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The previous lesson distinguished the causes of measurement error (systematic vs random). This lesson examines the consequences of those errors on the way a measurement is reported. Accuracy, precision, resolution and uncertainty are four related but distinct concepts, and AQA mark schemes regularly require candidates to apply each correctly. The everyday English usage of "accurate" and "precise" overlaps, but in physics they have separate, defined meanings — and confusing them costs marks.
This lesson sets out the formal definitions, the relationship between each concept and the error type it describes, and the conventions for reporting a measurement together with its uncertainty. We will also examine the rules for significant figures and decimal places, which control how precisely a result is stated on paper.
Spec mapping: This lesson addresses content from AQA 7408 §3.1 on the distinction between precision and accuracy, the meaning of resolution in measuring instruments, the conventions for reporting a measurement with its absolute and percentage uncertainty, and the use of significant figures (refer to the official AQA specification document for exact wording).
Synoptic links:
- Required practicals: every practical write-up demands a numerical statement of the absolute or percentage uncertainty on each measured quantity. Markschemes penalise candidates who write "g = 9.81 m s⁻²" without uncertainty.
- Quantum (§3.6): the Heisenberg uncertainty principle is a physical (not experimental) uncertainty; understanding its distinction from measurement uncertainty is part of A-Level reading.
- Significant figures across all numerical questions: quoting a final answer to too many or too few sig figs costs marks in every numerical AQA question.
Key definition. Accuracy describes how close a measured value (or the mean of a set of measurements) is to the true or accepted value. An accurate measurement has small systematic error.
Key definition. Precision describes how close repeated measurements of the same quantity are to each other. A precise set of measurements has small random error — the readings cluster tightly together, regardless of where that cluster sits relative to the true value.
The classic visualisation is a target with a bullseye representing the true value and the shots representing repeated measurements.
| Small random error (precise) | Large random error (not precise) | |
|---|---|---|
| Small systematic error (accurate) | All shots tight cluster on bullseye — best case | Shots scattered widely but centred on bullseye |
| Large systematic error (not accurate) | Tight cluster, off bullseye — repeatable but wrong | Wide scatter, off bullseye — unreliable in every respect |
A class of students measures the boiling point of water at standard pressure. The true value is 100.0 °C. Three groups submit data:
Group A is accurate and precise: the mean is on the true value (small systematic error) and the spread is small (small random error).
Group B is precise but not accurate: the spread is the same 0.4 °C, so they are equally precise, but the mean is 4.9 °C below the true value — a clear systematic error (perhaps an uncalibrated thermometer).
Group C is accurate but not precise: the mean of 100.2 °C is close to the true value, but individual readings scatter widely. A large random error — perhaps the students struggled to determine the instant of boiling.
Exam tip. If asked "how can the student improve accuracy?", look for ways to remove a systematic error (calibrate, change technique). If asked "how can the student improve precision?", look for ways to reduce the random scatter (repeat, use a more sensitive instrument).
Key definition. Resolution is the smallest change in the measured quantity that the instrument can detect, usually equal to the smallest division on the scale (for analogue instruments) or the smallest digit it can display (for digital instruments).
| Instrument | Typical resolution |
|---|---|
| Standard 30 cm ruler | 1 mm |
| Vernier calliper | 0.1 mm (some 0.02 mm) |
| Micrometer | 0.01 mm |
| Digital stopwatch | 0.01 s |
| Digital thermometer | 0.1 °C |
| Digital multimeter (3½ digit) | 0.01 V on 20 V range |
Resolution sets a hard lower limit on the uncertainty of a single reading. A 1 mm ruler cannot meaningfully report a length to "0.1 mm" — the digit would be meaningless.
Resolution and precision are related but not identical. A high-resolution digital voltmeter showing 2.4731 V offers fine resolution; but if the displayed value fluctuates between 2.4720 and 2.4742 over repeated readings, its effective precision is limited by the noise (about ±0.002 V) not by the displayed resolution (0.0001 V). In this case the uncertainty is set by the scatter, not by the display.
Key definition. Uncertainty is a quantitative statement of the range within which the true value of a measurement is expected to lie. It is reported in the form value ± uncertainty (e.g. 25.4 ± 0.2 cm).
The absolute uncertainty Δx has the same units as the measurement x.
The percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value:
%Δx = (Δx / x) × 100%
The percentage uncertainty has no units and is the form in which uncertainties are combined for products, quotients and powers (covered in the next lesson).
A student measures the diameter of a wire five times with a micrometer:
0.43 mm, 0.42 mm, 0.44 mm, 0.43 mm, 0.43 mm.
Mean = 0.430 mm. Maximum = 0.44 mm. Minimum = 0.42 mm. Range = 0.02 mm. Half-range = 0.01 mm.
