Uncertainties in the IB Chemistry IA: A Physicist's Guide
The maths in the IB Chemistry Internal Assessment worries many of my students more than the chemistry does. I asked Pietro Meloni, a physicist who has handled experimental uncertainties in research as well as in teaching, to explain it the way he explains it to his own students. Rose
In short: IB Chemistry asks for three things about uncertainty. First, give every measurement an uncertainty. Second, carry it through your calculation with two rules: when you add or subtract, add the absolute uncertainties, and when you multiply or divide, add the percentage uncertainties. Third, compare your final percentage uncertainty with your percentage error against the literature value. That comparison tells you whether your result is limited by random or by systematic error, and it gives your evaluation something specific to say.
Why does uncertainty matter so much in IB Chemistry?
Every number you measure in a lab is a range, not a point. A burette reading of 24.50 cm³ means "somewhere between 24.45 and 24.55 cm³". The uncertainty is simply the honest width of that range.
In the current IB Chemistry course (first assessment May 2025), this sits in the "Tools for chemistry", under Tool 3: Mathematics. It is not a separate topic you can skip. It runs through every practical, through the data questions in the exam, and above all through the scientific investigation, the Internal Assessment, which is worth 20% of the final grade. In the IA, your data analysis, your conclusion and your evaluation all depend on handling uncertainty correctly.
The good news is that the maths involved is small. Most of it is addition and percentages. What students usually lack is not the maths but a clear method, so here it is.
What is the uncertainty of a single measurement?
Start from the instrument. The usual conventions are:
- Volumetric glassware (pipettes, volumetric flasks, burettes): use the tolerance printed on the glass. It is the manufacturer's statement and the most defensible number you can quote.
- Digital instruments (balances, digital thermometers, pH meters): plus or minus one unit in the last digit shown.
- Analogue scales (a measuring cylinder, a liquid-in-glass thermometer): plus or minus half of the smallest division, unless you can justify a different value.
Typical values you will meet in a school lab:
| Equipment | Typical uncertainty |
|---|---|
| Burette, each reading | ±0.05 cm³ |
| 25.00 cm³ pipette | ±0.03 cm³ (check the glass) |
| 50 cm³ measuring cylinder | ±0.5 cm³ |
| Balance to 2 decimal places | ±0.01 g |
| Digital thermometer to 0.1 °C | ±0.1 °C |
One detail catches many students out. A burette volume is the difference of two readings, and each reading carries its own ±0.05 cm³. So the volume delivered has an uncertainty of ±0.10 cm³, not ±0.05 cm³. The same applies to a temperature change: two readings, so twice the uncertainty.
How do you propagate uncertainties in a calculation?
IB Chemistry needs just two rules.
- Adding or subtracting quantities: add the absolute uncertainties.
- Multiplying or dividing quantities: add the percentage uncertainties, then convert back to an absolute uncertainty at the end.
Percentage uncertainty is the absolute uncertainty divided by the value, times 100. Here is a titration worked from start to finish.
Worked example: the concentration of an acid by titration
A 25.00 cm³ sample of hydrochloric acid is titrated with 0.1000 mol dm⁻³ sodium hydroxide. The burette reads 0.50 cm³ at the start and 25.00 cm³ at the end point. We treat the uncertainty in the sodium hydroxide concentration as negligible here. In your own IA, include it if you prepared the solution yourself.
Titre: 25.00 − 0.50 = 24.50 cm³, uncertainty 0.05 + 0.05 = ±0.10 cm³, which is 0.10 / 24.50 = 0.41%
Pipette: 25.00 ± 0.03 cm³, which is 0.03 / 25.00 = 0.12%
Concentration of HCl (1:1 reaction): 0.1000 × 24.50 / 25.00 = 0.09800 mol dm⁻³
Total percentage uncertainty: 0.41% + 0.12% = 0.53%
Absolute uncertainty: 0.53% of 0.09800 = 0.00052, so the result is 0.0980 ± 0.0005 mol dm⁻³
Notice how the result is written. A common convention, and the one I recommend, is to round the absolute uncertainty to one significant figure and then round the result to the same decimal place. A value of 0.098000 ± 0.00052 claims a precision that the burette never delivered.
Which uncertainty actually limits your result?
This is the habit I would most like Chemistry students to borrow from physics: once you have the percentage uncertainty of each measurement, look for the largest one. It usually dominates the total, and it tells you exactly where the experiment can be improved.
Worked example: the enthalpy of neutralisation
50.0 cm³ of 1.00 mol dm⁻³ HCl is mixed with 50.0 cm³ of 1.00 mol dm⁻³ NaOH in a polystyrene cup. Each volume is measured with a 50 cm³ measuring cylinder. The temperature rises from 21.0 °C to 27.2 °C.
