9702 · AS & A Level · Papers 3 & 5

A Level Physics 9702 — Uncertainties Trainer

Master absolute and percentage uncertainties, combining rules, and graph-based uncertainty — the skills Cambridge tests on every practical paper.

About this tool

Cambridge A Level Physics uncertainty rules for Papers 3 and 5 — reference, combining rules, and drill in one tool

Uncertainty calculations appear on every Cambridge A Level Physics 9702 practical paper. Paper 3 (AS Level) tests absolute uncertainties from scale readings, percentage uncertainties, combining uncertainties when adding or subtracting and when multiplying or dividing, and reading the uncertainty in a gradient from a graph. Paper 5 (A Level) applies the same skills within a full analysis question — the uncertainty in the gradient is typically derived from the maximum and minimum gradient lines drawn through the error bars, then expressed as a percentage. Examiner reports across multiple series highlight the same recurring errors: omitting the factor of two when using half-range from repeated readings, confusing absolute and percentage forms when combining, and quoting percentage uncertainties to too many significant figures.

This tool combines a reference section and a drill. The reference covers what uncertainty means and why every measurement has one, reading instrument uncertainties (ruler, vernier caliper, micrometer, stopwatch, ammeter, voltmeter), the four combining rules (addition/subtraction; multiplication/division; raising to a power; functions), finding gradient uncertainty from maximum and minimum gradient lines, and Cambridge’s significant-figure conventions for uncertainty expressions. The drill generates numerical uncertainty problems with step-by-step worked solutions so you can practice the calculation mechanics until they are automatic.

Use the reference to build the mental model, then switch to drill once you can work through the combining rules without prompting. Focus particularly on the gradient-uncertainty method and percentage uncertainty combining — these appear on almost every Paper 5 Q2 and are among the most frequently dropped mark points.

What is Uncertainty?
The foundation — before combining anything, you must understand what uncertainty means
Core Idea
Every measurement has an uncertainty

When you measure something, you cannot be perfectly precise. The absolute uncertainty tells you the range within which the true value lies. It always has the same units as the measurement.

measurement = (value ± absolute uncertainty) unit
e.g. length = (24.5 ± 0.5) mm

The percentage uncertainty expresses that uncertainty relative to the measurement itself:

% uncertainty = (absolute uncertainty ÷ value) × 100%
e.g. (0.5 ÷ 24.5) × 100% = 2.0%
Cambridge convention: Quote % uncertainties to 1 or 2 significant figures. Never more.
Reading Uncertainty from Instruments
How to determine absolute uncertainty from a scale or measuring device
Rule
Analogue instruments (rulers, thermometers, ammeters)

For any analogue scale, the uncertainty is half the smallest division.

absolute uncertainty = ½ × smallest division
e.g. ruler with 1 mm divisions → uncertainty = ±0.5 mm
e.g. thermometer with 0.2 °C divisions → uncertainty = ±0.1 °C
For a length measured with a ruler: you take a reading at each end, so uncertainty in the length = 2 × (½ × smallest division) = ±1 division.
Rule
Digital instruments

The uncertainty is ±1 in the last displayed digit.

e.g. voltmeter reads 3.45 V → uncertainty = ±0.01 V
e.g. stopwatch reads 12.3 s → uncertainty = ±0.1 s
Rule
Vernier callipers & micrometers
Vernier calliper (0.02 mm graduation) → uncertainty = ±0.02 mm
Micrometer screw gauge → uncertainty = ±0.005 mm
(Stated precision is taken as the uncertainty)
Combining Uncertainties
The rules Cambridge tests most frequently — know these cold
Rule 1
Addition and Subtraction → Add absolute uncertainties

If z = x + y or z = x − y:

Δz = Δx + Δy
Always add — never subtract — the absolute uncertainties
x = (12.0 ± 0.5) cm, y = (4.0 ± 0.5) cm
z = x − y = 8.0 cm, Δz = 0.5 + 0.5 = 1.0 cm
z = (8.0 ± 1.0) cm
Key point: Subtracting two similar values makes the % uncertainty very large — this is why Cambridge often penalises poor experimental design where a small difference between large numbers is taken.
Rule 2
Multiplication and Division → Add percentage uncertainties

