Two lengths a = (12.0 0.2)\ cm and b = (8.0 0.2)\ cm are added. State the result with its absolute uncertainty. [2 marks]

A quantity is found from P = V^2R with V known to 3\% and R to 2\%. Find the percentage uncertainty in P. [2 marks]

A cube has side s = (2.00 0.02)\ cm. Find the volume and its percentage uncertainty. [3 marks]

SEAB Original-style practice: The density of a cylinder is found from = mπ r^2 L. The measurements are m = (45.0 0.1)\ g, r = (5.0 0.1)\ mm and L = (60.0 0.5)\ mm. Determine the percentage uncertainty in the density.

For a product or quotient, add the fractional (percentage) uncertainties, and double the one that is squared. Mass: 0.145.0 × 100 = 0.22\%. Radius (appears as r^2): 0.15.0 × 100 = 2.0\%, doubled to 4.0\%. Length: 0.560.0 × 100 = 0.83\%. Total percentage uncertainty: 0.22 + 4.0 + 0.83 = 5.05\% ≈ 5\%. Markers reward converting each absolute uncertainty to a percentage, doubling the radius contribution because of the square, and summing to a final percentage. The π is exact and contributes nothing.

SingaporePhysicsSyllabus dot point

How do the uncertainties in measured quantities combine when those quantities are added, multiplied or raised to a power?

Combine uncertainties in derived quantities by adding absolute uncertainties for sums and differences and adding fractional uncertainties for products, quotients and powers

A focused answer to the H2 Physics Measurement learning outcome on propagating uncertainty. Adding absolute uncertainties for sums and differences, adding fractional uncertainties for products and quotients, and handling powers.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
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What this dot point is asking

SEAB wants you to propagate measurement uncertainties through a calculation: add absolute uncertainties when quantities are added or subtracted, and add fractional (or percentage) uncertainties when quantities are multiplied, divided or raised to a power. This is examined in the Paper 3 data-based question and is the analysis backbone of Paper 4.

The answer

Absolute, fractional and percentage uncertainty

For a measured quantity $x \pm \Delta x$ :

$\Delta x$ is the absolute uncertainty (same units as $x$ ).
$\dfrac{\Delta x}{x}$ is the fractional uncertainty (no units).
$\dfrac{\Delta x}{x} \times 100\%$ is the percentage uncertainty.

Rule 1: sums and differences add absolute uncertainties

If $y = a + b$ or $y = a - b$ , then:

\Delta y = \Delta a + \Delta b

Note that the absolute uncertainties always add, even for a difference. A difference of two close values can therefore have a very large fractional uncertainty, which is why subtracting nearly equal measurements is poor experimental design.

Rule 2: products and quotients add fractional uncertainties

If $y = \dfrac{a \, b}{c}$ , then:

\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}

The fractional uncertainties add regardless of whether a quantity is in the numerator or denominator.

Rule 3: powers multiply the fractional uncertainty

If $y = a^n$ , then:

\frac{\Delta y}{y} = |n| \, \frac{\Delta a}{a}

So a squared quantity contributes twice its fractional uncertainty, a cubed quantity three times, and a square root half. This is just Rule 2 applied to repeated factors.

Getting to the final absolute uncertainty

After combining fractional uncertainties, convert back to an absolute uncertainty by multiplying the fractional uncertainty by the calculated value of $y$ :

\Delta y = \frac{\Delta y}{y} \times y

Then round the uncertainty to one significant figure and the value to match.

Worked example

The acceleration of free fall is found from $g = \dfrac{4\pi^2 L}{T^2}$ , with $L = (0.900 \pm 0.002)\ \text{m}$ and $T = (1.90 \pm 0.02)\ \text{s}$ . Find $g$ and its absolute uncertainty.

Step 1: Calculate the value of g

g = \frac{4\pi^2 (0.900)}{(1.90)^2} = \frac{35.53}{3.61} = 9.84\ \text{m s}^{-2}

Step 2: Find the fractional uncertainty in each measured quantity

Length: $\dfrac{0.002}{0.900} = 0.00222$ (that is $0.222\%$ ).

Period appears as $T^2$ , so its contribution is doubled: $2 \times \dfrac{0.02}{1.90} = 2 \times 0.01053 = 0.02105$ (that is $2.105\%$ ).

