Measurements and their errors: A-level Physics revision
The skills every AQA Physics answer depends on: SI units, prefixes, standard form, uncertainty, significant figures and order-of-magnitude estimates. This sheet covers the whole of section 3.1 with worked examples and a self-check.
Work through it in three passes
Get the units right
Learn the SI base units and prefixes, then practise converting between units and writing values in standard form.
Understand uncertainty
Distinguish random and systematic errors, and know when to add, combine or multiply percentage uncertainties.
Sense-check your answers
Use significant figures sensibly and make order-of-magnitude estimates to spot mistakes.
- Recall the SI base quantities and their units, and use prefixes from tera to femto.
- Use standard form and convert between units, including joule ↔ electronvolt and joule ↔ kilowatt-hour.
- Identify random and systematic errors and describe precision, accuracy, repeatability, reproducibility and resolution.
- Calculate absolute, fractional and percentage uncertainty and combine uncertainties for addition, subtraction, multiplication, division and powers.
- Give answers to an appropriate number of significant figures.
- Make and justify order-of-magnitude estimates of physical quantities.
The units every calculation needs
AQA expects you to use SI units consistently. The six base quantities assessed in AQA Physics are shown below. The wider SI system also includes luminous intensity and the candela, which are outside this specification. The derived units used here are built from the assessed base units.
| Base quantity | SI unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Amount of substance | mole | mol |
| Temperature | kelvin | K |
| Electric current | ampere | A |
Always write units in the correct index form. For example, acceleration is m s−2, not m/s/s.
Scaling units up and down
Prefixes let you write very large or very small numbers neatly. You must know the powers of ten for each prefix in the table below.
| Prefix | Symbol | Power of ten | Prefix | Symbol | Power of ten |
|---|---|---|---|---|---|
| tera | T | 1012 | centi | c | 10−2 |
| giga | G | 109 | milli | m | 10−3 |
| mega | M | 106 | micro | μ | 10−6 |
| kilo | k | 103 | nano | n | 10−9 |
| pico | p | 10−12 | |||
| femto | f | 10−15 |
Standard form means writing a number as a × 10n where 1 ≤ a < 10 and n is an integer. It keeps powers of ten visible, which makes prefix conversions much safer.
1 eV = 1.60 × 10−19 J 1 kWh = 3.6 × 106 J 1 year ≈ 3.15 × 107 s
Random, systematic and how to reduce them
Random errors
Scatter in readings due to unpredictable variation. They affect precision.
Reduce by: repeating readings and taking a mean.
Systematic errors
A consistent bias in readings, such as a zero offset or calibration scale error, often caused by apparatus or method. They affect accuracy.
Reduce by: calibration, checking zero errors, improving the method.
| Term | What it means |
|---|---|
| Precision | How close repeated readings are to each other; small scatter means high precision. |
| Accuracy | How close a measured value is to the true value. |
| Repeatability | Similar results obtained when the same experimenter repeats the measurement using the same method and equipment. |
| Reproducibility | Similar results obtained when a different experimenter, method or equipment is used. |
| Resolution | The smallest change an instrument can detect; the smallest interval on its scale. |
High precision does not mean high accuracy. Precise repeated readings can all be wrong by the same systematic offset.
Absolute, fractional, percentage and combined
The absolute uncertainty, written ±Δx, estimates the spread of values reasonably attributable to a measurement. An instrument-only starting estimate is often half the smallest division for an analogue reading or one least-significant digit for a digital reading. Use any stated instrument specification and include the actual measurement method: reaction time, repeated-reading scatter and two ruler-end readings can make the uncertainty larger. Resolution alone does not guarantee accuracy.
Fractional uncertainty
Divide the absolute uncertainty by the measured value.
Percentage uncertainty
Multiply the fractional uncertainty by 100.
Combining uncertainties
These are the usual approximate maximum-uncertainty rules for small relative uncertainties, rather than statistical standard-error propagation. Exact constants contribute no uncertainty. For a repeated measurement, half the range is a useful estimate of scatter; compare it with the instrumental and method uncertainties. For graph gradients, draw the best-fit line and the steepest and shallowest lines consistent with the error bars. Use their range to estimate the gradient uncertainty; use a large triangle on each line.
| Operation | Rule | Example |
|---|---|---|
| Addition or subtraction | Δq = Δa + Δb | If q = a + b, add absolute uncertainties. |
| Multiplication or division | %Δq = %Δa + %Δb | If q = ab or a/b, add percentage uncertainties. |
| Powers | %Δq ≈ |n| × %Δa | If q = an, multiply the percentage uncertainty by the absolute exponent |n|. For q = a−2, double it. |
For more worked practice in these transferable methods, see the OCR A uncertainty and data-analysis guide. The physics methods transfer; its specification references belong to OCR A.