Temperature change: 6.2 ± 0.2 °C (two readings of ±0.1 °C), which is 3.2%
Volume of solution: 100.0 ± 1.0 cm³ (two cylinders of ±0.5 cm³), which is 1.0%, taken as 100.0 g of solution
Heat released: q = mcΔT = 100.0 × 4.18 × 6.2 = 2592 J
Moles of water formed: 0.0500 mol, so ΔH = −2592 / 0.0500 = −51.8 kJ mol⁻¹
Total percentage uncertainty: 3.2% + 1.0% = 4.2%, so ΔH = −52 ± 2 kJ mol⁻¹
The temperature change contributes three times as much as the volumes. Buying a better balance or pipette would barely change the result. A larger temperature change would: more concentrated solutions, or a thermometer with a finer scale. That is a specific, justified improvement, the kind an evaluation should contain, far stronger than "use more precise equipment".
What is the difference between percentage error and percentage uncertainty?
The two sound alike and answer different questions.
- Percentage uncertainty comes from your equipment. It is the spread you expect from random error alone.
- Percentage error compares your result with an accepted value: the difference between them divided by the accepted value, times 100.
Put them side by side. The accepted value for a strong acid with a strong base is about −57.1 kJ mol⁻¹. Our result of −51.8 kJ mol⁻¹ is 9.3% away, while our uncertainty is only 4.2%.
When the percentage error is clearly larger than the percentage uncertainty, random error cannot explain the gap: something systematic is going on. Here the direction confirms it, since the measured value is smaller in magnitude, which is exactly what heat lost to the cup and the air would produce. When the error is within the uncertainty, the result agrees with the accepted value, and the way to improve it is to reduce the random error. This one comparison turns a vague evaluation into a reasoned one.
How do uncertainties work with graphs?
Many investigations end in a graph, and the result is a gradient. Take the Arrhenius equation: plotting ln k against 1/T gives a straight line whose gradient equals −Ea/R.
To estimate the uncertainty of a gradient without any statistics software:
- Draw uncertainty bars on your points where the uncertainty is significant.
- Draw the line of best fit, then the steepest and the shallowest lines that still pass through the uncertainty bars.
- The spread between the steepest and shallowest gradients gives the uncertainty of your gradient.
Best-fit gradient: −6500 K, so Ea = 6500 × 8.31 = 54.0 kJ mol⁻¹
Steepest and shallowest lines: −7000 K and −6000 K, giving 58.2 and 49.9 kJ mol⁻¹
Result: Ea = 54 ± 4 kJ mol⁻¹
Significant figures and logarithms: the pH trap
pH has its own rule, and it costs marks every year. With a logarithm, the number of decimal places in the pH equals the number of significant figures in the concentration. A hydrogen ion concentration of 0.0980 mol dm⁻³ has three significant figures, so the pH is 1.009, not 1.01 and not 1.0088. The digits before the decimal point only locate the power of ten; the precision lives in the decimals.
A checklist for your IA
- Every raw measurement in your tables has an uncertainty with units, stated once in the column header.
- Differences of two readings carry double the reading uncertainty.
- Additions and subtractions add absolute uncertainties; multiplications and divisions add percentages.
- The final result is rounded to match its uncertainty.
- You have named the largest source of uncertainty and proposed an improvement that targets it.
- You have compared percentage error with percentage uncertainty and said what that means for random and systematic error.
- The evaluation discusses your actual numbers, not generic sources of error.
Precise wording matters as much as the numbers: "evaluate", "discuss" and "justify" each ask for something different, as Rose explains in her guide to IB command words.
About the author
Pietro Meloni is a PhD physicist and a tutor of IB Mathematics and Physics, based in Europe and teaching international-school students online worldwide.
He earned his PhD in Physics at Roma Tre University, with research on CYGNO, a directional dark matter experiment of the Italian National Institute for Nuclear Physics (INFN), where measurements and their uncertainties were daily work. He teaches IB Mathematics AA and AI and IB Physics, SL and HL, one to one. More at pietromeloni.com.
Questions
How do you calculate uncertainty in an IB Chemistry titration?
The titre is the difference between two burette readings, so its uncertainty is twice the reading uncertainty, usually ±0.10 cm³. Convert it to a percentage, add the percentage uncertainty of the pipette, and apply the total percentage to the final concentration.
Do you add absolute or percentage uncertainties?
Add absolute uncertainties when you add or subtract quantities. Add percentage uncertainties when you multiply or divide them.
What does it mean if my percentage error is bigger than my percentage uncertainty?
Random error alone cannot explain the difference from the accepted value, so there is a systematic error in the method, such as heat loss in calorimetry. Identify it and suggest a change that addresses it.
How many significant figures should an uncertainty have?
A common convention is one significant figure for the absolute uncertainty, with the result rounded to the same decimal place, as in 0.0980 ± 0.0005 mol dm⁻³.
Pietro Meloni and Rose Kurian are independent tutors; neither pays the other for this article. Published September 2026. For help with IB Chemistry itself, see Rose's IB Chemistry tutoring.