If z = x × y or z = x ÷ y:

%Δz = %Δx + %Δy
Convert to % first, add them, convert back if needed
V = I × R, %ΔI = 2%, %ΔR = 3%
%ΔV = 2 + 3 = 5%
If V = 6.0 V: ΔV = 5% × 6.0 = 0.3 V → V = (6.0 ± 0.3) V
Rule 3
Powers → Multiply percentage uncertainty by the power

If z = xⁿ:

%Δz = n × %Δx
The power multiplies the % uncertainty — including fractional powers (roots)
z = x³, %Δx = 2% → %Δz = 3 × 2% = 6%
z = √x = x^(½), %Δx = 4% → %Δz = ½ × 4% = 2%
Rule 4
Constants → contribute zero uncertainty

Pure numbers and defined constants (π, 2, 4π², etc.) have no uncertainty and do not contribute to the combined uncertainty.

z = 2πr → %Δz = %Δr (2 and π contribute nothing)
Identifying the Largest Uncertainty Source
A Cambridge favourite — which measurement limits the precision most?
Strategy
Compare percentage contributions

Convert each measurement's uncertainty to a percentage, then compare. The largest % uncertainty is the dominant source — improving this instrument would give the biggest gain in precision.

For z = (x² × y) / w
%Δz = 2(%Δx) + %Δy + %Δw
Compare each term — the biggest term dominates
Note: Powers amplify — a 2% uncertainty in x becomes 4% in x², so powered terms are often the dominant source even when the raw % uncertainty looks small.
Uncertainty in Gradient and Intercept
Paper 5 — uncertainty from best-fit and worst-acceptable lines
Method
Best-fit and worst-acceptable lines

Draw the best-fit line (passing through or near all error bars, or the line of best fit through the points).

Draw the worst-acceptable line — the steepest or shallowest line that still passes through all error bars (or is consistent with the scatter).

gradient uncertainty = |gradient(best) − gradient(worst)|
intercept uncertainty = |intercept(best) − intercept(worst)|
m_best = 2.40 cm/s², m_worst = 2.10 cm/s²
Δm = |2.40 − 2.10| = 0.30 cm/s²
gradient = (2.40 ± 0.30) cm/s²
Always use large triangles when calculating gradients — at least half the drawn line length. Small triangles amplify reading errors.
% Uncertainty
Percentage uncertainty in a gradient
% uncertainty in gradient = (Δm / m_best) × 100%
Same formula as for any other quantity
Quick Reference Summary
All four combining rules at a glance
Operation Rule Example
z = x + y or x − y Δz = Δx + Δy Add absolute uncertainties
z = x × y or x / y %Δz = %Δx + %Δy Add percentage uncertainties
z = xⁿ %Δz = n × %Δx Multiply % by power
z = √x %Δz = ½ × %Δx Power is ½
z = constant × x %Δz = %Δx Constants add nothing
Graph gradient Δm = |m_best − m_worst| Best-line vs worst-line
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Uncertainties trainer P3 & P5
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Frequently asked questions

Uncertainties, quickly answered

How do you calculate percentage uncertainty in physics?

Divide the absolute uncertainty by the measured value and multiply by 100. For a single reading, the absolute uncertainty is usually half the smallest scale division, or the instrument's stated resolution.

How do you combine uncertainties?

When quantities are multiplied or divided, add their percentage uncertainties. When they are added or subtracted, add their absolute uncertainties. For a power, multiply the percentage uncertainty by that power.

What is the difference between absolute and percentage uncertainty?

Absolute uncertainty is the ± value in the same units as the measurement; percentage uncertainty expresses that same uncertainty as a percentage of the measured value.

How are uncertainties tested in A Level Physics?

On the practical papers — 9702 Paper 3 and Paper 5 — through error bars, percentage-uncertainty calculations, and finding the uncertainty in a gradient using the worst acceptable line.