Uncertainty Propagation Methods

Overview

Many physical quantities are not measured directly. Instead, they are calculated from measured quantities using formulas.

Examples:

• speed from distance and time
• density from mass and volume
• resistance from voltage and current
• gradient from graph data

When measured quantities contain uncertainty, the final calculated result must also contain uncertainty.

This page explains the standard H2 Physics methods for combining uncertainties.

See also:

• Measurement
• Measurement Uncertainty and Errors

Why It Matters

Most useful experimental results are calculated from measured quantities, so uncertainty must be carried through the calculation rather than ignored.

Definition

Uncertainty propagation is the method used to combine uncertainties from measured quantities into the uncertainty of a calculated result.

Key Representations

$$\Delta Y=\Delta A+\Delta B$$ $$\frac{\Delta Y}{Y}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$

Core Principle

If measured quantities are uncertain, any derived quantity is also uncertain.

Example:

$$v=\frac{s}{t}$$

If $s$ and $t$ have uncertainty, then $v$ must also have uncertainty.

Notation

Measured quantity:

$$x\pm \Delta x$$

Where:

• $x$ = measured value
• $\Delta x$ = absolute uncertainty

Fractional uncertainty:

$$\frac{\Delta x}{x}$$

Percentage uncertainty:

$$\frac{\Delta x}{x}\times 100\%$$

1. Addition and Subtraction

Rule

For:

$$Q=A+B$$

or

$$Q=A-B$$

Add absolute uncertainties:

$$\Delta Q=\Delta A+\Delta B$$

Why?

Worst-case uncertainty occurs when both values shift in the direction that maximises the error.

Example 1: Addition

$$A=12.4\pm 0.1$$ $$B=5.2\pm 0.2$$

Then:

$$Q=A+B=17.6$$

Uncertainty:

$$\Delta Q=0.1+0.2=0.3$$

Final answer:

$$Q=17.6\pm 0.3$$

Example 2: Subtraction

$$Q=A-B=12.4-5.2=7.2$$

Uncertainty still adds:

$$\Delta Q=0.1+0.2=0.3$$

So:

$$Q=7.2\pm 0.3$$

2. Multiplication and Division

Rule

For:

$$Q=AB$$

or

$$Q=\frac{A}{B}$$

Add fractional uncertainties:

$$\frac{\Delta Q}{Q}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$

Example 3: Multiplication

$$A=4.0\pm 0.1$$ $$B=3.0\pm 0.2$$

Value:

$$Q=AB=12.0$$

Fractional uncertainty:

$$\frac{\Delta Q}{Q}=\frac{0.1}{4.0}+\frac{0.2}{3.0}=0.025+0.0667=0.0917$$

Absolute uncertainty:

$$\Delta Q=0.0917(12.0)=1.10$$

Final answer:

$$Q=12.0\pm 1.1$$

Example 4: Division

$$v=\frac{s}{t}$$

Where:

$$s=2.00\pm 0.02\text{m}$$ $$t=0.80\pm 0.01\text{s}$$

Value:

$$v=2.50{\text{m s}}^{-1}$$

Fractional uncertainty:

$$\frac{\Delta v}{v}=\frac{0.02}{2.00}+\frac{0.01}{0.80}=0.010+0.0125=0.0225$$

Absolute uncertainty:

$$\Delta v=0.0225(2.50)=0.056$$

So:

$$v=2.50\pm 0.06{\text{m s}}^{-1}$$

3. Powers

Rule

For:

$$Q=A^n$$

Multiply the fractional uncertainty by the power:

$$\frac{\Delta Q}{Q}=n\frac{\Delta A}{A}$$

Example 5: Area of Square

Side length:

$$L=5.0\pm 0.1\text{cm}$$

Area:

$$A=L^2=25.0{\text{cm}}^2$$

Fractional uncertainty:

$$\frac{\Delta A}{A}=2\left(\frac{0.1}{5.0}\right)=0.040$$

Absolute uncertainty:

$$\Delta A=0.040(25.0)=1.0$$

Final answer:

$$A=25.0\pm 1.0{\text{cm}}^2$$

Example 6: Volume of Sphere (radius term)

Since:

$$V\propto r^3$$

Then:

$$\frac{\Delta V}{V}=3\frac{\Delta r}{r}$$

4. Combined Expressions

For expressions like:

$$Q=\frac{A^{2}B}{C^3}$$

Use:

$$\frac{\Delta Q}{Q}=2\frac{\Delta A}{A}+\frac{\Delta B}{B}+3\frac{\Delta C}{C}$$

(Signs in formula do not matter for uncertainty addition.)

