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

Usually quote to:

• 1 significant figure
• occasionally 2 if first digit is 1 or 2

Measured Value

Round to same decimal place as 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

Add / Subtract

Use absolute uncertainties:

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

Multiply / Divide

Use fractional uncertainties:

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

Power

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

Complex Case

Use max-min method.

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:

$$\frac{\Delta z}{z}=0.02+0.02=0.04$$

Absolute:

$$\Delta z=0.08$$

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