Step 3: Add the fractional uncertainties

\frac{\Delta g}{g} = 0.00222 + 0.02105 = 0.02327 \approx 2.3\%

Step 4: Convert to an absolute uncertainty and report

$\Delta g = 0.02327 \times 9.84 = 0.229\ \text{m s}^{-2}$ , round to $0.2\ \text{m s}^{-2}$ .

So $g = (9.8 \pm 0.2)\ \text{m s}^{-2}$ . The dominant contribution is the period, because it is squared.

Examples in context

Example 1. The danger of subtracting close values. Measuring a small temperature rise as $T_2 - T_1 = 41.0 - 40.0 = 1.0\ ^\circ\text{C}$ , each with $\pm 0.5\ ^\circ\text{C}$ , gives an absolute uncertainty of $1.0\ ^\circ\text{C}$ on a difference of $1.0\ ^\circ\text{C}$ : a $100\%$ uncertainty. This is why experiments are designed to measure large differences.

Example 2. Identifying the dominant error. In the pendulum example above, the period contributed $2.1\%$ while the length contributed only $0.2\%$ . Recognising that the squared, larger-fractional-uncertainty quantity dominates tells a student exactly where to improve their technique: time more oscillations to shrink the period uncertainty.

Try this

Q1. Two lengths $a = (12.0 \pm 0.2)\ \text{cm}$ and $b = (8.0 \pm 0.2)\ \text{cm}$ are added. State the result with its absolute uncertainty. [2 marks]

Cue. $a + b = (20.0 \pm 0.4)\ \text{cm}$ (absolute uncertainties add).

Q2. A quantity is found from $P = \dfrac{V^2}{R}$ with $V$ known to $3\%$ and $R$ to $2\%$ . Find the percentage uncertainty in $P$ . [2 marks]

Cue. $2 \times 3\% + 2\% = 8\%$ (square doubles the voltage contribution).

Q3. A cube has side $s = (2.00 \pm 0.02)\ \text{cm}$ . Find the volume and its percentage uncertainty. [3 marks]

Cue. $V = s^3 = 8.00\ \text{cm}^3$ ; percentage uncertainty $= 3 \times \dfrac{0.02}{2.00} \times 100 = 3\%$ , so $V = (8.0 \pm 0.2)\ \text{cm}^3$ .

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original4 marksThe density of a cylinder is found from

\rho = \dfrac{m}{\pi r^2 L}

. The measurements are

m = (45.0 \pm 0.1)\ \text{g}

r = (5.0 \pm 0.1)\ \text{mm}

and

L = (60.0 \pm 0.5)\ \text{mm}

. Determine the percentage uncertainty in the density.

Show worked answer →

For a product or quotient, add the fractional (percentage) uncertainties, and double the one that is squared.

Mass: $\dfrac{0.1}{45.0} \times 100 = 0.22\%$ .

Radius (appears as $r^2$ ): $\dfrac{0.1}{5.0} \times 100 = 2.0\%$ , doubled to $4.0\%$ .

Length: $\dfrac{0.5}{60.0} \times 100 = 0.83\%$ .

Total percentage uncertainty: $0.22 + 4.0 + 0.83 = 5.05\% \approx 5\%$ .

Markers reward converting each absolute uncertainty to a percentage, doubling the radius contribution because of the square, and summing to a final percentage. The $\pi$ is exact and contributes nothing.

Original3 marksA resistance is found from two voltage readings

V_1 = (6.20 \pm 0.05)\ \text{V}

and

V_2 = (2.10 \pm 0.05)\ \text{V}

across a component, where the voltage drop used is

V_1 - V_2

. Determine the voltage drop and its absolute uncertainty.

Show worked answer →

For a difference, add the absolute uncertainties.

Voltage drop: $V_1 - V_2 = 6.20 - 2.10 = 4.10\ \text{V}$ .

Absolute uncertainty: $0.05 + 0.05 = 0.10\ \text{V}$ .

So the voltage drop is $(4.10 \pm 0.10)\ \text{V}$ .

Markers reward adding (not subtracting) the absolute uncertainties for a difference, and presenting the result with value and uncertainty to matching decimal places. Note the fractional uncertainty of a difference can be large when the two values are close.