# Uncertainty Analysis

> A-Level Physics · CIE 9702
> Source: https://www.owlsprep.com/study/cie-9702-u15-uncertainty-analysis/

This module covers core uncertainty analysis for CIE AS Physics practical assessments, including error types, absolute/percentage uncertainty, and combining uncertainties for calculations. It addresses requirements for Paper 2 and Paper 3.

**Prerequisites:** [Basic experimental measurement and error types](https://www.owlsprep.com/study/cie-9702-u15-measurement-errors/); Arithmetic and scientific notation

## Learning objectives

- Distinguish between random and systematic errors, and absolute and percentage uncertainty
- Calculate uncertainty from repeated measurements
- Combine uncertainties for all arithmetic operations and powers
- Calculate uncertainty in gradient for graphical analysis

## Error Types and Core Uncertainty Definitions

All experimental measurements have quantifiable uncertainty, which is not a mistake but a range within which the true value is expected to lie. Uncertainty arises from two broad categories of error.

**Random error** — Unpredictable variation between measurements that causes scatter around the true value. Reduced by repeated measurements and averaging.

*Example:* Variation in stopwatch timings for a falling object between repeats

**Systematic error** — Consistent, repeatable error offset that shifts all measurements away from the true value in the same direction. Cannot be reduced by averaging.

*Example:* A thermometer that always reads 0.5°C higher than the true temperature

**Absolute uncertainty** — The absolute margin of error in a measurement, with the same units as the measurement. Equal to half the range of repeats or the smallest instrument division.

*Notation:* \Delta x

*Example:* $10 \pm 0.1$ g has an absolute uncertainty of 0.1 g

**Percentage uncertainty** — Absolute uncertainty expressed as a percentage of the measured value, useful for comparing uncertainty between measurements of different sizes.

*Example:* A 10 g mass with 0.1 g absolute uncertainty has a 1% percentage uncertainty

**Worked example:** A student measures wire diameter 3 times, getting 0.42 mm, 0.46 mm, 0.44 mm. Calculate the absolute and percentage uncertainty.

1. Calculate the mean diameter:

   $$d = \frac{0.42 + 0.46 + 0.44}{3} = 0.44 \text{ mm}$$
2. Find the range of measurements:

   $$\text{Range} = 0.46 - 0.42 = 0.04 \text{ mm}$$
3. Absolute uncertainty is half the range:

   $$\Delta d = \frac{0.04}{2} = 0.02 \text{ mm}$$
4. Calculate percentage uncertainty:

   $$\text{Percentage uncertainty} = \frac{0.02}{0.44} \times 100 \approx 4.5\%$$
5. Final result: $d = 0.44 \pm 0.02$ mm (or 0.44 mm ± 4.5%)

**Exam command terms**

CIE command terms for this topic:

- **Calculate uncertainty** — Apply the appropriate combination rule to find total uncertainty *(Expect to show your working for absolute/percentage conversion)*

- **Distinguish between random and systematic error** — State the key difference and give an example of each

> **Exam tip:** CIE requires half the range for absolute uncertainty from repeats, not the full range. Round uncertainty to 1 significant figure unless leading digit is 1.

*Calculator:* allowed

## Combining Uncertainties: Addition and Subtraction

When adding or subtracting measured quantities, always add the absolute uncertainties of each quantity to get the total uncertainty. This rule applies regardless of whether you add or subtract the measured values, because uncertainties always increase total error.

$$Q = a \pm b \implies \Delta Q = \Delta a + \Delta b$$

**Worked example:** An empty beaker is $50.0 \pm 0.1$ g. Total mass with water is $126.4 \pm 0.2$ g. Find the mass of water and its uncertainty.

1. Calculate mass of water by subtraction:

   $$m = 126.4 - 50.0 = 76.4 \text{ g}$$
2. Add the absolute uncertainties of both measurements:

   $$\Delta m = 0.1 + 0.2 = 0.3 \text{ g}$$
3. Final result: $m = 76.4 \pm 0.3$ g

**Check your understanding**

Test your understanding

1. If $x = 10 \pm 1$ and $y = 5 \pm 0.5$, what is the absolute uncertainty in $x - y$?

   - 0.5
   - 1.5
   - 5.5

   *Answer:* 1.5

   *Why:* Correct! We always add absolute uncertainties for addition and subtraction, so 1 + 0.5 = 1.5. Even when subtracting values, uncertainties add up.

*Calculator:* allowed

## Combining Uncertainties: Multiplication, Division and Powers

For multiplication, division, and powers, we work with percentage uncertainty and add them for all variables. For powers, multiply the percentage uncertainty by the exponent of the variable.

$$\text{For } Q = \frac{ab}{c} \implies \frac{\Delta Q}{Q} \times 100 = \frac{\Delta a}{a} \times 100 + \frac{\Delta b}{b} \times 100 + \frac{\Delta c}{c} \times 100
\\
\text{For } Q = a^n \implies \frac{\Delta Q}{Q} \times 100 = n \left( \frac{\Delta a}{a} \times 100 \right)$$

**Worked example:** Power $P = V^2 / R$, where $V = 10.0 \pm 0.5$ V and $R = 4.0 \pm 0.1$ Ω. Find the percentage uncertainty in $P$.

