Uncertainties in the IB IA: absolute, percentage and propagation, with worked examples

An uncertainty is the range within which the true value of a measurement probably lies. In an IB IA you record an absolute uncertainty (±, in the same units) for every measured quantity, convert to percentage uncertainties whenever you multiply or divide, carry them through your calculations with a few simple rules, and then use the final uncertainty to judge your result: in error bars, in the comparison with a literature value, and in the evaluation.

That last step is where the marks are. In the Biology, Chemistry and Physics IA, the Data analysis criterion looks for an appropriate consideration of uncertainties, and the guide's clarification for the Conclusion criterion says a fully consistent conclusion interprets the processed data including its uncertainties. Calculating them and never mentioning them again misses half the point.

What is the difference between absolute, relative and percentage uncertainty?

Absolute uncertainty tells you how far off a single reading might be. Percentage uncertainty tells you how much that matters. ±0.10 cm³ is negligible on a 23.35 cm³ titre and enormous on a 1.20 cm³ one (8.3%). That is why small measured differences (a tiny mass of fuel burned, a 2 °C temperature rise) are usually the weak point of an IA.

What uncertainty should I give each instrument?

Use the instrument's own limits for a single reading, then check whether your repeats vary by more than that.

• Analogue scales (thermometer, measuring cylinder, ruler, burette): the usual convention is ± half the smallest division. A thermometer marked every 1 °C gives ±0.5 °C. A burette marked every 0.1 cm³ gives ±0.05 cm³ per reading.
• Digital instruments (balance, digital thermometer, pH meter, data logger): ± one unit in the last digit shown. A balance reading 2.03 g gives ±0.01 g.
• Glassware with a printed tolerance (volumetric pipettes and flasks): use the tolerance printed on it rather than the half-division rule.
• Timing by hand: a stopwatch shows 0.01 s, but your reaction time is a few tenths of a second, so ±0.01 s is not honest. Either justify a realistic value or, better, take it from the spread of your repeats.

Readings that are differences

A burette titre and a temperature change are each the difference between two readings, so both readings contribute. A titre read at 0.50 and 23.85 cm³ is 23.35 ± 0.10 cm³ (0.05 + 0.05). A digital thermometer reading 20.5 and 38.7 °C gives ΔT = 18.2 ± 0.2 °C.

Repeated readings: half the range

For three to five repeats, the usual IB convention is to quote the mean, with an uncertainty of half the range: (largest − smallest) ÷ 2. Five repeat times for catalase to produce 20.0 cm³ of oxygen (48.2, 51.6, 46.9, 50.3 and 49.0 s) give a mean of 49.2 s and an uncertainty of (51.6 − 46.9) ÷ 2 = 2.35 s. Rounded for reporting, that is 49 ± 2 s; keep the unrounded 49.2 and 2.35 for any further calculation.

How do you propagate uncertainties?

These are the rules used in IB science. They are "worst case" rules: they assume all the errors push in the same direction, which slightly overestimates the final uncertainty. That's fine for an IA.

Two habits prevent most errors. First, never add an absolute uncertainty to a percentage one; convert first. Second, keep full precision in your spreadsheet and round only the final answer.

Worked example 1: a titration (Chemistry)

An HCl solution is titrated with standard NaOH to find its concentration. The data and their uncertainties:

1. Moles of NaOH = 0.1000 × 23.35 ÷ 1000 = 2.335 × 10⁻³ mol. The reaction is 1:1, so this is also the moles of HCl.
2. Concentration of HCl = 2.335 × 10⁻³ ÷ 0.02500 = 0.09340 mol dm⁻³.
3. Total percentage uncertainty = 0.43 + 0.50 + 0.24 = 1.17%, roughly 1.2%.
4. Absolute uncertainty = 0.09340 × 0.0117 = 0.0011 mol dm⁻³.

Result: 0.0934 ± 0.0011 mol dm⁻³. Notice which term is biggest: the concentration of the standard, the one students most often leave out.

Worked example 2: enthalpy of combustion with q = mcΔT (Chemistry)

A spirit burner of ethanol heats 100.0 cm³ of water in a copper can.

