Measurements and their errors

Section 3.1

Year 1

3 specification points

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against AQA 7408 section 3.1

Checked against AQA 7408 section 3.1. Review basis: the qualification registry sourced from the AQA A-level Physics (7408) specification; registry verification recorded 11 July 2026.

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In the exam: Data and formulae booklet provided · calculator allowed in every paper

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3.1.1

Year 1

Use of SI units and their prefixes

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

The required base quantities and SI units are mass in kilograms, length in metres, time in seconds, amount of substance in moles, temperature in kelvin and electric current in amperes; candela is excluded.
Derived units combine base units, for example $\text{N}=\text{kg m s}^{-2}$ .
Prefixes required are $\text{T}$ , $\text{G}$ , $\text{M}$ , $\text{k}$ , $\text{c}$ , $\text{m}$ , $\mu$ , $\text{n}$ , $\text{p}$ and $\text{f}$ , used with standard form.
Conversions also include units of the same quantity, such as joules and electronvolts or kilowatt-hours.
Base-quantity definitions and dimensional analysis are not required.

Common SI prefixes step through powers of one thousand around the base unit.

Worked example

Convert an area of $0.36\,\text{mm}^2$ to square metres.

1. $1\,\text{mm}=10^{-3}\,\text{m}$ .
2.Square the whole conversion: $1\,\text{mm}^2=10^{-6}\,\text{m}^2$ .
3. $0.36\times10^{-6}=3.6\times10^{-7}$ .

Answer: The area is 3.6 × 10⁻⁷ m².

Common mistakes

Don't square a measurement and fail to square its prefix conversion.
Don't use grams rather than kilograms as the SI base unit of mass.
Don't write a numerical conversion without retaining the physical unit.

Exam tip

Convert every quantity to a consistent unit system before substitution and show each prefix multiplier.

Tier 1 · Easy

ORIGINAL

Convert

4.7\,\mu\text{m}

into metres.

[1 mark]

Total for this question: 1

Tier 2 · Standard

ORIGINAL

A heater transfers

2.4\,\text{kW h}

of energy. Calculate this energy in joules. Use

1\,\text{kW h}=3.60\times10^6\,\text{J}

[2 marks]

Total for this question: 2

Tier 3 · Hard

ORIGINAL

A pulse transfers charge

5.4\,\text{nC}

through a sensor of area

0.36\,\text{mm}^2

12\,\mu\text{s}

. Calculate the mean current density in

\text{A m}^{-2}

[4 marks]

Total for this question: 4

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3.1.2

Year 1

Limitation of physical measurements

Notes

Evidence from your answers: none yet

Your confidence:

Definition

A random error produces scatter about the true value and is reduced by repeating and averaging; a systematic error shifts every reading the same way and is removed by correcting the method or the instrument. Precision is the closeness of repeated readings to each other; accuracy is closeness to the true value; resolution is the smallest change an instrument can show. Uncertainty is quoted as absolute ( $\pm0.2\,\text{s}$ ), fractional or percentage: absolute uncertainties add for a sum or difference, percentage uncertainties add for a product or quotient, and a power $n$ multiplies the percentage uncertainty by $n$ .

Explanation

Random errors cause scatter and are reduced by repeats and averaging; systematic errors shift results consistently and require correction or removal. Accuracy means closeness to the true value, precision concerns spread, resolution is the smallest detectable change, repeatability keeps method and operator fixed, and reproducibility changes them.
Uncertainty may be absolute, fractional or percentage. For addition and subtraction, combine absolute uncertainties; for multiplication and division, combine percentage uncertainties; for a power $x^n$ , multiply percentage uncertainty by $|n|$ .
Graph points may carry error bars.
Steepest and shallowest acceptable lines give gradient uncertainty, with corresponding intercept limits.
Report value and absolute uncertainty to compatible decimal places and significant figures.

Error bars constrain the steepest and shallowest acceptable straight lines.

Worked example

A student times

20

oscillations of a pendulum five times and records

19.0\,\text{s}

19.3\,\text{s}

19.2\,\text{s}

19.4\,\text{s}

and

19.1\,\text{s}

. Determine the period

T

of the pendulum and the percentage uncertainty in

T

[3 marks]

Total for this question: 3

1.Mean time for $20T$ : $\dfrac{19.0+19.3+19.2+19.4+19.1}{5}=19.2\,\text{s}$ , so $T=\dfrac{19.2}{20}=0.960\,\text{s}$ .
2.Uncertainty from half the range: $\dfrac{19.4-19.0}{2}=0.2\,\text{s}$ in $20T$ , i.e. $0.01\,\text{s}$ in $T$ .
3.Percentage uncertainty $=\dfrac{0.2}{19.2}\times100=1.04\%\approx1.0\%$ (the same for $T$ : $0.01/0.960$ ).

