AQA A-level Physics (7408) · 3.1 Measurements and their errors
Mini-Lesson
Measurements and their errors
This mini-lesson covers AQA 3.1 — Measurements and their errors: the SI system of base and derived units, prefixes and standard form, random and systematic errors, precision vs accuracy, and how to calculate and combine uncertainties. These skills are worth marks in every single paper.
Section 3.1 runs through the whole course — it is assessed in every paper.
Where this sits: AQA sections 3.1–3.8 are compulsory for every A-level Physics student. (Sections 3.9–3.13 are the five options — you study just one of those.)
Work through each screen, answer the questions as you go (many are full A-level calculations) and collect ⭐ stars. Press Start when you're ready.
3.1.1 · SI units
The SI system: base and derived units
Every physical quantity is expressed as a number × a unit. All units in physics are built from just six SI base units you need at A-level:
metre (m) — length · kilogram (kg) — mass · second (s) — time
ampere (A) — electric current · kelvin (K) — temperature · mole (mol) — amount of substance
Everything else is a derived unit — a combination of base units, sometimes given a special name.
N = kg m s−2 · J = N m = kg m2 s−2 · W = J s−1V = J C−1 = kg m2 s−3 A−1 · Pa = N m−2 = kg m−1 s−2
Worked example — checking an equation by homogeneity
Is v² = u² + 2as homogeneous? LHS: (m s−1)² = m² s−2.
RHS second term: (m s−2)(m) = m² s−2. Both sides match — the equation is homogeneous.
Exam skill: a homogeneous equation can still be wrong (a missing factor of ½ is dimensionless), but a non-homogeneous equation is definitely wrong. Use this to spot slips in derivations.
3.1.1 · prefixes & standard form
Prefixes, standard form and orders of magnitude
SI prefixes are multiplying factors. Learn these to convert quickly and safely:
T (tera) 1012 · G (giga) 109 · M (mega) 106 · k (kilo) 103
c (centi) 10−2 · m (milli) 10−3 · µ (micro) 10−6 · n (nano) 10−9 · p (pico) 10−12 · f (femto) 10−15
An order of magnitude is a power-of-ten estimate. Physicists compare wildly different scales by taking ratios:
Scale check
Atom ≈ 10−10 m · nucleus ≈ 10−15 m → ratio = 10−10 ÷ 10−15 = 105.
The atom is five orders of magnitude bigger than its nucleus — which is why the atom is mostly empty space.
Trap: when a prefix is squared or cubed, so is its factor. 1 cm² = (10−2 m)² = 10−4 m², and 1 mm³ = 10−9 m³. Cross-sectional areas of wires in mm² catch people out every year.
Sort it
Base unit, derived unit or prefix?
Tap a card, then tap the box it belongs in.
🧱 SI base unit
🔧 Derived unit
🔢 Prefix
Quick check
Think it through
?Which of these is not an SI base unit?
3.1.2 · errors
Random error, systematic error, precision and accuracy
Two different things can spoil a measurement, and AQA wants you to tell them apart:
Random error — readings scatter either side of the true value (e.g. reaction time when starting a stopwatch). Reduce it by repeating and averaging, and by using instruments with better resolution.
Systematic error — every reading is shifted the same way (e.g. an ammeter that reads 0.10 A with nothing connected — a zero error). Repeating does not help; you must recalibrate or subtract the offset.
precise ≠ accurateprecise = readings agree with each other · accurate = readings agree with the true value
Resolution — the smallest change an instrument can detect (a micrometer: 0.01 mm; a metre rule: 1 mm).
Repeatability — same person, same method, same equipment gets the same result. Reproducibility — a different person or method gets the same result.
Classic exam line: a set of readings can be very precise but very inaccurate — that is the signature of a systematic error (e.g. an uncalibrated balance). Random error alone makes results imprecise but they still average out around the true value.
Quick check
Think it through
?A student measures the current in a circuit five times. All five readings are within 0.01 A of each other, but every reading is 0.20 A higher than the true value. What does this show?
3.1.2 · uncertainty
Absolute, fractional and percentage uncertainty
Every measurement should be quoted as value ± uncertainty, e.g. (45.0 ± 0.5) mm.
% uncertainty = (absolute uncertainty ÷ measured value) × 100fractional uncertainty = absolute uncertainty ÷ value (no % sign)
For a single reading from an analogue scale, take the uncertainty as ± half the smallest division (metre rule, 1 mm divisions → ±0.5 mm).
For a digital instrument, take ± the last digit (a balance reading 12.34 g → ±0.01 g).
For repeated readings, uncertainty = ½ × (max − min) — the half-range — about the mean.
A length measured between two marks (both ends of a rule) uses the uncertainty twice: ±1 mm in total.
