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.
Outcome codes: CH-11WS-04 (Year 11) and CH-12WS-04 (Year 12) — Processing data and information. Same outcome wording in both years, with a separate code for each. Wording on this page is paraphrased for teaching use — check the official syllabus before treating any line 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:
- Scientific notation and significant figures — maths refresher
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 volume of \(25.0 \pm 0.5\) mL is a claim that the true value very likely lies between 24.5 mL and 25.5 mL — not a claim that 25.0 mL 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 judging an endpoint or reading a scale 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 balance that wasn't zeroed, a thermometer that reads 1 °C high, heat always being lost from an uninsulated calorimeter. 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). Calorimetry is the classic chemistry example — results are often very consistent, and consistently too small, because heat always escapes.
Uncertainty in a single reading¶
For a single reading from an analogue instrument (measuring cylinder, thermometer, gas syringe), the standard convention is half the smallest division: a thermometer marked in 1 °C steps has an uncertainty of \(\pm 0.5\) °C on any single reading. For a digital instrument, the convention is usually half the last displayed digit, unless the manufacturer states otherwise — though many school balances are best treated as \(\pm 0.01\) g, and volumetric glassware has its tolerance printed on it (e.g. a 25 mL pipette marked ±0.03 mL).
Absolute vs percentage uncertainty¶
- Absolute uncertainty has the same unit as the measurement: \(25.0 \pm 0.5\) mL.
- Percentage uncertainty expresses that same margin as a fraction of the measurement:
For \(25.0 \pm 0.5\) mL: \(\dfrac{0.5}{25.0} \times 100\% = 2\%\).
Percentage uncertainty is what lets you compare how "good" two very different measurements are — \(\pm 0.5\) mL is fine for 250 mL (0.2%) but poor for 5 mL (10%).
Combining uncertainties through a calculation¶
Almost nothing in a chemistry prac is reported as a single raw measurement — you calculate a result (a concentration, an enthalpy change) 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 \(b\):
The most common chemistry case: a temperature change. Two readings of \(\pm 0.5\) °C each give \(\Delta T\) an uncertainty of \(\pm 1.0\) °C.
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:
Molar masses and constants (like \(c = 4.18\) for water) are usually treated as exact — only measured quantities carry uncertainty.
Worked example — uncertainty in the concentration of a standard solution
\(1.33 \pm 0.01\ \text{g}\) of anhydrous sodium carbonate (\(MM = 105.99\)) is made up to \(250.0 \pm 0.2\ \text{mL}\) in a volumetric flask (see Measures of Concentration). Find the concentration and its uncertainty.
Percentage uncertainties: \(\%\Delta m = \dfrac{0.01}{1.33} \times 100\% = 0.75\%\); \(\%\Delta V = \dfrac{0.2}{250.0} \times 100\% = 0.08\%\)
Total: \(0.75\% + 0.08\% = 0.83\%\), and \(0.83\%\) of \(0.0502\) is about \(0.0004\).
Report as \(C = 0.0502 \pm 0.0004\ \text{mol L}^{-1}\).
Notice almost all the uncertainty came from the mass, not the volume. To improve this, weigh a larger mass (and use a bigger flask) — the volumetric flask is already doing its job.
Worked example — uncertainty in heat energy from calorimetry
In a neutralisation, \(100 \pm 1\ \text{g}\) of solution warms from \(18.3\) °C to \(24.6\) °C, using a thermometer read to \(\pm 0.5\) °C.
Temperature change (subtracting → add absolute uncertainties): \(\Delta T = 24.6 - 18.3 = 6.3 \pm 1.0\ ^\circ\text{C}\)
Heat (multiplying → add percentage uncertainties): \(q = mc\Delta T = 100 \times 4.18 \times 6.3 = 2633\ \text{J}\)
\(\%\Delta m = 1\%\); \(\%\Delta(\Delta T) = \dfrac{1.0}{6.3} \times 100\% = 16\%\); total \(\approx 17\%\)
Report as \(q = 2.6 \pm 0.4\ \text{kJ}\). The thermometer dominates — a small temperature change measured with a ±0.5 °C thermometer is very uncertain. A digital thermometer reading to 0.1 °C, or using more concentrated solutions to get a bigger \(\Delta T\), would help most.
Mean and spread of repeated trials¶
For a set of repeated trials, the mean is the best single estimate of the true value:
A simple, commonly-used estimate of the uncertainty in that mean, suitable for a small number of repeats, is half the range:
For a more rigorous measure of how spread out the repeats are, use the standard deviation, \(\sigma\):
A small standard deviation means the repeats clustered tightly (good precision); a large one means they were scattered. Most scientific calculators have a built-in standard deviation function (usually the \(\sigma_{n-1}\) key).
Worked example — mean and half-range
Five repeats of the disappearing-cross time at 30 °C: 35.2 s, 34.8 s, 35.0 s, 35.6 s, 34.9 s.
Report as \(t = 35.1 \pm 0.4\) 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. In the disappearing-cross prac, the stopwatch reads to 0.01 s, but the repeats spread over almost a second — because judging when the cross "disappears" varies far more than the stopwatch does. When repeats are scattered more widely than the instrument's resolution would suggest, 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 syllabus for this course — 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:
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." 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 covering the two rules (add absolute for +/−, add percentage for ×/÷) with a worked example. Essential — this is the most mechanically dense page in the toolkit.
Check yourself¶
-
A mass is measured as \(2.45 \pm 0.01\) g. Calculate the percentage uncertainty.
Answer
\(\dfrac{0.01}{2.45} \times 100\% \approx 0.41\%\)
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A gas syringe reads \(2.0 \pm 0.5\) mL at the start of a reaction and \(48.0 \pm 0.5\) mL at the end. Find the volume of gas produced and its uncertainty.
Answer
Subtracting → add absolute uncertainties. Volume \(= 48.0 - 2.0 = 46.0\) mL, uncertainty \(= 0.5 + 0.5 = 1.0\) mL. So \(46.0 \pm 1.0\) mL.
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A solution contains \(0.0250 \pm 0.0002\) mol of solute in \(0.250 \pm 0.002\) L. Calculate its concentration and absolute uncertainty.
Answer
\(C = \dfrac{0.0250}{0.250} = 0.100\ \text{mol L}^{-1}\)
\(\%\Delta n = \dfrac{0.0002}{0.0250} \times 100\% = 0.8\%\); \(\%\Delta V = \dfrac{0.002}{0.250} \times 100\% = 0.8\%\); total \(1.6\%\)
\(1.6\%\) of \(0.100\) is \(0.0016\), so report \(C = 0.100 \pm 0.002\ \text{mol L}^{-1}\).