Answer: $T=0.960\,\text{s}$ with a percentage uncertainty of $1.0\%$

Common mistakes

Don't add absolute uncertainties for a multiplication calculation.
Don't call a tightly clustered but offset set of readings accurate.
Don't quote the measured value to more decimal places than its absolute uncertainty.

Mark-scheme wording to use

measuring 20T reduces the effect of random error on T or reduces percentage/relative uncertainty — paraphrase of AQA 7408, June 2025 Paper 3, Q02.1
repeats check for, enable detection of, or enable elimination of anomalies — paraphrase of AQA 7408, June 2025 Paper 3, Q02.2
draw maximum- and minimum-gradient lines consistent with all error bars — paraphrase of AQA 7408, June 2024 Paper 3, Q01.8
combine as 2(%d)+(%L) — paraphrase of AQA 7408, June 2022 Paper 3, Q03.3
half-range uncertainty 0.0040 m — paraphrase of AQA 7408, November 2020 Paper 3, Q01.4

What the examiner wants

The November 2021 Paper 3 report found that most rejections of an inverse-square claim lacked a quantitative comparison — calculate the constant for at least two points and compare the numbers, never 'the values differ'. Reject an anomaly BEFORE the mean, take half the range as the uncertainty, and quote the value and its absolute uncertainty to the same decimal place. Give the reason with the procedure: 'repeat and average' scores only when you say it reduces random error.

Exam tip

State whether each operation needs absolute or percentage uncertainty before combining terms.

Tier 1 · Easy

ORIGINAL

A diameter is measured as

(82.0\pm0.5)\,\text{mm}

. Calculate its percentage uncertainty.

[2 marks]

Total for this question: 2

Tier 2 · Standard

ORIGINAL

The sides of a rectangular card are

(4.20\pm0.05)\,\text{cm}

and

(2.10\pm0.03)\,\text{cm}

. Determine its area and absolute uncertainty.

[3 marks]

Total for this question: 3

Tier 3 · Hard

ORIGINAL

A pendulum has length

(0.842\pm0.002)\,\text{m}

. The time for

20

oscillations is

(36.4\pm0.2)\,\text{s}

. Use

g=4\pi^2L/T^2

to calculate

g

with its absolute uncertainty.

[5 marks]

Total for this question: 5

3.1.3

Year 1

Estimation of physical quantities

Notes

Evidence from your answers: none yet

Your confidence:

Explanation

An order of magnitude is the nearest power of ten. A sound estimate chooses plausible approximate inputs, states assumptions such as representative size, density or operating fraction, and uses relevant physics to derive the requested quantity.
Intermediate values usually need only one significant figure.
The final numerical estimate is then compared with neighbouring powers of ten: the boundary between $10^n$ and $10^{n+1}$ is $\sqrt{10}\times10^n\approx3.2\times10^n$ .
Units and dimensions provide an essential reasonableness check.
Estimation is not guessing a remembered number; the awarded reasoning comes from transparent assumptions, a valid physical relationship and a final answer rounded to the nearest order of magnitude.

The nearest-order boundary lies at approximately 3.2 times the lower power of ten.

Worked example

Estimate the order of magnitude of the mass of air in an $8\,\text{m}\times6\,\text{m}\times3\,\text{m}$ room, using air density $1\,\text{kg m}^{-3}$ .

1.Volume is approximately $8(6)(3)=1.4\times10^2\,\text{m}^3$ .
2. $m=\rho V\approx1(1.4\times10^2)=1.4\times10^2\,\text{kg}$ .
3.This is nearer $10^2$ than $10^3$ .

Answer: The mass has order of magnitude $10^2\,\text{kg}$ .

Common mistakes

Don't report a multi-significant-figure value instead of a power-of-ten order of magnitude.
Don't use an assumed value without stating or justifying it.
Don't round every coefficient above one to the next power of ten.

Exam tip

Show the assumption, physics relationship and numerical estimate before stating the nearest power of ten.

Tier 1 · Easy

ORIGINAL

Estimate the order of magnitude of the mass of air in a room measuring

8\,\text{m}\times6\,\text{m}\times3\,\text{m}

. Take the density of air as

1\,\text{kg m}^{-3}

[1 mark]

Total for this question: 1

Tier 2 · Standard

ORIGINAL

Estimate the order of magnitude of the number of water molecules in

250\,\text{cm}^3

of water. Use density

1.0\,\text{g cm}^{-3}

, molar mass

18\,\text{g mol}^{-1}

and

N_A=6.0\times10^{23}\,\text{mol}^{-1}

[3 marks]

Total for this question: 3

Tier 3 · Hard

ORIGINAL

Estimate the order of magnitude of the total electrical power drawn by domestic kettles in a country of population

6.8\times10^7

. Assume

2.5

people per household, a

3\,\text{kW}

kettle in each household, and that

4\%

of kettles are operating at one time.

[5 marks]

Total for this question: 5

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