Measurement uncertainties

Physics· Unit 1: Physical quantities and units, 1.6 Measurement uncertainties· 25 min read

1. Types of Uncertainty★★☆☆☆⏱ 5 min

All experimental measurements have uncertainty, arising from limitations in instruments or technique. Uncertainties are split into two core categories that behave differently and are reduced in different ways.

📘 Definition

Random vs Systematic Uncertainty

Random uncertainties cause scatter of measurements around the true value, leading to a spread of results. They can be reduced by taking multiple readings and calculating a mean. Systematic uncertainties cause all measurements to shift consistently in one direction from the true value, resulting in bias that cannot be reduced by averaging, only by improving experimental technique.

Example:

A stopwatch with zero error that reads 0.2 s when started introduces a systematic uncertainty; human reaction time when starting/stopping the watch introduces a random uncertainty.

📐 Worked Example

Identify whether each uncertainty is random or systematic: (a) Variation in estimating readings between 1 mm marks on a ruler (b) A thermometer calibrated to read 1 °C too high at all values

1
For (a): Estimates between marks vary between readings, with no consistent shift. This is a random uncertainty.
2
For (b): All readings are shifted 1 °C higher in the same direction. This is a systematic uncertainty.

Exam tip:

CIE often asks how to reduce each uncertainty type – remember averaging only works for random uncertainty.

2. Absolute, Fractional and Percentage Uncertainty★★☆☆☆⏱ 7 min

Uncertainty can be expressed in three common forms, each used for different stages of calculation.

📘 Definition

Uncertainty Types

A measurement is written $x \pm Δ x$ , where $x$ is the measured value and $Δ x$ is absolute uncertainty

Absolute uncertainty: Margin of error in the same units as $x$ , equal to half the range of repeated readings, or the smallest division on analog instruments.
Fractional uncertainty: Unitless ratio $\frac{Δ x}{x}$
Percentage uncertainty: $\frac{Δ x}{x} \times 100%$

📐 Worked Example

A student measures current 4 times, getting readings: 1.2 A, 1.5 A, 1.3 A, 1.4 A. Find the mean current, absolute uncertainty and percentage uncertainty.

1
Calculate the mean current:
$I = \frac{1.2 + 1.5 + 1.3 + 1.4}{4} = 1.35 \text{ A}$
2
Find the range of readings:
$\text{Range} = 1.5 - 1.2 = 0.3 \text{ A}$
3
Calculate absolute uncertainty as half the range:
$\Delta I = \frac{0.3}{2} = 0.15 \approx 0.2 \text{ A}$
4
Calculate percentage uncertainty:
$\text{Percentage uncertainty} = \frac{0.2}{1.35} \times 100 \approx 15\%$

✓ Quick check

Check your understanding:

A digital ammeter displays current to 0.01 A. What is its absolute uncertainty?
- 0.005 A
- 0.01 A
- 0.05 A
Reveal answer
0.01 A —
For digital instruments, the absolute uncertainty equals the smallest displayed division.

3. Combining Uncertainties★★★☆☆⏱ 10 min

When you calculate a final quantity from multiple measured values, you combine uncertainties according to the mathematical operation used. Uncertainty always increases when combining measurements, so we always add uncertainties.

Addition / Subtraction: For $y = a \pm b$ , add absolute uncertainties: $Δ y = Δ a + Δ b$
Multiplication / Division: For $y = \frac{ab}{c}$ , add fractional uncertainties: $\frac{Δ y}{y} = \frac{Δ a}{a} + \frac{Δ b}{b} + \frac{Δ c}{c}$
Power: For $y = a x^{n}$ , fractional uncertainty is $\frac{Δ y}{y} = \frac{Δ a}{a} + n \frac{Δ x}{x}$

📐 Worked Example

The side length of a cube is $l = 5.0 \pm 0.2$ cm. Calculate the surface area and its absolute uncertainty.

