Random, systematic and reading uncertainties, absolute and percentage uncertainties, combining uncertainties, and presenting data with graphs, best-fit lines and error bars.

What this key area is asking

The SQA wants you to distinguish random, systematic and reading uncertainties, calculate absolute and percentage uncertainties, combine uncertainties when quantities are multiplied or divided, and present data with best-fit lines and error bars. This is the analytical heart of Advanced Higher and the project.

Types of uncertainty

Recognising the type matters because the cure differs: repeat and average to beat random uncertainty, but find and correct the cause to remove a systematic one. A measurement can be precise (small scatter) but inaccurate (a systematic offset), so both must be considered.

Absolute and percentage uncertainty

The percentage form lets you compare the quality of different measurements and is the form you combine. A small absolute uncertainty on a large value can be a tiny percentage, and vice versa. Quote a final result as value $\pm$ absolute uncertainty, rounded so the uncertainty has one (occasionally two) significant figures.

Combining uncertainties

This is the rule that decides the uncertainty in a calculated result. For $R = \frac{V}{I}$ you add the percentage uncertainties of $V$ and $I$; for an area $A = \pi r^2$ you double the percentage uncertainty of $r$. The largest percentage uncertainty usually dominates, which tells you which measurement to improve first, a key evaluation skill for the project.

Presenting data: best-fit lines and error bars

A straight-line graph is the most powerful tool in data analysis: arranging a relationship into the form $y = mx + c$ lets you find a physical quantity from the gradient or intercept. The spread of acceptable lines through the error bars gives the uncertainty in the gradient, and hence in the result. Good presentation, labelled axes with units, sensible scales, error bars and a best-fit line, is rewarded directly.

Examples in context

Measuring $g$ with a pendulum, the period is timed over many swings to cut the random uncertainty, and the result's uncertainty is dominated by whichever measurement has the largest percentage uncertainty. Calibrating an instrument removes a systematic zero error before readings are trusted. Straight-line analysis turns $T^2 = \frac{4\pi^2}{g}L$ into a graph of $T^2$ against $L$ whose gradient gives $g$. Every project report is marked partly on correct uncertainty handling and clear graphs with error bars.

Try this

Q1. State which type of uncertainty is reduced by taking more readings and averaging. [1 mark]

• Cue. Random uncertainty.

Q2. State how percentage uncertainties combine when two quantities are multiplied. [1 mark]

• Cue. They are added (in quadrature).

Q3. A length of $(20.0 \pm 0.5)\ \text{cm}$ has what percentage uncertainty? [1 mark]

• Cue. $\frac{0.5}{20.0} \times 100 = 2.5\%$.