Worked example
A wire diameter is (0.36 ± 0.01) mm from a micrometer.
% uncertainty = (0.01 ÷ 0.36) × 100 = 2.8%. A small absolute uncertainty on a small quantity can still be a large percentage — that is why the diameter usually dominates the error in a resistivity experiment.
Calculate
Your turn — calculation 1
1A length is measured with a metre rule as 45.0 mm ± 0.5 mm. Calculate the percentage uncertainty in the length, to 2 significant figures.
%
Hint: (0.5 ÷ 45.0) × 100.
Calculate
Your turn — calculation 2
2Five repeat readings of a length (in mm) are: 20.1, 20.5, 20.3, 20.7, 20.4. Calculate the mean length.
mm
Hint: Sum = 102.0 mm; mean = 102.0 ÷ 5.
Calculate
Your turn — calculation 3
3For the same five readings (20.1, 20.5, 20.3, 20.7, 20.4 mm), calculate the absolute uncertainty using the half-range method.
mm
Hint: Uncertainty = ½ × (max − min) = ½ × (20.7 − 20.1).
3.1.2 · combining uncertainties
Combining uncertainties — the three rules
Uncertainties propagate through calculations. AQA expects three rules:
ADD / SUBTRACT → add the absolute uncertaintiesMULTIPLY / DIVIDE → add the percentage uncertainties · RAISE TO POWER n → multiply the % uncertainty by n
x = a + b or a − b → Δx = Δa + Δb (absolute).
x = ab or a/b → %Δx = %Δa + %Δb.
x = an → %Δx = n × %Δa. (So a volume from a cube side has 3× the % uncertainty of the side; an area has 2×.)
Worked example — resistivity
ρ = RA/L. Suppose R has 2%, the diameter d has 3% (so A = πd²/4 has 2 × 3% = 6%) and L has 1%.
% uncertainty in ρ = 2% + 6% + 1% = 9%. The squared diameter dominates — measure it most carefully.
Graph skill: plot error bars, then draw the steepest and shallowest lines that still pass through every bar. Uncertainty in the gradient = ½ × (max gradient − min gradient).
Calculate
Your turn — calculation 4
4A resistance is found from R = V / I. The pd has a percentage uncertainty of 2% and the current 3%. Calculate the percentage uncertainty in R.
%
Hint: Dividing → ADD the percentage uncertainties: 2 + 3.
Calculate
Your turn — calculation 5
5A cube has sides measured as 2.0 cm ± 0.1 cm. Calculate the percentage uncertainty in its volume (V = side³).
%
Hint: % uncertainty in a side = (0.1 ÷ 2.0) × 100 = 5%. Cubed → 3 × 5%.
Calculate
Your turn — calculation 6
6Power is calculated using P = I²R. The current has a percentage uncertainty of 2% and the resistance 3%. Calculate the percentage uncertainty in P.
%
Hint: I is squared → 2 × 2% = 4%; then add R: 4% + 3%.
Match it
Match each description to the right term
Tap a card on the left, then its partner on the right.
Statement
Answer
3.1.3 · estimation
Estimation and Fermi-style physics
AQA expects you to make order-of-magnitude estimates from everyday knowledge — a skill examiners love because it shows physical judgement.
Worked estimate — power of a person climbing stairs
Mass ≈ 70 kg; a flight of stairs ≈ 3 m; time ≈ 5 s.
P = mgh/t = (70 × 9.81 × 3) ÷ 5 = 2060 ÷ 5 ≈ 4 × 102 W. Comparable to a kettle running for a few seconds — sensible.
Round every input to 1 significant figure — you are hunting the power of ten, not the third decimal place.
Always sanity-check the answer against something you know (a person is not a 10 MW power station).
Quote answers to a sensible number of significant figures: no more than the least precise piece of data you used.
Significant figures rule: if you measure 2.0 V (2 s.f.) and 0.35 A (2 s.f.), quote R = 5.7 Ω (2 s.f.) — not 5.714285 Ω. Writing every digit your calculator shows costs marks.
Quick check
Think it through
?The diameter of an atom is about 1 × 10−10 m and the diameter of its nucleus is about 1 × 10−15 m. By how many orders of magnitude is the atom larger?
Quick check
Think it through
?A student uses a stopwatch by hand to time 1 oscillation of a pendulum, and repeats it 10 times. Which change would most reduce the random uncertainty in the period?
Recap
The big ideas to know
SI units: six base units (m, kg, s, A, K, mol); everything else is derived, e.g. N = kg m s−2
Homogeneity: both sides of an equation must have the same base units
Errors: random = scatter (repeat & average) · systematic = same shift every time (recalibrate)
Precision vs accuracy: precise = readings agree with each other · accurate = agree with the true value