EduNinja

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 $(\Delta x/x)\times100\%$ .

z=x\pm y:\quad \Delta z=\Delta x+\Delta yz=\frac{x^a}{y^b}:\quad \frac{\Delta z}{z}=|a|\frac{\Delta x}{x}+|b|\frac{\Delta y}{y}

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.

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Calculate carefully

Rearrange symbolically, resolve vectors, convert units and use proportional reasoning before substituting.

Report evidence

Propagate uncertainty with the correct operation, then use labelled graphs, error bars, gradients, intercepts and areas to test the model.

S1.3.4—Uncertainties

Propagate Uncertainties

Retrieve Mathematical Practice

S1.3.4 Exam Analysis