All courses › Foundations of Physics › Measurement uncertainty

Measurement uncertainty

No measurement is completely exact, so it is written as a value plus or minus an uncertainty. The relative uncertainty tells you how large the error is compared with the value itself, often given as a percentage. When you multiply or divide quantities, you add their relative uncertainties to find the uncertainty of the result.

x = \bar x \pm \Delta x

measured value with uncertainty

\dfrac{\Delta x}{x}

relative uncertainty

\dfrac{\Delta A}{A} \approx \dfrac{\Delta a}{a} + \dfrac{\Delta b}{b}

relative uncertainties add for

A = ab

Symbols

$x$	measured value	varies
$\Delta x$	uncertainty of the measurement	same as x

Example

A plate is measured as $a = 2.50 \pm 0.02$ m and $b = 1.20 \pm 0.01$ m. The area is $A = ab = 3.00$ m². Relative uncertainty: $0.8\% + 0.8\% \approx 1.6\%$ , so $A = 3.00 \pm 0.05$ m².

Add the relative uncertainties when multiplying and dividing, not the absolute ones.

Practise quantities, units and measurement for free →

← Significant figures · Velocity and acceleration →

Part of Foundations of Physics: Quantities, units and measurement.