5. Max-Min Method

When Used

Useful when:

• formula is complicated
• involves trigonometric or logarithmic functions
• quick estimation needed

Method

1. Calculate central value of $Q$
2. Use largest possible measured values to find $Q_{\max }$
3. Use smallest possible measured values to find $Q_{\min }$
4. Estimate uncertainty:

$$\Delta Q=\frac{Q_{\max }-Q_{\min }}{2}$$

Example 7

$$R=\frac{V}{I}$$

Where:

$$V=6.0\pm 0.1$$ $$I=2.0\pm 0.1$$

Central value:

$$R=3.0\Omega$$

Maximum:

$$R_{\max }=\frac{6.1}{1.9}=3.21$$

Minimum:

$$R_{\min }=\frac{5.9}{2.1}=2.81$$

So:

$$\Delta R=\frac{3.21-2.81}{2}=0.20$$

Final:

$$R=3.0\pm 0.2\Omega$$

6. Percentage Uncertainty Shortcut

Sometimes easiest to convert all uncertainties into percentages.

Example:

$$L=4.0\pm 0.1$$

Percentage uncertainty:

$$2.5\%$$

If:

$$A=L^2$$

Then:

$$\%\Delta A=5.0\%$$

7. Rounding Rules

Uncertainty

• Quote to 1 significant figure
• Use 2 significant figures if rounding would be too large

Measured Value

• Round to the same decimal place as the uncertainty

Example

$$12.347\pm 0.62\to 12.3\pm 0.6$$

Common Exam Mistakes

• adding percentage uncertainties for addition
• adding absolute uncertainties for multiplication
• forgetting power multiplier
• rounding too early
• giving more precision than justified
• forgetting units

Fast Revision Summary (Pattern-Based)

Type of Mathematical Expression	Uncertainty Rule
$Q=A+B$ $Q=A-B$	$\Delta Q=\Delta A+\Delta B$
$Q=AB$ $Q=\frac{A}{B}$	$\frac{\Delta Q}{Q}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$
$Q=A^n$	$\frac{\Delta Q}{Q}=n\frac{\Delta A}{A}$
$Q=kA$ (constant $k$)	$\Delta Q=k\Delta A$
$Q=k{A}^m{B}^n$	$\frac{\Delta Q}{Q}=m\frac{\Delta A}{A}+n\frac{\Delta B}{B}$

Complex Case

Use max-min method.

---

How to Use This Table

1. Identify the form of the equation
2. Match it to the row
3. Apply the corresponding rule

---

Quick Examples

• If $Q=A+B$ → use absolute uncertainties
• If $Q=\frac{A}{B}$ → use fractional uncertainties
• If $Q=A^2$ → multiply fractional uncertainty by 2

---

Key Insight

• Addition / subtraction → absolute uncertainty
• Multiplication / division / powers → fractional uncertainty

Worked Mini Drill

If:

$$x=10.0\pm 0.2$$ $$y=5.0\pm 0.1$$

Find:

$$z=\frac{x}{y}$$

Value:

$$z=2.0$$

Fractional uncertainty:

$$\frac{\Delta z}{z}=\frac{0.2}{10.0}+\frac{0.1}{5.0}=0.02+0.02=0.04$$

Absolute uncertainty:

$$\Delta z=0.04\times 2.0=0.08$$

Final answer:

$$z=2.00\pm 0.08$$

Links

• Measurement
• Measurement Units and Dimensions
• Measurement Uncertainty and Errors
• Measurement Data Presentation
• Measurement Estimation and Experimental Design
• Kinematics