Uncertainty Propagation Methods

Branch note: This page deepens one part of Measurement.

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.

What these rules mean

The 9749 rules are simplified maximum/worst-case uncertainty assessments, not exact statistical propagation laws. Equal signs below state the assessed maximum uncertainty under this syllabus convention.

See also:

• Measurement
• Measurement Uncertainty and Errors

Core Ideas

• Addition and subtraction combine absolute uncertainties.
• Multiplication, division, and powers combine fractional or percentage uncertainties.
• The final uncertainty should be rounded sensibly and quoted with a matching final value.

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.

In syllabus wording, actual uncertainty means absolute uncertainty (for example, $\pm 0.2\mathrm{m}$), rather than fractional or percentage uncertainty.

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

Figure: Choose the method from the mathematical structure. Add absolute uncertainties for sums and differences; add fractional or percentage uncertainties for products, quotients and powers; use limiting-value numerical substitution for other functions. These are maximum-uncertainty assessments.

Rule

For:

$$Q=A+B$$

or

$$Q=A-B$$

Add absolute uncertainties:

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

More generally, for exact numerical constants $m$ and $n$,

$$Q=mA+nB\Rightarrow \Delta Q=|m|\Delta A+|n|\Delta B.$$

An exact numerical constant contributes no uncertainty of its own. If a quantity described as a “constant” was measured and has a stated uncertainty, that uncertainty must still be propagated.

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 magnitude of 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 the general product-of-powers form

$$Q=k{A}^m{B}^n,$$

where $k$ is exact, the assessed maximum fractional uncertainty is

$$\frac{\Delta Q}{Q}=|m|\frac{\Delta A}{A}+|n|\frac{\Delta B}{B}.$$

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

For each measured input, use its allowed upper and lower values and determine which combination actually gives $Q_{\max }$ and $Q_{\min }$. For a non-monotonic function, “put every input at its maximum” may not maximise the result; inspect the function over the stated uncertainty interval.

Method

1. Calculate the central result $Q_0$ from the stated central inputs.
2. Test the allowed input extremes that could produce $Q_{\max }$ and $Q_{\min }$.
3. Use one of these conventions explicitly; do not mix their centre and uncertainty:

Retain the central result and use the larger departure conservatively:

$$\Delta Q=\max \left(|Q_{\max }-Q_0|,|Q_0-Q_{\min }|\right),$$

then report $Q_0\pm \Delta Q$.

Or use the limiting interval’s midpoint and half-range:

$$Q_{\mathrm{mid}}=\frac{Q_{\max }+Q_{\min }}{2},\Delta Q=\frac{Q_{\max }-Q_{\min }}{2},$$

then report $Q_{\mathrm{mid}}\pm \Delta Q$.

Example 7

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

Where:

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

Central value:

$$R_0=\frac{6.0}{2.0}=3.00\Omega$$

The extra displayed zero is retained here as a guard digit so that the final central value can be quoted to the same decimal place as its uncertainty.

Maximum:

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

Minimum:

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

So:

$$\Delta R=\max (3.21-3.00,3.00-2.81)=0.21\Omega .$$

Retaining the central result, the conservative final statement is

$$R=(3.00\pm 0.21)\Omega .$$

Two uncertainty digits are retained here so that rounding does not reduce the stated maximum departure below $0.21\Omega$. The alternative midpoint/half-range convention would use $R_{\mathrm{mid}}=3.01\Omega$ and $\Delta R=0.20\Omega$; its centre must not be silently replaced by $3.00\Omega$.

Example 8: Numerical substitution for a trigonometric expression

Suppose $S=x\cos \theta$, where $x=(2.0\pm 0.2)\mathrm{cm}$ and $\theta =(53\pm 2{)}^{\circ }$. Over this interval, $S$ increases with $x$ and decreases with $\theta$, so

$$S_{\text{central}}=2.0\cos 53^{\circ }=1.204\mathrm{cm},$$ $$S_{\max }=2.2\cos 51^{\circ }=1.385\mathrm{cm},S_{\min }=1.8\cos 55^{\circ }=1.032\mathrm{cm}.$$

The departures from the central value are $0.181\mathrm{cm}$ and $0.172\mathrm{cm}$. A conservative one-significant-figure uncertainty is therefore $0.2\mathrm{cm}$, giving

$$S=(1.2\pm 0.2)\mathrm{cm}.$$

If limiting values are asymmetric about the central result, state the convention used: take the larger departure conservatively, or report a midpoint and half-range if that is the required convention.

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

• In this course, quote uncertainty to 1 significant figure normally.
• Retain 2 significant figures when the leading digit is 1, or when a stated examination/data convention requires it, to avoid an unnecessarily coarse rounded uncertainty.
• Other sensible conventions may be accepted when applied consistently; do not switch conventions merely to improve agreement with an expected value.

Measured Value

• Round to the same decimal place as the uncertainty

Example

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

Exam Relevance

In exams, propagation questions usually test whether you choose the correct uncertainty rule for the mathematical form of the calculated quantity.

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
$Q=A+B$ or $Q=A-B$	$\Delta Q=\Delta A+\Delta B$
$Q=AB$ or $Q=A/B$	$\Delta Q/Q=\Delta A/A+\Delta B/B$
$Q=A^n$	$\Delta Q/Q=
$Q=kA$ (constant $k$)	$\Delta Q=
$Q=k{A}^m{B}^n$	$\Delta Q/Q=

Complex Case

Use max-min method.

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How to Use This Table

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

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Quick Examples

• If $Q=A+B$ → use absolute uncertainties
• If $Q=A/B$ → use fractional uncertainties
• If $Q=A^2$ → multiply fractional uncertainty by 2

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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:

$$z=2.00$$

Fractional uncertainty:

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

Absolute uncertainty:

$$\Delta z=0.04\times 2.00=0.08$$

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