Measurement uncertainty and percentage error

Uncertainty of a reading

• For a scale you read by eye (ruler, measuring cylinder, thermometer), the uncertainty is usually half the smallest division: a measuring cylinder graduated in 0.2 cm³ gives ±0.1 cm³.
• For a digital display, it is usually ± the last digit (a balance reading to 0.01 g gives ±0.01 g).
• When a value comes from two readings (a burette titre = final reading minus initial reading, or a length from two ends of a ruler), add the uncertainties: a burette read to ±0.05 cm³ gives a titre of ±0.10 cm³.

Percentage uncertainty and percentage error

Percentage uncertainty = (uncertainty / measured value) x 100

The same uncertainty is a smaller percentage of a larger reading, so measuring a larger volume, longer time or longer length lowers the percentage uncertainty.

Percentage error = (difference between measured and true value / true value) x 100

Percentage error compares your result with an accepted or known value (for example the known vitamin C content of a standard solution).

If a result is calculated by multiplying or dividing two measurements, add their percentage uncertainties.

Random and systematic error

• Random errors make repeated readings scatter either side of the true value (judging a colour end point, reaction time). Reduce their effect by repeating and calculating a mean, or by using a better method (colorimeter, data logger).
• Systematic errors shift every reading the same way (a balance not zeroed, reading a meniscus from above, a colorimeter not zeroed with a blank). Repeating does not remove them; find and correct the cause, or calibrate.
• Accuracy is how close a result is to the true value; precision is how close repeated results are to each other (small spread, small standard deviation).

Worked example

A burette is read to ±0.05 cm³. The initial reading is 0.50 cm³ and the final reading is 12.90 cm³. Calculate the percentage uncertainty in the titre.

1. Titre = 12.90 - 0.50 = 12.40 cm³
2. Two readings, so uncertainty = 2 x 0.05 = ±0.10 cm³
3. Percentage uncertainty = 0.10 / 12.40 x 100 = 0.81%

Answer: 0.81% (about 0.8%)

Check yourself

1. A 10 cm³ measuring cylinder is graduated in 0.2 cm³ divisions. Calculate the percentage uncertainty when measuring 5.0 cm³.
   Show answer

   2.0 %. Uncertainty = half of 0.2 = 0.1 cm³; 0.1 / 5.0 x 100 = 2%.

2. A root is measured as 24 mm with a ruler; the uncertainty is ±0.5 mm. Calculate the percentage uncertainty to 2 significant figures.
   Show answer

   2.1 %. 0.5 / 24 x 100 = 2.1%.

3. A student's titration gives a vitamin C concentration of 0.86 mg cm⁻³ for a standard that is really 0.80 mg cm⁻³. Calculate the percentage error to 2 significant figures.
   Show answer

   7.5 %. (0.86 - 0.80) / 0.80 x 100 = 7.5%.

4. A burette titre of 6.20 cm³ comes from two readings, each ±0.05 cm³. Calculate the percentage uncertainty to 2 significant figures.
   Show answer

   1.6 %. 2 x 0.05 = 0.10; 0.10 / 6.20 x 100 = 1.6%.

5. How could the percentage uncertainty in measuring the volume of oxygen collected be reduced?
   • Repeat with a smaller volume
   • Use a measuring cylinder with larger divisions
   • Round the answer to fewer figures
   • Collect gas for longer so a larger volume is measured
   Show answer

   Collect gas for longer so a larger volume is measured. A larger reading makes the same uncertainty a smaller percentage.

6. A colorimeter is not zeroed with a blank, so every absorbance is 0.05 too high. What type of error is this?
   • Random error
   • Systematic error
   • Percentage error
   • Anomalous error
   Show answer

   Systematic error. The same shift in every reading is systematic; repeats will not remove it.

7. Five repeats give 4.1, 4.2, 4.1, 4.2, 4.1 s, but the true value is 5.0 s. How are these results best described?
   • Accurate but not precise
   • Accurate and precise
   • Neither accurate nor precise
   • Precise but not accurate
   Show answer

   Precise but not accurate. They are close together (precise) but far from the true value (not accurate).

8. Which method best reduces the effect of random error in judging a colour end point?
   • Zero the balance
   • Change the independent variable
   • Repeat several times and calculate a mean, or use a colorimeter
   • Use a smaller sample
   Show answer

   Repeat several times and calculate a mean, or use a colorimeter. Repeats average out random error; a colorimeter removes the judgement.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Biology specification (XBI11/YBI11, first teaching 2018): Unit 3, Unit 6 and Appendix 6 mathematical skills
• Pearson mark scheme WBI13/01, June 2024 (via Physics & Maths Tutor)

Last updated 8 November 2026. Spotted a mistake? Tell us.