assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties

Errors and Uncertainties

Objective

Assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties.

Key Concepts

• Absolute uncertainty – the ± value expressed in the same units as the measurement.
• Percentage (relative) uncertainty – the absolute uncertainty divided by the measured value, expressed as a percentage:

  \$\text{Percentage uncertainty} = \frac{\Delta x}{x}\times 100\%\$

• Derived quantity – a quantity calculated from two or more measured values (e.g., speed, density, resistance).

Rules for Combining Uncertainties

When a derived quantity is obtained by simple arithmetic operations, uncertainties are combined as follows:

Step‑by‑Step Procedure

1. Write down the measured values with their absolute uncertainties.
2. Identify the mathematical relationship between the measured quantities and the derived quantity.
3. Decide whether the operation is addition/subtraction or multiplication/division.
4. Apply the appropriate rule from the table above to obtain the uncertainty of the derived quantity.
5. Express the final result with its uncertainty, rounding the uncertainty to one (or at most two) significant figures and matching the derived quantity to the same decimal place.

Worked Example

Two lengths are measured:

• \$L_1 = 12.3\ \text{cm} \pm 0.2\ \text{cm}\$
• \$L_2 = 8.7\ \text{cm} \pm 0.1\ \text{cm}\$

Find the total length \$L = L1 + L2\$ and its uncertainty.

Since this is addition, absolute uncertainties add:

\$\Delta L = \Delta L1 + \Delta L2 = 0.2\ \text{cm} + 0.1\ \text{cm} = 0.3\ \text{cm}\$

The total length is:

\$L = 12.3\ \text{cm} + 8.7\ \text{cm} = 21.0\ \text{cm}\$

Result (rounded to one significant figure in the uncertainty):

\$L = 21.0\ \text{cm} \pm 0.3\ \text{cm}\$

Multiplication Example

Density \$\rho\$ is calculated from mass \$m\$ and volume \$V\$:

\$\rho = \frac{m}{V}\$

Given:

• \$m = 50.0\ \text{g} \pm 0.2\ \text{g}\$ (percentage uncertainty \$=0.4\%\$)
• \$V = 20.0\ \text{cm}^3 \pm 0.5\ \text{cm}^3\$ (percentage uncertainty \$=2.5\%\$)

Percentage uncertainties add:

\$\frac{\Delta\rho}{\rho}\times100\% = 0.4\% + 2.5\% = 2.9\%\$

Calculate \$\rho\$:

\$\rho = \frac{50.0\ \text{g}}{20.0\ \text{cm}^3}=2.50\ \text{g cm}^{-3}\$

Absolute uncertainty:

\$\Delta\rho = 2.9\%\times2.50\ \text{g cm}^{-3}=0.0725\ \text{g cm}^{-3}\approx0.07\ \text{g cm}^{-3}\$

Result:

\$\rho = 2.50\ \text{g cm}^{-3} \pm 0.07\ \text{g cm}^{-3}\$

Practice Questions

1. Two voltages are measured: \$V1 = 5.00\ \text{V} \pm 0.02\ \text{V}\$ and \$V2 = 3.00\ \text{V} \pm 0.01\ \text{V}\$. Find the total voltage \$V = V1 + V2\$ and its uncertainty.
2. A force \$F = 12.0\ \text{N} \pm 0.3\ \text{N}\$ acts over a distance \$d = 0.45\ \text{m} \pm 0.01\ \text{m}\$. Calculate the work done \$W = Fd\$ and its uncertainty.
3. The period \$T = 2.00\ \text{s} \pm 0.02\ \text{s}\$ and amplitude \$A = 0.10\ \text{m} \pm 0.005\ \text{m}\$ of a simple harmonic oscillator are measured. Determine the maximum speed \$v_{\max}=2\pi A/T\$ and its uncertainty.

Summary

• For addition or subtraction, add absolute uncertainties directly.
• For multiplication or division, add percentage (relative) uncertainties.
• Always express the final uncertainty with one or two significant figures and match the decimal place of the result.