# Measurement uncertainties

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u1-measurement-uncertainties/

This sub-topic covers types of measurement uncertainty, how to calculate different forms of uncertainty, and how to combine them for calculated quantities, a core skill for both theory and practical CIE papers.

**Prerequisites:** [Physical quantities and units](https://www.owlsprep.com/study/cie-9702-u1-physical-quantities-units/); [Significant figures](https://www.owlsprep.com/study/cie-9702-u1-significant-figures/)

## Learning objectives

- Distinguish between random and systematic uncertainties
- Calculate absolute, fractional and percentage uncertainties
- Combine uncertainties for calculated quantities
- Find uncertainty in gradient from graphical error bars

## Types of Uncertainty

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.

**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.

## Absolute, Fractional and Percentage Uncertainty

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

**Uncertainty Types** — - **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 $\frac{\Delta x}{x}$
- **Percentage uncertainty**: $\frac{\Delta x}{x} \times 100\%$

*Notation:* A measurement is written $x \pm \Delta x$, where $x$ is the measured value and $\Delta x$ is absolute uncertainty

**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:

   $$\text{Percentage uncertainty} = \frac{0.2}{1.35} \times 100 \approx 15\%$$

**Check your understanding**

Check your understanding:

1. A digital ammeter displays current to 0.01 A. What is its absolute uncertainty?

   - 0.005 A
   - 0.01 A
   - 0.05 A

   *Why:* For digital instruments, the absolute uncertainty equals the smallest displayed division.

## Combining Uncertainties

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.

1. Addition / Subtraction: For $y = a \pm b$, add **absolute uncertainties**: $\Delta y = \Delta a + \Delta b$
2. Multiplication / Division: For $y = \frac{ab}{c}$, add **fractional uncertainties**: $\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$
3. Power: For $y = ax^n$, fractional uncertainty is $\frac{\Delta y}{y} = \frac{\Delta a}{a} + n\frac{\Delta x}{x}$

**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:

   $$\Delta A = 0.08 \times 150 = 12 \text{ cm}^2 \implies A = 150 \pm 12 \text{ cm}^2$$

> **warning**
>
> Never subtract uncertainties, even when variables are divided. All uncertainties add regardless of the operation sign.

## Uncertainties in Graphs

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.

**Error Bars** — Vertical or horizontal bars drawn through each plotted point, extending from $x - \Delta x$ to $x + \Delta x$, 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 $m_{best}$.
2. 2. Draw the steepest possible line and shallowest possible line that both pass through all error bars, to get $m_{max}$ and $m_{min}$.
3. 3. 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.

## Common pitfalls

- **Wrong:** Subtracting uncertainties when dividing two quantities
  - Why it fails: Uncertainty always increases when combining measurements, so all uncertainties are added
  - Correct: Add fractional uncertainties for both multiplication and division, never subtract
- **Wrong:** Forgetting to multiply by the power for power terms
  - Why it fails: This is a common exam mistake that leads to an undercalculated uncertainty
  - Correct: Always multiply the fractional uncertainty by the power, e.g. 2 for area, 3 for volume
- **Wrong:** Claiming averaging reduces systematic uncertainty
  - Why it fails: Systematic uncertainty is a consistent bias that affects all readings equally
  - Correct: Averaging only reduces random uncertainty; systematic uncertainty is fixed via calibration or correcting zero error
- **Wrong:** Taking absolute uncertainty equal to the full range of repeated readings
  - Why it fails: This overestimates the uncertainty in the mean value
  - Correct: For repeated readings, absolute uncertainty is half the range of readings
- **Wrong:** Reporting uncertainties with more than two significant figures
  - Why it fails: Uncertainty is an estimate, extra significant figures are meaningless in exams
  - Correct: Round all uncertainties to 1 or 2 significant figures, always round up

## Cheatsheet

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

## 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.

- [Kinematics](https://www.owlsprep.com/study/cie-9702-u2-overview/)
- [Distance, displacement, speed, velocity and acceleration](https://www.owlsprep.com/study/cie-9702-u2-distance-displacement-speed-velocity-and/)
- [Graphical representation of motion](https://www.owlsprep.com/study/cie-9702-u2-graphical-representation-of-motion/)

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