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Percentage uncertainty and error propagation

Every reading has an uncertainty set by the apparatus: half a division on a scale, one in the last digit on a display. Turning that into a percentage, adding the percentages through a calculation and then seeing which one dominates is the whole of A-level error analysis.

One reading, one uncertainty

A piece of apparatus is read to the nearest division, so a single reading is uncertain by half a division either way. A burette graduated in 0.1 cm³ is read to ±0.05 cm³. A thermometer marked in 1 °C steps is ±0.5 °C. A digital display is read differently, to one in its last digit, because it has already rounded: a balance reading to 0.01 g is ±0.01 g. A question that states ±0.005 g a reading overrides that, and its value is the one used. A 25 cm³ pipette carries the tolerance stamped on it, ±0.06 cm³ for a grade B pipette, and a 50 cm³ measuring cylinder is ±0.5 cm³. These are the absolute uncertainties, in the units of the reading.

The percentage uncertainty is the absolute uncertainty divided by the value, times 100. It is the number that matters, because it says how much of the reading is doubt. ±0.5 cm³ on a 50 cm³ measurement is 1%; the same ±0.5 on 5 cm³ is 10%. The apparatus has not changed, the reading has.

A value found as the difference between two readings carries the uncertainty of both. A titre is a final burette reading minus an initial one, so its uncertainty is 2 × 0.05 = ±0.10 cm³. A mass weighed by difference, boat plus solid then empty boat, is ±0.02 g. A temperature rise is ±1 °C from a thermometer read to ±0.5 °C. The doubling is where most lost marks sit: the scheme awards the percentage only when it is built on the doubled figure.

Combining through a calculation

When values are multiplied or divided, the percentage uncertainties add. A concentration from a mass and a volume carries the sum of their two percentages; a moles value from that concentration and a titre adds a third. The sum is the percentage uncertainty in the final answer, and the absolute uncertainty in the answer is that percentage of it. Adding percentages is the A-level convention, and it is a maximum, the case where every reading is wrong in the direction that compounds the rest.

When values are added or subtracted, the absolute uncertainties add, which is where the doubling for a difference came from. Do not add a percentage to an absolute figure: convert first, and keep each percentage to two significant figures.

When a bigger sample helps, and when it does not

A percentage uncertainty falls as the measured value grows, because the absolute figure is fixed by the apparatus. A 10 cm³ titre from a burette is ±1%, a 25 cm³ titre is ±0.4%, which is why a titration is planned so the titre is around 25 cm³: the concentration of the solution in the burette is chosen to put it there. Weighing more solid, taking a larger temperature rise and timing a longer interval all work the same way.

It stops helping when the bigger sample brings a new error. Doubling the solid in a calorimeter doubles the heat lost to the surroundings as well as the rise; a titre over 50 cm³ needs the burette refilled, which adds two more readings. The other route is better apparatus: a pipette in place of a measuring cylinder takes the 1.0% in the example to 0.24%, and a thermometer marked every 0.1 °C, read to ±0.05 °C, takes the 15% to 1.5%.

A mark scheme awards one mark for the percentage from the doubled reading, one for naming the largest contribution, and ignores a suggestion that repeats the experiment, because repeating reduces random error and does nothing to the uncertainty of a reading.

The lessons that use this: Measurement, uncertainty and significant figures in chemistry · Measuring an enthalpy change: calorimetry · Concentration, volumetric solutions and titration calculations. Every skill the boards number, and where each is taught: maths skills.