So the reported diameter is d = 0.43 ± 0.01 mm. Percentage uncertainty = (0.01 / 0.43) × 100 ≈ 2.3%.
A digital voltmeter has a resolution of 0.01 V and reads 4.27 V.
Absolute uncertainty = ±0.005 V (half the resolution).
Percentage uncertainty = (0.005 / 4.27) × 100 ≈ 0.12%.
A metre rule with mm divisions reads 35.7 cm.
Absolute uncertainty conventionally taken as ±1 mm = ±0.1 cm (accounting for both the resolution and the difficulty of judging the meniscus / endpoint).
Percentage uncertainty = (0.1 / 35.7) × 100 ≈ 0.28%.
The number of significant figures (sig figs) reflects the precision with which a quantity is stated. A result quoted to too many sig figs implies a precision the measurement does not have; quoted to too few, it discards real information.
The final answer should be quoted to the same number of significant figures as the least precise input value used in the calculation.
Worked example. A student measures the resistance of a wire as R = 2.15 Ω (3 s.f.) and the length as L = 0.750 m (3 s.f.). The cross-sectional area is calculated from a diameter d = 0.28 mm (only 2 s.f.). The resistivity ρ = RA/L should be quoted to 2 s.f. because the diameter is the limiting input.
The number of sig figs in the value should be consistent with the size of the absolute uncertainty. If you quote a value as 4.27 V, the implicit precision is to the nearest 0.01 V. If the actual uncertainty is ±0.5 V, you should quote the value as 4 ± 0.5 V (1 s.f. in the value, matched to the uncertainty). The rule is: the last significant digit of the value should be at the same decimal place as the first significant digit of the uncertainty.
The standard format for a single quoted measurement is:
quantity = value ± absolute uncertainty unit
Examples:
Markschemes routinely deduct marks for:
A common A-Level required-practical task is to determine the diameter of a wire as accurately as possible. Two instruments are typically available: a vernier calliper (resolution 0.1 mm or 0.02 mm depending on the model) and a digital micrometer (resolution 0.01 mm or 0.001 mm). Reading each correctly, and quoting the absolute and percentage uncertainty for each, is examined directly on Paper 3.
Vernier calliper reading. Suppose a student measures the diameter of a wire and the main scale reads between 0.6 mm and 0.7 mm, with the vernier scale showing the 2nd vernier mark coinciding most clearly with a main-scale division. On a calliper with resolution 0.01 mm (a 50-division vernier), this corresponds to a reading of 0.6 + 0.02 = 0.62 mm.
The absolute uncertainty for a single vernier reading is conventionally taken as ± half the resolution, so Δd = ±0.005 mm. However, in practice the wire may be slightly oval and the contact between the jaws and the wire may not be exact, so the realistic uncertainty is closer to ±0.01 mm — equal to the smallest scale division rather than half of it. Students should justify their chosen uncertainty either way; A-Level markschemes accept both conventions provided the choice is stated.
So d = (0.62 ± 0.01) mm.
Percentage uncertainty = (0.01 / 0.62) × 100 = 1.6%.
Digital micrometer reading. The same wire measured on a digital micrometer of resolution 0.001 mm gives, say, 0.624 mm on a single reading. The instrumental absolute uncertainty is ±0.0005 mm (half the resolution). Repeated readings at different orientations might give 0.624, 0.623, 0.626, 0.625, 0.624 mm — mean 0.6244 mm, half-range 0.0015 mm. The reported uncertainty is then governed by the larger of the two: the half-range (0.0015 mm) exceeds the instrumental half-resolution (0.0005 mm), reflecting the real wire-to-wire variation rather than the instrument's display.
So d = (0.624 ± 0.002) mm (rounding the uncertainty up to a single significant figure and matching the value's last digit).
Percentage uncertainty = (0.002 / 0.624) × 100 = 0.32%.
Comparison. The digital micrometer reduces the percentage uncertainty by a factor of five (from 1.6% to 0.32%) — a substantial improvement that feeds directly into the percentage uncertainty in any quantity calculated from d² (such as cross-sectional area, Young's modulus or resistivity), where the factor-of-two power-rule contribution makes the diameter the dominant uncertainty in most A-Level practicals. The wider lesson is that the choice of instrument is itself an experimental-design decision, and selecting the higher-resolution micrometer is the standard markscheme-rewarded suggestion when asked "how could you reduce the uncertainty in d?".
The four terms precision, accuracy, resolution and sensitivity are easily confused, particularly because everyday English collapses several of them onto "accurate". Examiners regularly test the formal distinctions; the table below sets them out side by side with a worked example for each.
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