# Advanced Uncertainty Analysis

> CIE A-Level Physics · 9702 A2 Practical Skills
> Source: https://www.owlsprep.com/study/cie-9702-u30-advanced-uncertainty-analysis/

This subtopic covers advanced error propagation for multi-variable calculations, how to combine uncertainties for powers, products and mixed functions, find gradient/intercept uncertainties from graphs, and link uncertainty to significant figures for CIE Paper 5.

**Prerequisites:** [Basic uncertainty and error analysis](https://www.owlsprep.com/study/cie-9702-u20-basic-uncertainty-analysis/)

## Learning objectives

- Propagate absolute and percentage uncertainties through multi-step calculations
- Calculate total uncertainties for products, quotients, powers and mixed functions
- Find uncertainties in gradients and intercepts from graphical error bars
- Justify the number of significant figures for a final result based on uncertainty

## Core Rules of Uncertainty Propagation

When you calculate a final result from multiple measured quantities, each with their own uncertainty, you must combine these uncertainties to find the total uncertainty in your result. The rule you use depends on the type of mathematical relationship between your inputs and output.

**Error Propagation** — The process of combining individual uncertainties from measured inputs to find the total uncertainty in a calculated output.

*Example:* Finding uncertainty in density $\rho = m/V$ from uncertainties in mass $m$ and volume $V$

1. **Addition/Subtraction**: If $y = a \pm b$, then $\Delta y = \Delta a + \Delta b$ (add absolute uncertainties)
2. **Multiplication/Division**: If $y = \frac{ab}{c}$, then $\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$ (add percentage uncertainties)
3. **Powers**: If $y = k a^n$, then $\frac{\Delta y}{y} = n \frac{\Delta a}{a}$ (multiply percentage uncertainty by the power)

**Worked example:** A student measures a sphere of diameter $d = 2.0 \pm 0.1$ cm. Calculate the volume $V$ of the sphere and its uncertainty.

1. Write the formula for volume of a sphere:
2. $$V = \frac{4}{3}\pi \left(\frac{d}{2}\right)^3 = \frac{\pi d^3}{6}$$
3. Calculate the nominal value of V:
4. $$V = \frac{\pi (2.0)^3}{6} \approx 4.19 \text{ cm}^3$$
5. Find the percentage uncertainty in d first:
6. $$\frac{\Delta d}{d} \times 100\% = \frac{0.1}{2.0} \times 100\% = 5\%$$
7. Use the power rule: the exponent of d is 3, so multiply percentage uncertainty by 3:
8. $$\frac{\Delta V}{V} \times 100\% = 3 \times 5\% = 15\%$$
9. Convert percentage uncertainty to absolute uncertainty:
10. $$\Delta V = 0.15 \times 4.19 \approx 0.6 \text{ cm}^3$$
11. Final result is $V = 4.2 \pm 0.6$ cm³.

> **Exam tip:** CIE requires you to give your final uncertainty to 1 or 2 significant figures, and your nominal value to the same decimal place as the uncertainty.

## Uncertainty in Mixed Mathematical Relationships

Most practical calculations involve combinations of different operations, for example a quantity that is a product of a power of one variable and a quotient of another. You can combine the core rules step-by-step, always adding percentage uncertainties for multiplicative operations and absolute uncertainties for additive operations.

**Worked example:** Resistance $R = \frac{4 \rho L}{\pi d^2}$, where $\rho = 1.7 \times 10^{-8} \pm 0.1 \times 10^{-8}$ Ωm, $L = 1.000 \pm 0.001$ m, $d = 0.0010 \pm 0.0001$ m. Calculate the percentage uncertainty in R.

1. R is proportional to $\rho^1 L^1 d^{-2}$, so add each variable's percentage uncertainty multiplied by its power:
2. Calculate percentage uncertainty for each input:
3. $$\frac{\Delta \rho}{\rho} = \frac{0.1 \times 10^{-8}}{1.7 \times 10^{-8}} \approx 5.9\%, \quad \frac{\Delta L}{L} = 0.1\%, \quad \frac{\Delta d}{d} = 10\%$$
4. Sum the scaled percentage uncertainties:
5. $$\frac{\Delta R}{R} = (1 \times 5.9\%) + (1 \times 0.1\%) + (2 \times 10\%) = 26\%$$
6. Final percentage uncertainty is 26% (or 30% if rounded to 1 significant figure, as CIE often accepts).

> **tip**
>
> Exact constant terms (like 4, $\pi$) do not contribute to uncertainty, as their value is treated as exact for calculation purposes in CIE.

## Linking Uncertainty to Significant Figures

A common CIE Paper 5 question asks you to justify the number of significant figures for your final result. The golden rule is that the number of significant figures is determined by where the uncertainty falls in the nominal value, not by the number of significant figures in your inputs.

