In experiments we never get a perfect number. Uncertainty tells us how far our measured value could be from the true value. Think of it like a “fuzzy” bubble around the number.
Absolute uncertainty is the actual error in the same units as the measurement.
Example: Length = 5.0 m ± 0.1 m → absolute uncertainty = 0.1 m.
Percentage (relative) uncertainty is the absolute uncertainty divided by the measured value, expressed as a percent.
Example: 0.1 m ÷ 5.0 m = 0.02 → 2 %.
When you combine measured values to calculate a new quantity, the uncertainties combine in a simple way:
If Q = A ± B, then
\$\Delta Q = \Delta A + \Delta B\$
(add the absolute uncertainties directly).
If Q = A \times B or Q = A \div B, then
\$\frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\$
(add the relative uncertainties).
Tip: For any power law Q = A^n, the relative uncertainty is n × (ΔA/A).
Measure a rectangle:
Length L = 5.0 m ± 0.1 m
Width W = 3.0 m ± 0.2 m
Find the area A = L × W and its uncertainty.
| Step | Calculation | Result |
|---|---|---|
| Area | \$A = L \times W\$ | \$A = 5.0 \times 3.0 = 15.0\ \text{m}^2\$ |
| Relative uncertainty of L | \$\frac{0.1}{5.0} = 0.02\$ | \$2\%\$ |
| Relative uncertainty of W | \$\frac{0.2}{3.0} \approx 0.0667\$ | \$6.7\%\$ |
| Total relative uncertainty | \$0.02 + 0.0667 = 0.0867\$ | \$8.7\%\$ |
| Absolute uncertainty of A | \$15.0 \times 0.0867 \approx 1.3\$ | \$1.3\ \text{m}^2\$ |
| Final answer | \$A = 15.0 \pm 1.3\ \text{m}^2\$ |
Measure a mass of m = 2.50 kg ± 0.05 kg and a time of t = 4.0 s ± 0.1 s.
Calculate the average speed v = m / t and its uncertainty.
Answer: v = 0.625 kg/s ± 0.019 kg/s (rounded to one significant figure in the uncertainty).