1. q = mcΔT = 100.0 × 4.18 × 18.2 = 7608 J = 7.61 kJ. (The specific heat capacity, 4.18 J g⁻¹ K⁻¹, is treated as exact.)
2. Moles of ethanol = 0.46 ÷ 46.07 = 9.985 × 10⁻³ mol.
3. ΔH_c = −7.61 ÷ 9.985 × 10⁻³ = −762 kJ mol⁻¹.
4. Percentage uncertainty = 0.50 + 1.10 + 4.35 = 5.95%, about 6%. Absolute: 762 × 0.0595 = 45 kJ mol⁻¹.

Result: −760 ± 50 kJ mol⁻¹ (the mass of 0.46 g only supports two significant figures anyway). The literature value is −1367 kJ mol⁻¹. The section on systematic error below explains why that gap matters more than the ±50.

Worked example 3: a rate from a timer and a volume (Biology)

Catalase in yeast breaks down hydrogen peroxide; the oxygen is collected in an inverted measuring cylinder with 1 cm³ divisions. Rate is calculated as volume ÷ time.

• Volume: 20.0 ± 0.5 cm³ → 2.5%
• Time: 49.2 ± 2.35 s (half the range of five repeats, from above) → 4.8%
• Rate = 20.0 ÷ 49.2 = 0.4065 cm³ s⁻¹, with 2.5 + 4.8 = 7.3% uncertainty → ±0.030

Result: 0.41 ± 0.03 cm³ s⁻¹. The timing spread dominates, and in biology it usually will: variation between yeast samples is far larger than any instrument limit. That variation is a finding, not a nuisance. Report it as the standard deviation of your replicates and show it on your graph.

If you measure a time to an end point instead (amylase and the starch–iodine test, for example), rate is often taken as 1/t. A power of −1 keeps the same percentage uncertainty: t = 125 ± 5 s (4%) gives 1/t = (8.0 ± 0.3) × 10⁻³ s⁻¹.

How many decimal places and significant figures?

• Raw data: record to the precision of the instrument, the same number of decimal places down a column. If the balance reads to 0.01 g, write 4.20 g, not 4.2 g.
• Uncertainties: one significant figure, or two if the first digit is 1.
• Processed values: round to the same decimal place as the uncertainty.
• Headers: put the unit and uncertainty in the column header, e.g. "Time / s ± 0.2", so you don't repeat it in every cell.

Spreadsheets love eight decimal places. Use the cell format to fix the displayed precision; the "precise communication" part of Data analysis includes exactly this kind of convention.

Random or systematic error: what can uncertainties tell you?

Random errors scatter readings either side of the true value: reaction time, judging an end point, natural variation between organisms. Repeats reduce their effect, and your uncertainty measures them. Systematic errors push every reading the same way: heat lost to the air, a balance that isn't zeroed, a pipette rinsed with water instead of the solution. Repeats don't reduce them, and error bars don't show them.

The test that separates them in a Chemistry IA is to compare percentage error with percentage uncertainty:

percentage error = |experimental − literature| ÷ literature × 100

• Combustion example: |−762 − (−1367)| ÷ 1367 × 100 = 44%. The total uncertainty was 6%. Random error cannot explain a 44% gap, so systematic error dominates: heat warming the can and the air, incomplete combustion (soot on the can), ethanol evaporating from the wick. A more precise thermometer would change nothing; a draught shield, a lid and a cooling correction would.
• Titration example: if the teacher made the HCl at 0.0950 mol dm⁻³, the error is 1.7% against an uncertainty of 1.2%. The result is slightly outside its uncertainty, so a small systematic error is likely. A pipette left wet with water dilutes the acid and gives exactly this kind of low result.

How do uncertainties feed into the graph, conclusion and evaluation?

• Error bars: plot the uncertainty (or the standard deviation of replicates) as error bars and say in the caption which one you used. The guide to IA graphs covers this, plus how to get an uncertainty for a gradient from steepest and shallowest lines.
• Conclusion: state the final value with its uncertainty and say whether the literature value lies inside that range. For comparisons between conditions, ask whether the ranges overlap, and back that up with a test where it fits (see choosing a statistical test).
• Evaluation: rank your weaknesses using the numbers. The biggest percentage contribution is the measurement worth improving; a percentage error far above the uncertainty points to a systematic problem in the method. That turns a generic list into the specific, weighted evaluation the top band asks for.

To see uncertainties written into a full report, read the annotated science IA example: it shows instrument uncertainties in the headers, standard deviations in the processed table and error bars on the graphs.