CIE Insider · 9702 — Uncertainties Trainer

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

All questions complete 🎉