Uncertainty in a Measurement | CAIE 9702 Paper 1 Notes

Paper 1

Physics9702

Uncertainty in a Measurement

An error is what makes a reading differ from the true value. Usually the true error is not known.

An states the estimated range around a measured value within which the true value is expected to lie.

A length of 24.3 ± 0.1 cm means an expected range from 24.2 cm to 24.4 cm. It does not mean that the actual error is known to be exactly 0.1 cm.

The uncertainty of one reading

Analogue scale: when the mark can be judged between divisions and no better information is given, one reading is often taken as ± half the smallest division.
Digital display: when no instrument specification is given, use ± one unit in the last displayed digit.
A difference uses two readings. For a length found from two ruler positions, add the two reading uncertainties. Two ±0.5 mm readings give ±1.0 mm in the length.
Repeated readings: use half the range when this is larger than the instrument reading uncertainty. Identical repeats do not make the uncertainty zero.
Stopwatch: human judgement at the start and end is usually more important than the display resolution. Estimate it from the procedure or repeated timings; do not automatically use 0.01 s because the display shows hundredths.
Quote the uncertainty to 1 significant figure, then round the measured value to the same decimal place.

Swipe left or right to see the whole diagram →

The rod's end lands between marks. You read 24.3 cm, but the true edge could be anywhere inside about ±0.5 mm, half of the smallest division. A length has two ends, so reading both gives ±1 mm in total.

Worked example

Smallest case: repeats give the uncertainty

Two careful readings of the same length give 20.6 cm and 20.8 cm. Quote the result.

Mean: $\dfrac{20.6 + 20.8}{2} = 20.7 \text{ cm}$ .
Half the range: $\dfrac{20.8 - 20.6}{2} = 0.1 \text{ cm}$ .

Answer

20.7 \pm 0.1 \text{ cm}

Choose uncertainty from the limiting part of the measurement, not from the number of digits printed on the instrument.

Absolute and percentage uncertainty

The ± number itself is the absolute uncertainty. It carries the same unit as the value.
Dividing it by the value gives the percentage uncertainty, which has no unit. The next page shows why the percentage form is so useful.
In the formula below, $x$ is the measured value and $\Delta x$ is its absolute uncertainty.

\%\text{ uncertainty} = \dfrac{\Delta x}{x} \times 100\%

Symbols

$\Delta x$ = the ± number quoted with the reading (same unit as the value)
$x$ = the reading itself (same unit)
$\%\text{ uncertainty}$ = the doubt as a fraction of the value (no unit)

Worked example

Percentage uncertainty of a length

A length is 24.3 cm, read at both ends of a millimetre ruler, so the absolute uncertainty is ±0.1 cm. Find the percentage uncertainty.

$\%\text{ unc} = \dfrac{0.1}{24.3} \times 100\% = 0.4\%$ .

Answer

0.4\%

Worked example

One change: from percentage back to absolute

A current is 2.4 A with a percentage uncertainty of 5%. Quote the reading with its absolute uncertainty.

Absolute uncertainty: $5\% \times 2.4 = 0.12 \approx 0.1 \text{ A}$ (1 significant figure).

Answer

2.4 \pm 0.1 \text{ A}

Your turn9702-style · 2 marks

A stopwatch displays time to 0.01 s. Explain why 0.01 s may be an unrealistically small uncertainty when a student times one swing of a pendulum.

Show worked answer

The student must judge the start and end instants. Variation in this judgement is larger than one display step, so the procedure, not only the stopwatch resolution, sets the uncertainty.

Common mistake

An absolute uncertainty has the same unit as the measured value. A percentage uncertainty has no unit because it is a ratio. Do not assume that every analogue reading is exactly half a division or every digital reading is exactly one last digit. Those are default estimates only when the apparatus and procedure do not introduce a larger uncertainty.

Uncertainty in a Measurement

An error is what makes a reading differ from the true value. Usually the true error is not known.

An states the estimated range around a measured value within which the true value is expected to lie.

A length of 24.3 ± 0.1 cm means an expected range from 24.2 cm to 24.4 cm. It does not mean that the actual error is known to be exactly 0.1 cm.

The uncertainty of one reading

Analogue scale: when the mark can be judged between divisions and no better information is given, one reading is often taken as ± half the smallest division.
Digital display: when no instrument specification is given, use ± one unit in the last displayed digit.
A difference uses two readings. For a length found from two ruler positions, add the two reading uncertainties. Two ±0.5 mm readings give ±1.0 mm in the length.
Repeated readings: use half the range when this is larger than the instrument reading uncertainty. Identical repeats do not make the uncertainty zero.
Stopwatch: human judgement at the start and end is usually more important than the display resolution. Estimate it from the procedure or repeated timings; do not automatically use 0.01 s because the display shows hundredths.
Quote the uncertainty to 1 significant figure, then round the measured value to the same decimal place.

Swipe left or right to see the whole diagram →

Worked example

Smallest case: repeats give the uncertainty

Two careful readings of the same length give 20.6 cm and 20.8 cm. Quote the result.

Mean: $\dfrac{20.6 + 20.8}{2} = 20.7 \text{ cm}$ .
Half the range: $\dfrac{20.8 - 20.6}{2} = 0.1 \text{ cm}$ .

Answer

20.7 \pm 0.1 \text{ cm}

Choose uncertainty from the limiting part of the measurement, not from the number of digits printed on the instrument.

Absolute and percentage uncertainty

The ± number itself is the absolute uncertainty. It carries the same unit as the value.
Dividing it by the value gives the percentage uncertainty, which has no unit. The next page shows why the percentage form is so useful.
In the formula below, $x$ is the measured value and $\Delta x$ is its absolute uncertainty.

\%\text{ uncertainty} = \dfrac{\Delta x}{x} \times 100\%

Symbols

$\Delta x$ = the ± number quoted with the reading (same unit as the value)
$x$ = the reading itself (same unit)
$\%\text{ uncertainty}$ = the doubt as a fraction of the value (no unit)

Worked example

Percentage uncertainty of a length

A length is 24.3 cm, read at both ends of a millimetre ruler, so the absolute uncertainty is ±0.1 cm. Find the percentage uncertainty.

$\%\text{ unc} = \dfrac{0.1}{24.3} \times 100\% = 0.4\%$ .

Answer

0.4\%

Worked example

One change: from percentage back to absolute

A current is 2.4 A with a percentage uncertainty of 5%. Quote the reading with its absolute uncertainty.

Absolute uncertainty: $5\% \times 2.4 = 0.12 \approx 0.1 \text{ A}$ (1 significant figure).

Answer

2.4 \pm 0.1 \text{ A}

Your turn9702-style · 2 marks

A stopwatch displays time to 0.01 s. Explain why 0.01 s may be an unrealistically small uncertainty when a student times one swing of a pendulum.

Show worked answer

The student must judge the start and end instants. Variation in this judgement is larger than one display step, so the procedure, not only the stopwatch resolution, sets the uncertainty.

Common mistake