Units, prefixes and uncertainties: scale-reading, random and systematic uncertainties, mean and approximate random uncertainty, percentage uncertainty, and combining uncertainties by adding in quadrature.

What this key area is asking

Every measurement in physics carries doubt, and the SQA expects you to quote that doubt honestly. Across both the question paper and the assignment you must choose sensible units and prefixes, estimate the scale-reading uncertainty of an instrument, find a mean and an approximate random uncertainty from repeated readings, work out percentage uncertainties, and combine the uncertainties in several quantities into the uncertainty in a final result. These skills are not a separate area of content: they are tested wherever data appears.

Units, prefixes and how to quote a result

Higher Physics uses SI units and standard prefixes ($\text{n} = 10^{-9}$, $\mu = 10^{-6}$, $\text{m} = 10^{-3}$, $\text{k} = 10^{3}$, $\text{M} = 10^{6}$, $\text{G} = 10^{9}$). A result should be written with a unit, a sensible number of significant figures, and an uncertainty:

The three types of uncertainty

The key distinction the examiners test: random uncertainties scatter readings either side of the true value and shrink when you average more readings, while systematic uncertainties shift every reading the same way and survive averaging. You reduce a systematic uncertainty by finding and correcting its cause (re-zeroing, recalibrating, improving the technique), not by taking more readings.

Mean and approximate random uncertainty

When a measurement is repeated, quote the mean as the best estimate and the spread as the random uncertainty.

Dividing by $n$ (not just quoting the range, and not dividing by $n-1$) is the SQA convention for the approximate random uncertainty in the mean. The more readings you take, the smaller this becomes, which is exactly why repeating reduces random uncertainty.

Percentage uncertainty

Percentage uncertainties are the working currency of uncertainty analysis because they can be compared between different quantities and combined directly. A large value measured to the nearest millimetre may have a tiny percentage uncertainty, while a small value measured with the same ruler has a large one, which is why you measure the longest sensible length.

Combining uncertainties: adding in quadrature

When a final result is calculated by multiplying or dividing measured quantities, combine their percentage uncertainties by adding in quadrature.

Adding in quadrature, rather than simply adding the percentages, recognises that the separate uncertainties are independent and unlikely all to push the result the same way at once. A useful consequence: the largest percentage uncertainty dominates. If one quantity is known to $\pm 1\%$ and another to $\pm 8\%$, the combined value is $\sqrt{1^2 + 8^2} = 8.06\%$, barely more than the $8\%$ alone. This tells you which measurement to improve: chase down the biggest percentage uncertainty.

How this appears in the question paper and the assignment

In the question paper, uncertainty questions are usually short: read a scale-reading uncertainty off an instrument, find a mean and random uncertainty from a table, calculate a percentage uncertainty, or combine two percentage uncertainties for a final answer. You may also be asked to judge whether a result supports a relationship by checking that a theoretical value lies within the experimental uncertainty of the measured one.

In the assignment, the same skills are assessed in context: raw data must be recorded with units and appropriate precision, repeated readings averaged with a random uncertainty, percentage uncertainties calculated for the key quantities, and combined where more than one quantity feeds the final result. The evaluation must distinguish random from systematic effects and suggest how each could be reduced. Choosing apparatus that keeps percentage uncertainties small (measuring the longest time, the largest length) is itself a planning skill the markers reward.

Try this

Q1. A digital ammeter reads $0.45\ \text{A}$, with the last digit in hundredths. State the scale-reading uncertainty. [1 mark]

• Cue. One in the last digit shown, so $\pm 0.01\ \text{A}$.

Q2. Five timings give a range of $0.20\ \text{s}$ about a mean of $4.00\ \text{s}$. State the approximate random uncertainty in the mean. [1 mark]

• Cue. $\frac{\text{range}}{n} = \frac{0.20}{5} = 0.04\ \text{s}$, so $t = 4.00 \pm 0.04\ \text{s}$.

Q3. A speed is found from a distance measured to $\pm 1\%$ and a time measured to $\pm 5\%$. Calculate the percentage uncertainty in the speed. [2 marks]

• Cue. $\sqrt{1^2 + 5^2} = \sqrt{26} = 5.1\%$; the time uncertainty dominates.