Uncertainty and Error

$\text{Random errors}$ cause readings to scatter unpredictably around the true value — for example, human reaction time with a stopwatch. They are reduced by taking repeat readings and averaging.

$\text{Systematic errors}$ shift all readings by the same amount or in the same proportion — for example, a zero error on a balance or a metre rule with a worn end. Repeating does not reduce them; they must be identified and corrected.

$\text{Precision}$ describes how closely repeated readings agree. $\text{Accuracy}$ describes how close a result is to the true value. Readings can be precise but inaccurate if a systematic error is present.

For a single reading on an analogue scale, the uncertainty is usually half the smallest division; for a digital instrument, the resolution. A length measured with a rule needs two readings (each end), so its uncertainty is often taken as one full division.

For repeated readings, a common estimate of uncertainty is $\text{half the range}$: (largest − smallest)/2. Anomalous results are excluded before calculating the mean and range.

The $\text{percentage uncertainty}$ is the absolute uncertainty divided by the measured value, times 100%.

$\text{Adding or subtracting}$ quantities: add the $\text{absolute}$ uncertainties. If $d=x_2-x_1$, then $\Delta d=\Delta x_1+\Delta x_2$.

$\text{Multiplying or dividing}$: add the $\text{percentage}$ uncertainties. If $\rho =m/V$, then $\%\rho =\%m+\%V$.

$\text{Raising to a power}$: multiply the percentage uncertainty by the power. If $A=\pi r^2$, then $\%A=2\times \%r$. Constants such as $\pi$ carry no uncertainty.

State the result as value $\pm$ absolute uncertainty with a unit, for example $g=9.7\pm 0.3\mathrm{m}{\mathrm{s}}^{-2}$.

Round the uncertainty to one or two significant figures, then quote the value to the same decimal place. Writing $9.736\pm 0.3$ is inconsistent.

To judge accuracy, compare with an accepted value: if it lies within the uncertainty range, the result is consistent with it. The $\text{percentage difference}$ from the accepted value can also be compared with the percentage uncertainty.