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Percentage uncertainty questions

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

Practise

Timing several oscillations

10 complete oscillations take 27.6 s, with absolute uncertainty ±0.2 s in that total time. The count is exact. Find the percentage uncertainty of the period T = t / 10. Give 2 significant figures. Use the stated uncertainties.

quoted-total-timing

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

Hint

Dividing by an exact count divides the time and its absolute uncertainty by the same factor. Their ratio is unchanged.

Worked solution
  1. T = 27.6 / 10 s; ΔT = 0.2 / 10 s.
  2. ΔT / T = 0.2 / 27.6.
  3. Percentage uncertainty = (0.2 / 27.6) × 100%.
  4. Answer: 0.72 % (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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