At a glance
- Level: IB Diploma Chemistry (SL & HL); also useful for IGCSE Chemistry
- Best for: the error analysis section of an IB Internal Assessment (IA)
- Reading time: ~10 minutes
- Includes: a fully worked titration example and a downloadable PowerPoint
Every measurement you take in the lab carries some uncertainty — and turning that uncertainty into a meaningful number on your calculated answer is one of the most important practical skills in IGCSE and IB Chemistry. This post walks through the whole process, from reading the uncertainty off a piece of glassware to quoting your final answer as value ± uncertainty. The downloadable PowerPoint contains the same material in a teaching format, with a fully worked example you can drop straight into a lesson.
Why Uncertainty Matters
No piece of equipment is perfect. Every reading you take has a small range of values it could plausibly be — set by the smallest division on the scale and how confidently you can read it. We capture this as an absolute uncertainty (e.g. ±0.05 cm³), then convert it to a percentage error so that uncertainties from different pieces of equipment can be combined fairly.
The key formula is simple:
% error = (absolute uncertainty ÷ measured value) × 100
Typical Equipment Uncertainties
For analogue scales the rule of thumb is ± half the smallest division. The values below are the ones examiners expect to see in IGCSE and IB practical work:
| Equipment | Uncertainty (per reading) | Notes |
|---|---|---|
| Burette (50 cm³) | ± 0.05 cm³ | Two readings — see below |
| Volumetric pipette (25 cm³) | ± 0.06 cm³ | Single fixed-volume reading |
| Volumetric flask (250 cm³) | ± 0.30 cm³ | Single reading at the mark |
| Measuring cylinder (100 cm³) | ± 0.5 cm³ | Less precise — avoid for analysis |
| Top-pan balance (2 d.p.) | ± 0.005 g | Two readings if weighing by difference |
| Thermometer (1 °C scale) | ± 0.5 °C | Doubled if measuring a change |
Single Reading or Two Readings?
This is the part students forget most often, and it is a guaranteed mark on any IB or IGCSE markscheme. The uncertainty doubles whenever a value is found from the difference between two readings:
- Burette titre = final reading − initial reading → uncertainty = 2 × 0.05 = 0.10 cm³
- Mass by difference = mass of boat + solid, then mass of boat after transfer → uncertainty = 2 × 0.005 = 0.010 g
- Temperature change ΔT = final − initial → uncertainty = 2 × 0.5 = 1.0 °C
For a single fixed-volume reading (pipette, volumetric flask), use the equipment uncertainty as it stands.
Combining Percentage Errors
Once every measurement has its own % error, the rule for combining them is the easiest part of the whole topic:
When values are multiplied or divided to get a result, simply add their percentage errors.
This applies to almost every quantitative calculation in school chemistry — moles (n = c × V), concentration (c = n ÷ V), density (m ÷ V), rate (Δquantity ÷ Δtime). Constants such as molar mass have negligible error and can be ignored. Powers count multiple times — V² contributes its % error twice.
Converting Back to an Absolute Uncertainty
A percentage on its own does not tell anyone reading your report what the uncertainty actually is on the calculated value. Once you have a total % error, convert it back so the final answer can be quoted as value ± uncertainty:
absolute uncertainty = (% error ÷ 100) × final answer
Round the uncertainty to 1 (occasionally 2) significant figures, and match the precision of the answer to it.
The Five-Step Procedure
- Identify — list every measuring device used and its uncertainty.
- Adjust — double the uncertainty for any quantity that needs two readings.
- Convert — turn each absolute uncertainty into a % error using the actual reading taken.
- Combine — add the % errors of every quantity that appears in the calculation.
- Report — convert back to an absolute value and quote answer ± uncertainty.
Worked Example
Finding the Concentration of an Unknown HCl
A student is given a bottle of hydrochloric acid of unknown concentration. To standardise it, they prepare their own sodium carbonate primary standard and titrate against the unknown HCl using methyl orange indicator.