1. Calculate percentage uncertainty for V and R:

   $$\frac{\Delta V}{V} \times 100 = 5\%, \quad \frac{\Delta R}{R} \times 100 = 2.5\%$$
2. Multiply V's percentage uncertainty by its exponent (2):

   $$\text{Uncertainty from } V = 2 \times 5\% = 10\%$$
3. Add all percentage uncertainties:

   $$\text{Total percentage uncertainty} = 10\% + 2.5\% = 12.5\%$$
4. Convert to absolute uncertainty for final result:

   $$P = 25 \text{ W}, \quad \Delta P = 3 \text{ W}, \quad P = 25 \pm 3 \text{ W}$$

> **tip**
>
> Always convert to percentage uncertainty before combining for these operations. Never add absolute uncertainties here.

*Calculator:* allowed

## Uncertainty in Graphical Analysis

In practical assessments, you will often need to find the uncertainty in the gradient of a best-fit line. Error bars are used to show the absolute uncertainty for each plotted point.

To find gradient uncertainty, draw the steepest and shallowest possible lines of best fit that pass through almost all error bars. The absolute uncertainty in gradient is half the difference between the maximum and minimum gradient.

**Worked example:** Best-fit gradient is 2.4. Maximum possible gradient is 2.6, minimum is 2.2. Find the uncertainty in gradient.

1. Calculate the range between maximum and minimum gradient:

   $$\text{Range} = 2.6 - 2.2 = 0.4$$
2. Absolute uncertainty is half the range:

   $$\Delta m = 0.2$$
3. Final result: $m = 2.4 \pm 0.2$

> **Exam tip:** You will get most marks for correctly drawing error bars and maximum/minimum lines, even if your final uncertainty has a small arithmetic error.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using the full range of repeated measurements instead of half the range for absolute uncertainty
  - Why it fails: This incorrectly doubles the uncertainty, leading to a mark penalty
  - Correct: Always calculate absolute uncertainty as half the range of repeated measurements
- **Wrong:** Subtracting uncertainties when calculating the difference between two measurements
  - Why it fails: Uncertainties always add, even when you subtract measured values. This underestimates total error
  - Correct: Add absolute uncertainties for all addition and subtraction operations
- **Wrong:** Adding absolute uncertainties for multiplication and division calculations
  - Why it fails: Absolute uncertainty depends on measurement size, so adding them gives an incorrect total
  - Correct: Convert all uncertainties to percentage, add the percentages, then convert back to absolute if needed
- **Wrong:** Forgetting to multiply percentage uncertainty by the exponent when dealing with powers
  - Why it fails: This underestimates total uncertainty by a factor equal to the exponent
  - Correct: Always multiply percentage uncertainty by the exponent of the variable
- **Wrong:** Quoting uncertainty to two or more significant figures for leading digits greater than 1
  - Why it fails: Uncertainty is an approximate estimate, extra significant figures are meaningless
  - Correct: Round uncertainty to 1 significant figure for leading digits >1, use 2 only for leading digit 1

## Cheatsheet

| Operation | Uncertainty Rule |
| --- | --- |
| Addition/subtraction ($Q = a \pm b$) | Add absolute uncertainties: $\Delta Q = \Delta a + \Delta b$ |
| Multiplication/division ($Q = ab/c$) | Add percentage uncertainties: $\frac{\Delta Q}{Q} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$ |
| Power ($Q = a^n$) | Multiply percentage uncertainty by $n$: $\frac{\Delta Q}{Q} = n\left(\frac{\Delta a}{a}\right)$ |
| Repeated measurements | $\Delta = \frac{1}{2} \times (max - min)$ |
| Gradient uncertainty (graphs) | $\Delta m = \frac{1}{2} \times (m_{max} - m_{min})$ |

## What's next

Uncertainty analysis is the foundation of all practical work in physics, and forms a core part of all AS and A Level practical assessments, including Paper 3 (practical test) and Paper 2 (theory questions on practical skills). Mastery of these rules will allow you to correctly process experimental data, avoid common errors, and access full marks for calculation questions. These same rules are carried forward to A Level practical work, where they are applied to more complex experiments and extended analysis.

- [Evaluation of Results](https://www.owlsprep.com/study/cie-9702-u15-evaluation-of-results/)
- [Motion in a circle](https://www.owlsprep.com/study/cie-9702-u16-overview/)
- [Angular displacement and speed](https://www.owlsprep.com/study/cie-9702-u16-angular-displacement-and-speed/)

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