Every measurement you ever make is a bit wrong, and the honest thing to do is say by how much. Uncertainty is not an admission of failure — it is the part of the answer that tells the reader how much to trust it.
📚 What you need to know
Uncertainty is a range around a measurement within which the true value is expected to lie.
Uncertainty is not the same as error. An error is a mistake or a bias; uncertainty is unavoidable.
Analogue scale: uncertainty is ± half the smallest division.
Digital display: uncertainty is ± the last significant digit.
Repeated readings: uncertainty is half the range.
Three forms: absolute, fractional and percentage.
Adding or subtracting → add the absolute uncertainties.
Multiplying or dividing → add the percentage uncertainties.
Raising to a power (HL) → multiply the percentage uncertainty by the power.
R2 measures how well a line or curve fits the data, from 0 to 1.
Uncertainty is not error
These two words get used interchangeably in everyday speech, and in the exam that will cost you.
Uncertainty is a property of the instrument. A burette cannot be read more finely than half a division, however careful you are.
Error is a difference between your result and the true value, caused by the equipment or the technique. Heat escaping from a calorimeter is an error.
Reducing uncertainty means using better equipment or measuring a larger quantity. Reducing error means fixing the method.
Where the numbers come from
Type of measurement
Uncertainty
Example
A reading on an analogue scale
± half the smallest division
burette with 0.10 cm3 divisions → ±0.05 cm3
A measurement needing two readings
at least ±1 smallest division
a titre needs a start and an end → ±0.10 cm3
Repeated data
half the range
±½(largest − smallest)
A digital display
± the last significant digit
balance reading 24.31 g → ±0.01 g
The red line is the meniscus sitting exactly on a division. Even then the uncertainty is ±0.05, because that is the finest judgement the scale allows.
The three forms of uncertainty
Form
What it is
For the burette reading above
Absolute
The actual amount of doubt, with units
±0.05 cm3
Fractional
Absolute uncertainty ÷ the measurement
0.05 ÷ 19.60 = 0.00255
Percentage
Fractional uncertainty × 100
0.26%
Percentage uncertainty
absolute uncertainty ÷ measured value × 100
Percentage uncertainty is the useful one, because it lets you compare instruments fairly. ±0.05 cm3 sounds small, but on a 2.00 cm3 measurement it is 2.5% — ten times worse than the same uncertainty on 19.60 cm3.
Combining uncertainties
Real answers come from several measurements. Uncertainty has to travel through the calculation with them, and how it travels depends on what you did.
The middle rule is the one you use most, because almost every chemistry calculation is a chain of multiplications and divisions.
🧩 Propagating uncertainty through a calculation
List every measured value with its absolute uncertainty.
Convert each to a percentage uncertainty.
Look at the calculation. Multiplications and divisions mean you add the percentages.
Add up the percentages to get the total percentage uncertainty in the answer.
Convert back to an absolute uncertainty by taking that percentage of your final answer.
Quote the answer as value ± absolute uncertainty, rounded sensibly.
How to reduce percentage uncertainty. Two ways only: use equipment with a smaller absolute uncertainty, or measure a larger quantity. Weighing 5.00 g on the same balance halves the percentage uncertainty of weighing 2.50 g.
How well does the line fit? R2
When a spreadsheet draws a trend line it usually offers a value called the coefficient of determination, written R2. It is a measure of how closely the points follow that line.
R2 = 0 — the line tells you nothing. You cannot predict y from x at all.
Between 0 and 1 — the closer to 1, the better the fit.
R2 = 1 — every point sits exactly on the line.
An R2 of 1.00 for a straight line drawn through data that is really curved would still be a bad choice of model. Fit and correctness are different questions.
Worked examples
WORKED EXAMPLE
A burette has 0.10 cm3 divisions. The initial reading is 0.00 cm3 and the final reading is 19.60 cm3. Calculate the absolute and percentage uncertainty in the titre.
Uncertainty in one reading
Analogue scale, so half the smallest division.
0.10 ÷ 2 = ±0.05 cm3A titre uses two readings
It is a subtraction, so the absolute uncertainties add.
0.05 + 0.05 = ±0.10 cm3Convert to a percentage0.10 ÷ 19.60 × 100 = 0.510…Titre = 19.60 ± 0.10 cm3, or ±0.51%this is why forgetting the second reading halves your stated uncertainty and costs a mark
WORKED EXAMPLE
A liquid has mass 2.50 ± 0.01 g and volume 250.0 ± 0.5 cm3. Calculate its density and the absolute uncertainty in that density.
Convert both to percentagesmass: 0.01 ÷ 2.50 × 100 = 0.40%volume: 0.5 ÷ 250.0 × 100 = 0.20%Density is a division, so add the percentages0.40 + 0.20 = 0.60%Work out the density itself2.50 ÷ 250.0 = 0.0100 g cm–3Turn the percentage back into an absolute0.60% of 0.0100 = 6.0 × 10–5Density = 0.0100 ± 0.00006 g cm–3the mass contributes twice as much uncertainty as the volume — that is where to improve
WORKED EXAMPLE
A thermometer reads to ±0.1°C. The temperature rises from 21.4°C to 28.9°C. Give the temperature change with its uncertainty, and comment on how to reduce the percentage uncertainty.
Find the change28.9 − 21.4 = 7.5°CThis is a subtraction, so add the absolutes0.1 + 0.1 = ±0.2°CAs a percentage0.2 ÷ 7.5 × 100 = 2.7%ΔT = 7.5 ± 0.2°C, which is ±2.7%to reduce it, produce a bigger temperature rise — the ±0.2 stays fixed while the 7.5 grows
💡 Exam tip
Quote uncertainties to 1 significant figure, and round the measurement to match its decimal places.
For anything read twice — a titre, a mass by difference, a temperature change — double the uncertainty.
Convert everything to percentages before combining across a multiplication or division.
Finish by converting back to an absolute uncertainty with the right units.
When asked how to improve an experiment, “measure a larger quantity” is often the strongest answer.
If you quote R2, say what it means for this data rather than just stating the number.
⚠️ Common mix-up
Uncertainty and error. Uncertainty comes from the instrument; error comes from the method.
Percentage uncertainty and percentage error. Uncertainty uses the equipment; error compares to a literature value.
Adding percentages when you should add absolutes. Check whether the step was +/− or ×/÷.
Using one reading’s uncertainty for a titre. Two readings means double.
Quoting an uncertainty to three decimal places. One significant figure is enough.
Assuming a high R2 proves the relationship. It only says the line fits the points you have.
Up next: Graphing Skills — drawing graphs that earn full marks, and pulling gradients, intercepts and areas back out of them.
Want this explained one-to-one?
Book a free session with an experienced IB Chemistry tutor and get your trickiest topics made simple.