1
Write the formula for surface area of a cube:
$A = 6l^2$
2
Calculate the mean surface area:
$A = 6 \times (5.0)^2 = 150 \text{ cm}^2$
3
6 is a constant with no uncertainty, so apply the power rule:
$\frac{\Delta A}{A} = 2 \times \frac{\Delta l}{l} = 2 \times \frac{0.2}{5.0} = 0.08$
4
Find absolute uncertainty and write the final answer:
$\Delta A = 0.08 \times 150 = 12 \text{ cm}^2 \implies A = 150 \pm 12 \text{ cm}^2$

4. Uncertainties in Graphs★★★☆☆⏱ 8 min

In practical exam questions, you will often need to use error bars to find the uncertainty in the gradient and intercept of a line of best fit.

📘 Definition

Error Bars

Vertical or horizontal bars drawn through each plotted point, extending from $x - Δ x$ to $x + Δ x$ , to show the range of possible true values for each measurement.

📐 Worked Example

Calculate the uncertainty in the gradient of a straight line fit from experimental data.

1
1. Plot all points with error bars, draw the line of best fit and calculate its gradient $m_{b es t}$ .
2
1. Draw the steepest possible line and shallowest possible line that both pass through all error bars, to get $m_{ma x}$ and $m_{min}$ .
3
1. Calculate the absolute uncertainty:
$\Delta m = \frac{m_{max} - m_{min}}{2}$

Exam tip:

If a question asks you to show error bars, you must include them to get full marks, even if they are small.

5. Common Pitfalls

Wrong move:

Subtracting uncertainties when dividing two quantities

Why:

Uncertainty always increases when combining measurements, so all uncertainties are added

Correct move:

Add fractional uncertainties for both multiplication and division, never subtract

Wrong move:

Forgetting to multiply by the power for power terms

Why:

This is a common exam mistake that leads to an undercalculated uncertainty

Correct move:

Always multiply the fractional uncertainty by the power, e.g. 2 for area, 3 for volume

Wrong move:

Claiming averaging reduces systematic uncertainty

Why:

Systematic uncertainty is a consistent bias that affects all readings equally

Correct move:

Averaging only reduces random uncertainty; systematic uncertainty is fixed via calibration or correcting zero error

Wrong move:

Taking absolute uncertainty equal to the full range of repeated readings

Why:

This overestimates the uncertainty in the mean value

Correct move:

For repeated readings, absolute uncertainty is half the range of readings

Wrong move:

Reporting uncertainties with more than two significant figures

Why:

Uncertainty is an estimate, extra significant figures are meaningless in exams

Correct move:

Round all uncertainties to 1 or 2 significant figures, always round up

6. Quick Reference Cheatsheet

Operation/Type	Rule
Addition/Subtraction $y = a \pm b$	$Δ y = Δ a + Δ b$
Multiplication/Division $y = \frac{ab}{c}$	$\frac{Δ y}{y} = \sum \frac{Δ a _{i}}{a _{i}}$
Power $y = a x^{n}$	$\frac{Δ y}{y} = n \frac{Δ x}{x}$
Random Uncertainty	Reduced by averaging multiple readings
Systematic Uncertainty	Fixed by calibration/zero error correction
Gradient Uncertainty	$Δ m = \frac{m _{ma x} - m _{min}}{2}$

7. Frequently Asked

What is the difference between uncertainty and error?

In CIE A-Level Physics, uncertainty is the range of values within which the true value is expected to lie. An error is the difference between a measured value and the true value.

How do I round uncertainties for final answers?

Uncertainties are rounded up to 1 or 2 significant figures. Measured values are rounded to the same decimal place as their absolute uncertainty.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 11
Percentage uncertainty calculation
2023 · 22
Combine uncertainties for product
2024 · 12
Random vs systematic uncertainty

Going deeper

official guideCIE 9702 Practical Skills HandbookOfficial guidance for practical uncertainty questions

What's Next

Measurement uncertainties are a core skill for the entire CIE A-Level Physics course, appearing in every paper from multiple choice to the full practical assessment. You will apply uncertainty calculations to every experimental topic, from kinematics and forces to electricity and waves. Mastering this sub-topic early ensures you do not lose easy marks in exam questions involving experimental data, which make up ~15% of total marks across the qualification. This knowledge also forms the foundation for more advanced practical data analysis skills required for A2 level.