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Processing: Precision and Uncertainty

Syllabus mapping

Working Scientifically outcome — Processing (part 1 of 2). Covers quantifying measurement uncertainty, distinguishing random and systematic error, and combining uncertainties through a calculation. See Representing Data for the second half of Processing — tables, graphs, and linearising relationships.

Exact NESA outcome code TODO — confirm against the official syllabus PDF (Resources) before treating any wording on this page as verbatim NESA text.

You need to know: every measurement has a margin of doubt attached to it, that margin has a name (uncertainty) and a size you can calculate, and that size changes predictably when you use the measurement in a calculation.

Assumed knowledge check

Before this page makes sense, you should be comfortable with:

Core

Why every measurement has uncertainty

No measurement is perfectly exact — it's limited by the resolution of the instrument, the skill of whoever's reading it, or genuine tiny variations in what's being measured. Uncertainty is a stated, quantified estimate of that doubt, written as \(\pm\) something. A length of \(12.4 \pm 0.1\) cm is a claim that the true value very likely lies between 12.3 cm and 12.5 cm — not a claim that 12.4 cm is exactly right.

Random vs systematic error, revisited

Conducting introduced these at the bench; here's the fuller picture and what each one does to your data:

  • Random error scatters results on both sides of the true value — some readings too high, some too low, for reasons like reaction time or reading precision that vary trial to trial. Repeating trials and averaging reduces its effect on your final reported value (though it doesn't shrink the uncertainty of any individual reading).
  • Systematic error shifts every result the same way, by roughly the same amount — a ruler with a worn zero-end, a stopwatch that runs consistently fast, a balance that wasn't zeroed. No amount of repeating and averaging fixes this, because every repeat is wrong in the same direction. Catching it requires checking the equipment itself (calibration) or cross-checking with an independent method.

This distinction matters for how you report a result: random error is why you quote an uncertainty at all; systematic error is why a result can be precise (tightly clustered repeats) while still being inaccurate (consistently off from the true value).

Uncertainty in a single reading

For a single reading from an analogue instrument (ruler, protractor, analogue meter), the standard convention is half the smallest division: a ruler marked in mm has an uncertainty of \(\pm 0.5\) mm on any single reading. For a digital instrument, the convention is usually half the last displayed digit, unless the manufacturer states otherwise — a stopwatch reading to 0.01 s has an uncertainty of \(\pm 0.005\) s from resolution alone (though, as Planning points out, human reaction time is usually the far larger uncertainty in practice).

Absolute vs percentage uncertainty

  • Absolute uncertainty has the same unit as the measurement: \(12.4 \pm 0.1\) cm.
  • Percentage uncertainty expresses that same margin as a fraction of the measurement:
\[ \%\text{ uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\% \]

For \(12.4 \pm 0.1\) cm: \(\dfrac{0.1}{12.4} \times 100\% \approx 0.8\%\).

Percentage uncertainty is what lets you compare how "good" two very different measurements are — \(\pm 0.1\) cm sounds identical to \(\pm 0.1\) mm until you check what fraction of the actual measurement each one represents.

Combining uncertainties through a calculation

Almost nothing in a physics prac is reported as a single raw measurement — you calculate a result from several measured quantities, and the uncertainty has to carry through that calculation. Two rules cover the great majority of Year 11 cases:

Add the absolute uncertainties.

If \(x = a + b\) (or \(a - b\)), then the absolute uncertainty in \(x\) is the sum of the absolute uncertainties in \(a\) and \(b\):

\[ \Delta x = \Delta a + \Delta b \]

Add the percentage uncertainties — and if a quantity is raised to a power \(n\), its percentage uncertainty is multiplied by \(n\) first.

If \(x = \dfrac{a \cdot b}{c}\), then:

\[ \%\Delta x = \%\Delta a + \%\Delta b + \%\Delta c \]

If \(x = a^n\), then \(\%\Delta x = n \times \%\Delta a\) — this is why a squared or cubed term in a formula (like \(t^2\) below) usually dominates the total uncertainty.

Worked example — uncertainty in g from free fall

Using \(g = \dfrac{2s}{t^2}\) (from \(s = \tfrac{1}{2}gt^2\), rearranged — see Motion Relationships), with measured drop height \(s = 1.20 \pm 0.01\) m and time \(t = 0.49 \pm 0.02\) s:

Percentage uncertainties:

\[ \%\Delta s = \frac{0.01}{1.20} \times 100\% \approx 0.8\% \]
\[ \%\Delta t = \frac{0.02}{0.49} \times 100\% \approx 4.1\%, \quad \text{so } \%\Delta(t^2) = 2 \times 4.1\% \approx 8.2\% \]

Total percentage uncertainty in g:

\[ \%\Delta g = \%\Delta s + \%\Delta(t^2) \approx 0.8\% + 8.2\% = 9.0\% \]
\[ g = \frac{2(1.20)}{(0.49)^2} \approx 9.996 \approx 10.0\ \text{m s}^{-2} \]

Absolute uncertainty: \(9.0\%\) of \(10.0\) is about \(0.9\), so report \(g = 10.0 \pm 0.9\ \text{m s}^{-2}\).

