S1.3.4—Uncertainties

Uncertainty states a range, not a mistake

Write a measured result as $x\pm\Delta x$ in matching units. The absolute uncertainty is $\Delta x$; fractional uncertainty is $\Delta x/x$; percentage uncertainty is $(\Delta x/x)\times100\%$.

Worked propagation

For $L=(2.00\pm0.01)\,\text{m}$ and $W=(1.00\pm0.01)\,\text{m}$, $A=LW=2.00\,\text{m}^2$. The fractional uncertainty is $0.01/2.00+0.01/1.00=0.015$, or $1.5\%$. Therefore $\Delta A=0.015\times2.00=0.03\,\text{m}^2$, so $A=(2.00\pm0.03)\,\text{m}^2$.

Powers multiply fractional uncertainty

If $V\propto r^3$ and $r$ has $2\%$ uncertainty, $V$ has $3\times2\%=6\%$ uncertainty under the syllabus propagation rule. For addition or subtraction, add absolute uncertainties instead—never percentage uncertainties.

Report sensible precision

Round the uncertainty first, then round the measured value to the same decimal place. A small difference between two measured values can have a large percentage uncertainty even when both original measurements look precise.