**Significant Figure Justification** — The process of matching the least significant digit of the nominal value to the magnitude of the uncertainty, then justifying this choice in an exam answer.

**Worked example:** A calculated result is $12.345$ m s⁻¹ with an absolute uncertainty of $0.2$ m s⁻¹. State the result with correct significant figures and justify.

1. The uncertainty 0.2 affects the first decimal place of the nominal value 12.345.
2. Round the nominal value to the first decimal place:
3. $$12.3 \pm 0.2 \text{ m s}^{-1}$$
4. Justification: The uncertainty falls in the first decimal place, so the nominal value is quoted to the first decimal place, giving 3 significant figures.

**Exam command terms**

- **Justify the number of significant figures** — You must explicitly link your significant figure choice to the size of the uncertainty, not just state the number of significant figures *(Always write 'Uncertainty is in the X decimal place, so result is quoted to X decimal places' to get full marks.)*

## Uncertainties in Gradients and Intercepts

When drawing a graph with error bars, you need to find the uncertainty in the best-fit gradient and y-intercept using the maximum and minimum (worst acceptable) lines that pass through all error bars.

**Worked example:** A best-fit line has gradient 2.4. The maximum possible gradient through error bars is 2.7, and the minimum possible gradient is 2.1. Find the uncertainty in the gradient.

1. Calculate the difference between the maximum gradient and the best gradient:
2. $$\Delta m_{max} = 2.7 - 2.4 = 0.3$$
3. Calculate the difference between the best gradient and the minimum gradient:
4. $$\Delta m_{min} = 2.4 - 2.1 = 0.3$$
5. The uncertainty is equal to the maximum of these two differences:
6. $$\Delta m = 0.3, \quad \text{so } m = 2.4 \pm 0.3$$
7. The exact same method is used to find the uncertainty in the y-intercept.

> **tip**
>
> Draw your worst acceptable line to go through the opposite corners of all error bars to get the maximum possible deviation from your best-fit line.

## Common pitfalls

- **Wrong:** Adding absolute uncertainties when multiplying quantities instead of adding percentage uncertainties.
  - Why it fails: Absolute uncertainties of quantities with different units cannot be added meaningfully, leading to a drastically incorrect total uncertainty.
  - Correct: Always convert to percentage uncertainties before combining for multiplication, division and power operations.
- **Wrong:** Forgetting to multiply percentage uncertainty by the exponent when calculating uncertainty for powers.
  - Why it fails: Uncertainty scales with the power of the quantity, so failing to scale leads to an understated total uncertainty.
  - Correct: For any term $a^n$, the percentage uncertainty is $n \times$ percentage uncertainty in $a$, even if $n$ is negative.
- **Wrong:** Calculating gradient uncertainty as (max gradient - min gradient) instead of the deviation from the best gradient.
  - Why it fails: This gives an uncertainty twice the correct value, as the full range between max and min includes deviations on both sides of the best estimate.
  - Correct: Uncertainty = maximum deviation between the best estimate and the worst acceptable estimate.
- **Wrong:** Quoting the nominal value to more decimal places than the uncertainty allows.
  - Why it fails: Extra digits beyond the position of the uncertainty are not significant and gain no extra information, leading to lost marks in CIE.
  - Correct: Round the nominal value to the same decimal place as the absolute uncertainty, then justify this from the size of the uncertainty.
- **Wrong:** Including uncertainty from constant terms like $\pi$ or $g$ in the total uncertainty calculation.
  - Why it fails: Constants used in CIE calculations are treated as exact, so their uncertainty is negligible compared to measurement uncertainties.
  - Correct: Only include measured quantities when summing percentage uncertainties, ignore exact constants.

## Cheatsheet

| Relationship Type | Uncertainty Rule |
| --- | --- |
| Addition / Subtraction ($y = a \pm b$) | $\Delta y = \Delta a + \Delta b$ (add absolute uncertainties) |
| Multiplication / Division ($y = \frac{ab}{c}$) | $\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$ (add % uncertainties) |
| Power ($y = k a^n$) | $\frac{\Delta y}{y} = n \frac{\Delta a}{a}$ (multiply % uncertainty by $n$) |
| Gradient / Intercept | $\Delta m = \|m_{\text{worst}} - m_{\text{best}}\|$ |
| Significant Figures | Round nominal value to the decimal place of the absolute uncertainty |

## What's next

Advanced uncertainty analysis is a core skill that makes up 30-40% of the total marks for CIE A-Level Physics Paper 5. Mastering these rules lets you correctly process raw experimental data, report reliable final results, and earn full marks for significant figure justification questions. These skills underpin all practical assessment tasks at A2 level, from graphical analysis to experimental planning and evaluation.

- [Procedure evaluation](https://www.owlsprep.com/study/cie-9702-u30-procedure-evaluation/)

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