Reaction: Na₂CO₃ + 2HCl → 2NaCl + H₂O + CO₂ (1 : 2 ratio)
Procedure
- Weigh approximately 1.3 g of anhydrous sodium carbonate by difference on a 2 d.p. balance.
- Dissolve in distilled water, transfer to a 250 cm³ volumetric flask and make up to the mark.
- Pipette 25.00 cm³ of the standard into a conical flask and add a few drops of methyl orange.
- Titrate with the unknown HCl from a burette to the orange-red endpoint. Repeat for concordant titres.
Recorded Data
| Measurement | Value |
|---|---|
| Mass of weighing boat + Na₂CO₃ | 3.482 g |
| Mass of boat after transfer | 2.157 g |
| Mass of Na₂CO₃ used (by difference) | 1.325 g |
| Volume of standard solution prepared | 250.0 cm³ |
| Volume of standard pipetted | 25.00 cm³ |
| Mean titre (concordant only) | 24.50 cm³ |
Step A: Concentration of the Sodium Carbonate Standard
n(Na₂CO₃) = 1.325 ÷ 105.99 = 1.250 × 10⁻² mol
c(Na₂CO₃) = (1.250 × 10⁻²) ÷ 0.2500 = 0.05000 mol dm⁻³
Step B: Concentration of the HCl
n(Na₂CO₃) in pipette = 0.05000 × 0.02500 = 1.250 × 10⁻³ mol
From the 1 : 2 reaction ratio: n(HCl) = 2 × 1.250 × 10⁻³ = 2.500 × 10⁻³ mol
c(HCl) = (2.500 × 10⁻³) ÷ 0.02450 = 0.1020 mol dm⁻³
Step C: Percentage Error for Each Measurement
| Measurement | Working | % error |
|---|---|---|
| Mass (balance, by difference) | (2 × 0.005 ÷ 1.325) × 100 | 0.76 % |
| Volumetric flask (250 cm³) | (0.30 ÷ 250.0) × 100 | 0.12 % |
| Pipette (25 cm³) | (0.06 ÷ 25.00) × 100 | 0.24 % |
| Burette titre (two readings) | (2 × 0.05 ÷ 24.50) × 100 | 0.41 % |
| Molar mass (constant) | — | 0.00 % |
| Total % error in c(HCl) | — | 1.53 % |
Step D: Convert Back to an Absolute Uncertainty
absolute uncertainty = (1.53 ÷ 100) × 0.1020 = 0.001561 mol dm⁻³ ≈ 0.002 mol dm⁻³ (1 s.f.)
Final answer: c(HCl) = 0.102 ± 0.002 mol dm⁻³
Notice that the balance — not the volumetric glassware — is the largest single source of error here. That is a useful prompt for evaluation: if the student wanted to reduce the uncertainty further, weighing a larger mass of Na₂CO₃ (still using the same balance) would shrink the dominant term.
Watch out
Common Pitfalls
- Forgetting to double burette uncertainty. Every titre is a subtraction of two readings — markschemes always check this.
- Using the equipment maximum. Always divide by the actual reading taken, not the apparatus capacity.
- Including the rough titre in the mean. Only concordant titres (within ±0.10 cm³ of each other) belong in the mean.
- Quoting too many s.f. on the uncertainty. Round to 1 (sometimes 2) s.f., then match the precision of the answer.
- Adding absolute uncertainties for ×/÷. Absolute values are only added when quantities are added or subtracted. For multiplication or division, use percentage errors.
In one paragraph
Summary
Every device has an absolute uncertainty (usually ± half the smallest division). Double it for any quantity found from two readings. Convert each one to a percentage error using the actual measured value, add them up wherever the calculation involves multiplication or division, and convert back to an absolute value to give a final answer in the form value ± uncertainty, both quoted to a sensible number of significant figures.
Download the PowerPoint
The slide deck below presents this same material in a teaching-ready format — feel free to adapt it for your own classroom.
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