Notice how much of the total uncertainty came from \(t\), not \(s\) — because \(t\) is squared, its percentage uncertainty is doubled before it's added in. This is a completely general pattern: whichever measured quantity is squared (or worse, cubed) in a formula is usually the one worth measuring most carefully.

Mean and spread of repeated trials

For a set of repeated trials, the mean is the best single estimate of the true value:

\[ \bar{x} = \frac{\sum x_i}{n} \]

A simple, commonly-used estimate of the uncertainty in that mean, suitable for a small number of repeats, is half the range:

\[ \Delta \bar{x} = \frac{x_{max} - x_{min}}{2} \]

For a more rigorous measure of how spread out the repeats are, use the standard deviation, \(\sigma\):

\[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]

A small standard deviation means the repeats clustered tightly (good precision); a large one means they were scattered (poor precision, or a genuinely variable system). Most scientific calculators have a built-in standard deviation function (usually the \(\sigma_{n-1}\) key) — you're not expected to expand the sum by hand for anything beyond a small worked example.

Worked example — mean and half-range

Five repeats of a pendulum's period: 1.98 s, 2.01 s, 1.95 s, 2.03 s, 1.99 s.

\[ \bar{T} = \frac{1.98+2.01+1.95+2.03+1.99}{5} = 1.992 \approx 1.99\ \text{s} \]
\[ \Delta T = \frac{2.03 - 1.95}{2} = 0.04\ \text{s} \]

Report as \(T = 1.99 \pm 0.04\) s.

Advanced

Which uncertainty is the "real" one? A single reading's instrument uncertainty (half the smallest division) and the spread of repeated trials (half-range or standard deviation) are two different estimates of uncertainty, and they don't always agree. If your repeats are tightly clustered well within the instrument's resolution, the instrument uncertainty is the limiting factor and should be reported. If your repeats are scattered more widely than the instrument's resolution would suggest, something else — technique, an uncontrolled variable, genuine variability in the system — is the dominant source, and the spread of repeats is the more honest uncertainty to report. The rule of thumb: report whichever uncertainty is larger, since the smaller one is already "hidden" inside it.

Extension

Beyond the Physics 11–12 syllabus — won't appear in the HSC, included for interest / depth study inspiration.

📎 Depth study idea

The half-range method above is a simple estimate of the uncertainty in a single reading, but it's not the same question as "how confident am I in the mean of many repeats?" That second question is answered by the standard error of the mean:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

Notice it shrinks as \(n\) grows — averaging more repeats genuinely narrows your confidence in the mean, even though it doesn't shrink the underlying spread (\(\sigma\)) of individual readings at all. This is the mathematical justification for "more repeats is better," and it's directly usable in a depth study: for any investigation with a reasonably large number of repeats (10+), reporting standard error rather than half-range is a more defensible, university-level treatment of uncertainty in the mean.

For a full worked example, and how standard error feeds into a formal significance test, see the Extension boxes on Trends and Relationships.

Video/visual resources

  • 🎥 Khan Academy — TODO: source an uncertainty/error propagation explainer. Must cover the two propagation rules (add absolute for +/−, add percentage for ×/÷/powers) with a worked example — this is the most mechanically dense page in the whole toolkit and benefits most from a worked-through video. Essential.
  • 🎥 Physics High — TODO: check — this is genuinely exam-relevant physics content (uncertainty appears throughout HSC prac questions), so a match here is plausible unlike most other Working Scientifically pages.

Check yourself

  1. A length is measured as \(45.2 \pm 0.5\) cm. Calculate the percentage uncertainty.

    Answer

    \(\dfrac{0.5}{45.2} \times 100\% \approx 1.1\%\)

  2. Two lengths are measured as \(a = 15.0 \pm 0.2\) cm and \(b = 8.0 \pm 0.1\) cm. Find the absolute uncertainty in \(a + b\) and in \(a - b\).

    Answer

    Both use the same rule — add absolute uncertainties. \(\Delta(a+b) = \Delta(a-b) = 0.2 + 0.1 = 0.3\) cm. So \(a+b = 23.0 \pm 0.3\) cm and \(a - b = 7.0 \pm 0.3\) cm.

  3. A student measures resistance using \(R = V/I\), with \(V = 6.0 \pm 0.1\) V and \(I = 0.40 \pm 0.02\) A (see Electric Circuits). Calculate \(R\) and its absolute uncertainty.

    Answer

    \(R = 6.0 / 0.40 = 15.0\ \Omega\)

    \(\%\Delta V = \dfrac{0.1}{6.0}\times100\% \approx 1.7\%\), \(\%\Delta I = \dfrac{0.02}{0.40}\times100\% = 5.0\%\)

    \(\%\Delta R = 1.7\% + 5.0\% = 6.7\%\), which is \(6.7\%\) of \(15.0\ \Omega \approx 1.0\ \Omega\)

    Report as \(R = 15.0 \pm 1.0\ \Omega\). Note how much more the current's uncertainty contributes than the voltage's — worth knowing which meter to buy the extra care with, next time.