Percentage uncertainty and how to calculate uncertainty

Percentage uncertainty tells you how big the uncertainty in a measurement is compared with the measurement itself. The formula is percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%. A length of 20.0 cm measured to ±0.1 cm has a percentage uncertainty of 0.5%. To combine uncertainties, add the absolute uncertainties when you add or subtract values, and add the percentage uncertainties when you multiply or divide.

Absolute and percentage uncertainty

Every measurement has some doubt in it. The absolute uncertainty is the range either side of the value that the true value is likely to lie within, in the same units as the measurement. You write it with a ± sign, such as (2.40 ± 0.01) V.

The percentage uncertainty compares that range with the size of the measurement:

The first two rows use the same ruler, with the same absolute uncertainty. But the shorter length has ten times the percentage uncertainty. That is why measuring bigger quantities, such as the time for 10 swings of a pendulum rather than one, makes an experiment more precise.

How to find the absolute uncertainty

From the instrument's scale

AQA's A-level practical handbooks say the uncertainty in a reading is no smaller than plus or minus half of the smallest division. A thermometer marked every 1 °C gives readings to ±0.5 °C. So a reading of 23 °C has a percentage uncertainty of 0.5 ÷ 23 × 100 = 2.2%.

From two readings

Many measurements are the difference between two readings, and each reading has its own uncertainty. A titre is the final burette reading minus the initial one. If each reading is ±0.05 cm³, the titre has an uncertainty of ±0.10 cm³. The same applies to a temperature change found from a start and an end temperature. A length measured with a ruler counts as two readings too, the zero and the far end, so AQA's handbook gives it an uncertainty of one whole division: ±1 mm on most rulers.

From repeated readings

If you repeat a measurement, the uncertainty is half the range of the results. For four timings of 2.41 s, 2.47 s, 2.38 s and 2.44 s:

• Mean = (2.41 + 2.47 + 2.38 + 2.44) ÷ 4 = 2.425 s
• Range = 2.47 − 2.38 = 0.09 s, so the uncertainty is half of that, ±0.045 s
• Percentage uncertainty = 0.045 ÷ 2.425 × 100 = 1.9%

The result is (2.43 ± 0.05) s, rounding the uncertainty to match the precision of the readings. Leave out any anomalous result before you work out the range.

How to calculate percentage uncertainty: worked examples

Example 1: a titre

The initial burette reading is 1.20 cm³ and the final reading is 24.85 cm³. Each reading is ±0.05 cm³. Find the titre and its percentage uncertainty.

Titre = 24.85 − 1.20 = 23.65 cm³. The uncertainty is 0.05 + 0.05 = ±0.10 cm³.

A bigger titre would give a smaller percentage uncertainty, which is why titrations are designed to use a good fraction of the burette.

Example 2: a calculated resistance

A student measures V = (6.0 ± 0.1) V and I = (0.50 ± 0.02) A, and calculates R = V ÷ I. Find R and its uncertainty.

R = 6.0 ÷ 0.50 = 12 Ω. For a division, add the percentage uncertainties:

• Percentage uncertainty in V = 0.1 ÷ 6.0 × 100 = 1.7%
• Percentage uncertainty in I = 0.02 ÷ 0.50 × 100 = 4%
• Percentage uncertainty in R = 1.7% + 4% = 5.7%

The absolute uncertainty in R is 5.7% of 12 Ω, which is 0.68 Ω. So R = (12.0 ± 0.7) Ω.

Example 3: a squared quantity

The radius of a disc is (2.50 ± 0.05) cm. Find the percentage uncertainty in its area.

The percentage uncertainty in r is 0.05 ÷ 2.50 × 100 = 2%. Area = πr², and r is squared, so multiply the percentage uncertainty by 2: the area has a percentage uncertainty of 4%. The area is 19.6 cm², so its absolute uncertainty is 4% of 19.6, about 0.8 cm². π has no uncertainty, so it adds nothing.

Combining uncertainties: the rules

Uncertainties always add, even when you subtract the values. Subtracting two readings never cancels their uncertainties out.

Here is a quick check.

Percentage uncertainty and percentage error

These sound alike but answer different questions.

• Percentage uncertainty comes from your own measurements: how precise they are.
• Percentage error compares your result with an accepted value: how accurate it is. Percentage error = (your value − accepted value) ÷ accepted value × 100%.

If an experiment gives g = 9.6 m/s² and the accepted value is 9.81 m/s², the percentage error is (9.81 − 9.6) ÷ 9.81 × 100 = 2.1%. If your percentage uncertainty was 3%, the accepted value lies within your uncertainty, and the result is consistent with it. If your uncertainty was only 0.5%, there must be a systematic error you have not accounted for. The Percent Error Calculator works out the percentage error for you.

How to reduce percentage uncertainty

• Measure a larger quantity. Time 10 oscillations instead of 1: a ±0.1 s timing uncertainty on 14.2 s is 0.7%, but on 1.42 s it is 7%.
• Use an instrument with a finer scale, such as a micrometer instead of a ruler, or a balance reading to 0.001 g.
• Use larger amounts in chemistry: a larger titre or mass gives a smaller percentage uncertainty for the same apparatus.
• Repeat and average, which reduces the effect of random errors, though not systematic ones.

Common mistakes

• Subtracting uncertainties when you subtract values. Absolute uncertainties always add.
• Adding absolute uncertainties when you multiply. For multiplying or dividing, add the percentages.
• Forgetting the second reading. A titre or a length from a ruler involves two readings, so two uncertainties.
• Mismatched decimal places. AQA's handbooks say to quote the uncertainty to the same number of decimal places as the value: (2.43 ± 0.05) s, not (2.43 ± 0.045) s.
• Confusing uncertainty with error. Uncertainty is the spread you expect; error is how far you are from the true value.
• Including an anomaly in the range. Leave it out, and say why.

Where this comes up

Uncertainty is part of the practical skills assessed in A-level Chemistry and A-level Physics, and it appears in written exam questions about the required practicals. AQA's practical handbooks for both subjects set out the rules on this page: half the smallest division for a reading, half the range for repeats, adding absolute uncertainties for sums and differences, adding percentage uncertainties for products and quotients, and multiplying by the power for powers. The Practical Data Lab plots your results with error bars and helps you analyse them.

Practise in a short timed paper

A-level Chemistry: Year 1 and A-level Physics: Year 1 each give you a short timed paper every day, with 8 questions worth 25 to 35 marks, including questions on practical work and uncertainties.

Read next

Physics · Practical skillsWorst fit line calculator: gradient uncertainty, error barsFree worst fit line calculator: average repeat readings, draw error bars, and find the gradient and intercept with their uncertainties for a write-up.