Physics 9702 · Paper 2

Uncertainty in a Measurement

No reading is perfect. Every measured value has some doubt in it, and physics writes that doubt down as a number.
The doubt is called the , written after a ± sign: length = 24.3 ± 0.1 cm.

The uncertainty of one reading

From a scale: the uncertainty of one reading is about half the smallest division.
A length has two ends, so you read the scale twice. Two half-divisions make one whole division of uncertainty (±1 mm on a millimetre ruler).
From repeated readings: the uncertainty is about half the range (largest minus smallest, divided by two).
With a stopwatch: your reaction time is about 0.1 s, so the uncertainty is at least 0.1 s even if the display shows 0.01 s steps.
Quote an uncertainty to 1 significant figure.

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}

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.

\%\text{ uncertainty} = \dfrac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%

Symbols

$\text{absolute uncertainty}$ = the ± number quoted with the reading (same unit as the value)
$\text{measured value}$ = 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 turn— tap to reveal the worked answer (9702-style)

A student times a single swing of a pendulum with a stopwatch that shows steps of 0.01 s. Explain why the uncertainty in the time is about 0.1 s, not 0.01 s.

Answer: the student must judge when the swing starts and when it ends. Human reaction time is about 0.1 s, and that doubt is much bigger than the 0.01 s steps of the display. The instrument is not the only source of uncertainty. (9702-style)

Common mistake

Two errors to avoid: never put a unit on a percentage uncertainty (it is a ratio), and never quote an uncertainty smaller than the smallest division you could actually read.

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Uncertainty in a Measurement

No reading is perfect. Every measured value has some doubt in it, and physics writes that doubt down as a number.
The doubt is called the , written after a ± sign: length = 24.3 ± 0.1 cm.

The uncertainty of one reading

From a scale: the uncertainty of one reading is about half the smallest division.
A length has two ends, so you read the scale twice. Two half-divisions make one whole division of uncertainty (±1 mm on a millimetre ruler).
From repeated readings: the uncertainty is about half the range (largest minus smallest, divided by two).
With a stopwatch: your reaction time is about 0.1 s, so the uncertainty is at least 0.1 s even if the display shows 0.01 s steps.
Quote an uncertainty to 1 significant figure.

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}

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.

\%\text{ uncertainty} = \dfrac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%

Symbols

$\text{absolute uncertainty}$ = the ± number quoted with the reading (same unit as the value)
$\text{measured value}$ = 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 turn— tap to reveal the worked answer (9702-style)

A student times a single swing of a pendulum with a stopwatch that shows steps of 0.01 s. Explain why the uncertainty in the time is about 0.1 s, not 0.01 s.

Common mistake

Two errors to avoid: never put a unit on a percentage uncertainty (it is a ratio), and never quote an uncertainty smaller than the smallest division you could actually read.