When you multiply two quantities, the relative spread in the result depends on the relative spread in each input. Adding percentage uncertainties handles this correctly; adding absolute uncertainties does not.
Sense-check your answers
An order-of-magnitude estimate is a calculation rounded to the nearest power of ten. It is useful for checking whether an answer is sensible and for answering estimation questions.
Mass of an electron ≈ 10−30 kg, mass of a proton ≈ 10−27 kg, height of a room ≈ 3 m, speed of sound in air ≈ 330 m s−1, density of water ≈ 1000 kg m−3.
Model a classroom as roughly 10 m × 5 m × 3 m, so its volume is about 150 m3. Using an air density of about 1 kg m−3 gives a mass of about 150 kg, or 102 kg to the nearest order of magnitude. The important skill is combining plausible physical dimensions with a known density, not recalling an exact classroom value.
Test your unit and uncertainty skills
1. Convert 4.7 × 105 m into kilometres and write your answer in standard form.
2. A length L = (25.0 ± 0.5) cm and a width W = (10.0 ± 0.5) cm. What is the percentage uncertainty in the area A = L × W?
3. A digital stopwatch reads 12.46 s. Assume an instrument-only absolute uncertainty of ±0.01 s and ignore reaction time. What is the percentage uncertainty?
Two exam-style calculations
A capacitor stores 47 μC of charge. Write this charge in coulombs in standard form.
Method: micro means 10−6.
A ball falls through a height h = (1.25 ± 0.01) m and takes t = (0.505 ± 0.001) s to reach the ground. Calculate the percentage uncertainty in g = 2h/t2.
Method: Find %Δh and %Δt, then double %Δt because of the t2 term.
%Δt = (0.001 / 0.505) × 100 = 0.198%
%Δg = %Δh + 2 × %Δt = 0.80 + 2(0.198) = 1.20% ≈ 1.2%
Mistakes that cost marks
- Adding absolute uncertainties when multiplying. Always convert to percentage uncertainties first, then add.
- Forgetting to multiply the percentage uncertainty by the absolute exponent. If q = t2, then %Δq = 2 × %Δt.
- Confusing precision and accuracy. Precision is about scatter; accuracy is about closeness to the true value.
- Using too many significant figures. A calculated value cannot be more precise than the data it came from.
- Treating eV as a voltage. The electronvolt is a unit of energy, 1 eV = 1.60 × 10−19 J.
Try these, then reveal the answer
Convert 3.3 × 108 m to kilometres and megametres.
3.3 × 105 km and 3.3 × 102 Mm.
A force is (12.0 ± 0.5) N and a distance is (0.80 ± 0.02) m. Find the percentage uncertainty in the work done.
Work done = force × distance. %ΔF = 4.17% and %Δd = 2.50%, so %ΔW = 6.67% ≈ 6.7%.
Explain the difference between repeatability and reproducibility.
Repeatability refers to the same experimenter using the same equipment and method. Reproducibility refers to different experimenters, methods or equipment giving consistent results.
Why do repeats reduce random error but not systematic error?
Repeats allow you to average out random fluctuations. A systematic error is present in every reading, so averaging does not remove it.
Before you move on
- I can list the six SI base quantities and their units.
- I can convert between prefixes using powers of ten and standard form.
- I can distinguish random and systematic errors and describe precision, accuracy, repeatability, reproducibility and resolution.
- I can calculate absolute, fractional and percentage uncertainty.
- I can combine uncertainties for addition, subtraction, multiplication, division and powers.
- I give final answers to a sensible number of significant figures.
- I can make and explain order-of-magnitude estimates.
Where to go next
- Particles and radiation revision sheet — the next AQA module.
- Waves revision sheet — covers AQA section 3.3.
- A Level Physics MCQ practice — quick topic tests.
- Interactive physics tools — simulations and calculators.