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Percentage uncertainty questions

Calculate percentage uncertainty, work backwards to an absolute uncertainty, and use repeated measurements.

Practise

Repeated measurements

Repeated times are 5.29, 5.23, 5.31, 5.25 s. Use their mean as the measured value and half their range as the absolute uncertainty. Calculate the percentage uncertainty. Give 2 significant figures. Use the stated uncertainties.

half-range

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

Hint

Find the mean of all four times. Half the difference between the largest and smallest gives the stated estimate of absolute uncertainty.

Worked solution
  1. Mean = (5.29 + 5.23 + 5.31 + 5.25) / 4 = 5.27 s.
  2. Half-range = (5.31 − 5.23) / 2 = 0.04 s.
  3. Percentage uncertainty = (0.04 / 5.27) × 100%.
  4. Answer: 0.76 % (2 significant figures).
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Question type: Mixed practice
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How to calculate percentage uncertainty

A length is (250 ± 2) mm. Percentage uncertainty = (2/250) × 100 = 0.8%. If the same absolute uncertainty applies to a 50 mm length, the percentage is 4%. A larger measured value reduces the percentage uncertainty only when the absolute uncertainty stays the same.

Keep measurement and uncertainty in the same unit. An uncertainty of 0.2 cm is 2 mm, not 0.2 mm. The percentage calculation does not decide what the absolute uncertainty should be; that must come from the measurement method and stated convention.

Combine uncertainty in a calculated result

Percentage uncertainty

Compare the absolute uncertainty with the measured value. A length of (20.0 ± 0.2) cm has fractional uncertainty 0.2 / 20.0 = 0.01, or percentage uncertainty 1.0%.

Percentage uncertainty = (absolute uncertainty / measured value) × 100%

The uncertainty and value must use the same unit. An uncertainty of 1 mm in a length of 20 cm means 0.1 / 20 × 100 = 0.5%, not 5%.

Working backwards

Absolute uncertainty = measured value × percentage uncertainty / 100. If the absolute and percentage uncertainties are known, measured value = absolute uncertainty × 100 / percentage uncertainty.

Repeated readings and timing

Where a question specifies half-range, subtract the smallest reading from the largest, then divide by two. Use the mean of all the readings as the measured value. Half-range describes the spread of that sample; it is not a guaranteed confidence interval.

For N oscillations, T = t / N. If N is exact, ΔT = Δt / N. Dividing both by N leaves the percentage uncertainty unchanged.

A ruler distance uses two positions, so both reading uncertainties contribute. Instrument resolution is one possible contribution; use the uncertainty quoted in each question.

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 (partial); OCR A H556: 2.2.1(c), 1.1.4(d) (partial). Tasks cover the selected skills listed above.

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

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