Propagating uncertainty through a calculation

Covers AS91521

Key ideas

When a result is calculated from several measured quantities, the uncertainty in each measurement contributes to the uncertainty in the final answer. This is called propagating uncertainty.
Work in percentage uncertainty — it makes combining uncertainties from different quantities (with different units) straightforward.
Multiplying or dividing quantities: add their percentage uncertainties.

$\%\,\delta z = \%\,\delta x + \%\,\delta y \quad \text{for } z = xy \text{ or } z = \frac{x}{y}$

Raising to a power: multiply the percentage uncertainty by the power.

$\%\,\delta z = n \times \%\,\delta x \quad \text{for } z = x^n$

Adding or subtracting quantities: add their absolute uncertainties (not percentages).

$\delta z = \delta x + \delta y \quad \text{for } z = x + y \text{ or } z = x - y$

Once the final percentage uncertainty is known, convert back to an absolute uncertainty on the final answer, and round the answer to match.

Worked ExamplePropagating uncertainty into a calculated area

A rectangle has length $l = 12.0 \pm 0.2$ cm and width $w = 5.0 \pm 0.1$ cm. Find its area with its uncertainty.

Step 1 — Calculate the percentage uncertainty in each measurement

$\%\,\delta l = \frac{0.2}{12.0}\times 100 = 1.67\%, \qquad \%\,\delta w = \frac{0.1}{5.0}\times 100 = 2.0\%$

Step 2 — Add the percentage uncertainties (multiplication rule)

Since area $A = l \times w$ is a product:

$\%\,\delta A = 1.67\% + 2.0\% = 3.67\% \approx 3.7\%$

Step 3 — Calculate the area and convert the uncertainty back to absolute

$A = 12.0 \times 5.0 = 60.0 \text{ cm}^2$

$\delta A = \frac{3.67}{100}\times 60.0 \approx 2.2 \text{ cm}^2$

$\textbf{A = 60.0 ± 2.2 cm}^2$

Practice question

Speed is calculated as $v = d/t$ , where $d = 2.00 \pm 0.02$ m and $t = 0.50 \pm 0.02$ s. Find $v$ with its percentage and absolute uncertainty.

Worked solution: $\%\delta d = \dfrac{0.02}{2.00}\times100 = 1.0\%$ . $\%\delta t = \dfrac{0.02}{0.50}\times100 = 4.0\%$ . Division rule: $\%\delta v = 1.0\% + 4.0\% = 5.0\%$ . $v = 2.00/0.50 = 4.0$ m s⁻¹. $\delta v = 0.05 \times 4.0 = 0.20$ m s⁻¹. So $v = 4.0 \pm 0.2$ m s⁻¹.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A length is measured as $8.0 \pm 0.1$ cm. Calculate its percentage uncertainty.

Merit

A calculation uses $z = x^3$ , where $x$ has a percentage uncertainty of $2.0\%$ . Find the percentage uncertainty in $z$ , and explain why raising to a power increases uncertainty so much.

Excellence

Kinetic energy is calculated as $E_k = \tfrac{1}{2}mv^2$ , where $m = 2.00 \pm 0.01$ kg (0.5%) and $v = 3.0 \pm 0.3$ m s⁻¹ (10%). Find the percentage uncertainty in $E_k$ and explain which measurement the student should improve first to make the biggest difference to the final uncertainty.

Propagating uncertainty through a calculation

Covers AS91521

Key ideas

When a result is calculated from several measured quantities, the uncertainty in each measurement contributes to the uncertainty in the final answer. This is called propagating uncertainty.
Work in percentage uncertainty — it makes combining uncertainties from different quantities (with different units) straightforward.
Multiplying or dividing quantities: add their percentage uncertainties.

$\%\,\delta z = \%\,\delta x + \%\,\delta y \quad \text{for } z = xy \text{ or } z = \frac{x}{y}$

Raising to a power: multiply the percentage uncertainty by the power.

$\%\,\delta z = n \times \%\,\delta x \quad \text{for } z = x^n$

Adding or subtracting quantities: add their absolute uncertainties (not percentages).

$\delta z = \delta x + \delta y \quad \text{for } z = x + y \text{ or } z = x - y$

Once the final percentage uncertainty is known, convert back to an absolute uncertainty on the final answer, and round the answer to match.

Worked ExamplePropagating uncertainty into a calculated area

A rectangle has length $l = 12.0 \pm 0.2$ cm and width $w = 5.0 \pm 0.1$ cm. Find its area with its uncertainty.

Step 1 — Calculate the percentage uncertainty in each measurement

$\%\,\delta l = \frac{0.2}{12.0}\times 100 = 1.67\%, \qquad \%\,\delta w = \frac{0.1}{5.0}\times 100 = 2.0\%$

Step 2 — Add the percentage uncertainties (multiplication rule)

Since area $A = l \times w$ is a product:

$\%\,\delta A = 1.67\% + 2.0\% = 3.67\% \approx 3.7\%$

Step 3 — Calculate the area and convert the uncertainty back to absolute

$A = 12.0 \times 5.0 = 60.0 \text{ cm}^2$

$\delta A = \frac{3.67}{100}\times 60.0 \approx 2.2 \text{ cm}^2$

$\textbf{A = 60.0 ± 2.2 cm}^2$

Practice question

Speed is calculated as $v = d/t$ , where $d = 2.00 \pm 0.02$ m and $t = 0.50 \pm 0.02$ s. Find $v$ with its percentage and absolute uncertainty.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A length is measured as $8.0 \pm 0.1$ cm. Calculate its percentage uncertainty.

Merit

A calculation uses $z = x^3$ , where $x$ has a percentage uncertainty of $2.0\%$ . Find the percentage uncertainty in $z$ , and explain why raising to a power increases uncertainty so much.

Excellence