainty calculation
a + b or a − bAdd absolute uncertainties: Δa + Δb
ab or a / bAdd percentage uncertainties
aⁿMultiply percentage uncertainty by |n|

Subtracting measurements does not subtract their uncertainties. If two positions have uncertainties ±0.1 cm and ±0.2 cm, their difference has uncertainty ±0.3 cm by this method.

Powers and several factors

In ρ = m / (πr²L), add the percentage uncertainty in mass, twice that in radius and that in length. Denominator terms still increase the uncertainty. Exact constants such as π contribute none.

These are approximate maximum-uncertainty rules for the stated independent inputs. They do not calculate statistical confidence intervals. Shared systematic errors or correlations can need a different analysis.

Return to an absolute uncertainty

Multiply the calculated value by its percentage uncertainty and divide by 100. A resistance of 200 Ω with 3.0% uncertainty has absolute uncertainty 6.0 Ω.

Give uncertainty magnitudes to 2 significant figures in these exercises. This is a calculation-practice choice; follow any rounding instruction in your exam question. A more precise entry is accepted if it rounds to the same answer. Use the stated uncertainties.

A-level relevance

AQA 7408: 3.1.2; MS 1.5 (partial); OCR A H556: 2.2.1(c) (partial). Tasks cover the selected skills listed above.

AQA specification · OCR A specification · AQA practical handbook · OCR practical skills handbook

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