Errors and Uncertainties

A $\text{random error}$ causes readings to scatter unpredictably above and below the true value — caused by things like reaction time in stopwatch timing, or reading a scale from a slightly different angle each time. Random errors can be reduced by taking repeated readings and averaging, since the scatter tends to cancel out.

A $\text{systematic error}$ shifts every reading consistently in the same direction by roughly the same amount — for example a top-pan balance that hasn’t been zeroed, adding a fixed offset to every mass reading. Repeating the measurement does nothing to reduce a systematic error, because every repeat is shifted the same way; you can only remove it by finding its cause and correcting for it (or recalibrating the instrument).

$\text{Precision}$ describes how close repeated measurements are to each other (small random error, but not necessarily close to the true value). $\text{Accuracy}$ describes how close a measurement is to the true, accepted value (small overall error, both random and systematic). It’s entirely possible to have precise but inaccurate readings — tightly clustered, but clustered around the wrong value because of a systematic error.

The $\text{resolution}$ of an instrument is the smallest change in the quantity it can detect — for a ruler with millimetre markings, the resolution is 1 mm. Resolution sets a lower limit on the uncertainty of any single reading: you cannot claim a smaller uncertainty than the instrument’s resolution allows.

The $\text{absolute uncertainty}$ is the actual size of the possible error in a measurement, in the same units as the measurement itself (e.g. $\pm 0.5$ cm). The $\text{fractional uncertainty}$ is the absolute uncertainty divided by the measured value (no units), and the $\text{percentage uncertainty}$ is the fractional uncertainty expressed as a percentage.

When quantities are $\text{added or subtracted}$, add their absolute uncertainties. When quantities are $\text{multiplied or divided}$, add their percentage (or fractional) uncertainties. When a quantity is raised to a power $n$, multiply its percentage uncertainty by $n$.

When you take several repeated readings of the same quantity, a good working estimate of the uncertainty in their mean is half the $\text{range}$: half the difference between the largest and smallest reading. This gives a simple, defensible uncertainty without needing formal statistics.

When a quantity is found from the gradient of a graph, uncertainty is estimated using the $\text{best-line, worst-line}$ method: draw the best-fit line through the data, then the steepest and shallowest lines that still reasonably fit within the error bars on the data points. The uncertainty in the gradient is then half the difference between the best-fit gradient and the worst-fit gradient.