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Every measurement you will ever take in this course — and every one you take in the twelve required practicals — comes with some built-in doubt. Reporting a result honestly means reporting that doubt alongside it, and knowing exactly how that doubt behaves when you combine several measurements into a final calculated answer.
What you'll be able to do
A causes readings to scatter unpredictably above and below the true value, often due to things outside the experimenter’s full control — reaction time when starting a stopwatch, or a slightly different line of sight each time a scale is read. Repeating the reading and averaging reduces the effect of random error, since the scatter tends to cancel out.
A shifts every single reading by roughly the same amount, in the same direction — for example, a metre ruler with a worn-down end that makes every length read 2 mm too long. No amount of repeating the measurement removes a systematic error; it can only be found and corrected by recognising its cause (or recalibrating the instrument).
Tip — If repeating a measurement many times and averaging doesn’t bring your result closer to the accepted value, suspect a systematic error, not "not enough readings".
describes how tightly repeated measurements cluster together (a small random error) — it says nothing about whether they cluster around the correct value. describes how close a measurement is to the true, accepted value (small overall error, both random and systematic). A set of readings can therefore be precise without being accurate, if a systematic error has shifted them all consistently away from the truth.
The of an instrument is the smallest change in the quantity that it can actually detect — for digital vernier callipers reading to 0.01 mm, the resolution is 0.01 mm. Resolution sets a hard lower limit on how small the uncertainty in any single reading can be claimed to be.
Tip — Picture an archery target: tightly grouped arrows off to one side are precise but inaccurate; widely scattered arrows centred on the bullseye are accurate but imprecise.
The is the size of the possible error in a measurement, in the measurement’s own units (e.g. mm). The expresses that same uncertainty as a fraction of the measured value, useful for comparing measurements of very different sizes and for combining uncertainties through a calculation.
When quantities are added or subtracted, their absolute uncertainties add. When quantities are multiplied or divided, their percentage uncertainties add. When a quantity is raised to a power , its percentage uncertainty is multiplied by .
Tip — Powers are unforgiving: a squared quantity doubles its percentage uncertainty, and a cubed quantity triples it — the radius in the example above contributes far more uncertainty to the volume than the height does, despite being measured more precisely in absolute terms.
When several repeated readings are taken, a simple and defensible estimate of the uncertainty in their mean is half the : half the difference between the largest and smallest reading obtained.
Tip — Always quote the uncertainty to the same number of decimal places as the mean — "16.3 ± 0.2 s", never "16.3 ± 0.2214 s".
Equation recap
Common mistakes to avoid
Key takeaways
Test yourself
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