Calculating Percentage Uncertainty

No measurement is perfect. Every balance, burette and thermometer has a limit to how precisely it can be read, and every result calculated from those measurements inherits that limitation. Uncertainty is how scientists express that limit honestly. It turns “the concentration is 0.0987 mol/dm³” into “the concentration is 0.0987 ± 0.0010 mol/dm³”, which is far more useful, because it tells you how much to trust the number.

Uncertainty calculations appear in almost every practical assessment. This guide shows how to do them step by step.

Absolute and percentage uncertainty

• Absolute uncertainty is the range either side of a measurement, in the same units: 25.00 ± 0.06 cm³.
• Percentage uncertainty expresses that range as a percentage of the measured value:

percentage uncertainty = (absolute uncertainty ÷ measured value) × 100

Percentage uncertainty lets you compare the precision of very different measurements, such as a mass and a volume.

Finding the absolute uncertainty of an instrument

Common conventions (check your own course’s rules, as they vary slightly):

Instrument	Typical uncertainty per reading
Balance reading to 0.01 g	±0.005 g (half the last digit) or ±0.01 g
Balance reading to 0.001 g	±0.0005 g or ±0.001 g
Thermometer with 1 °C divisions	±0.5 °C
Digital thermometer to 0.1 °C	±0.05 °C or ±0.1 °C
Burette	±0.05 cm³ per reading
25.00 cm³ volumetric pipette	±0.06 cm³ (Class B) or as marked
250.0 cm³ volumetric flask	±0.3 cm³ (Class B) or as marked
50 cm³ measuring cylinder	±0.5 cm³
Stopwatch (human reaction time)	often ±0.5 s or more, not the display’s 0.01 s

For glassware, use the tolerance printed on it where possible. See glassware tolerances.

Two readings, double the uncertainty

Many measurements are a difference between two readings:

• a burette titre (final − initial)
• a temperature change (final − initial)
• a mass by difference (before − after)

Each reading carries its own uncertainty, so the uncertainty of the difference is the sum of both:

• Burette titre: ±0.05 + ±0.05 = ±0.10 cm³
• Temperature rise with a ±0.5 °C thermometer: ±1.0 °C
• Mass by difference on a ±0.005 g balance: ±0.01 g

Note: taring a balance to zero doesn’t remove the second reading; the zero is itself a reading. Many courses therefore double the balance uncertainty for tared masses too. Follow your course’s convention.

Worked example 1: a single volume

A 25.00 cm³ pipette has an uncertainty of ±0.06 cm³. What is the percentage uncertainty?

0.06 ÷ 25.00 × 100 = 0.24%

Worked example 2: a titre

A titre is 18.45 cm³. Each burette reading has an uncertainty of ±0.05 cm³.

• Absolute uncertainty = ±0.10 cm³
• Percentage uncertainty = 0.10 ÷ 18.45 × 100 = 0.54%

Worked example 3: a temperature change

The temperature rises from 21.0 °C to 27.5 °C. The thermometer reads to ±0.5 °C.

• ΔT = 6.5 °C, absolute uncertainty = ±1.0 °C
• Percentage uncertainty = 1.0 ÷ 6.5 × 100 = 15%

That’s large. It shows why a digital thermometer reading to ±0.1 °C (uncertainty ±0.2 °C, or 3.1% here) makes a huge difference in energy change experiments. See enthalpy of neutralisation.

Combining uncertainties

When a result is calculated by multiplying or dividing measured quantities, the percentage uncertainties add.

If R = A × B or R = A ÷ B, then %U(R) ≈ %U(A) + %U(B).

When quantities are added or subtracted, the absolute uncertainties add (as with the two burette readings above).

These are the simple rules used in most school and college courses. More rigorous statistical methods combine uncertainties “in quadrature” (square root of the sum of squares), which gives a slightly smaller total; see propagating uncertainty.

Worked example 4: concentration from a titration

25.00 cm³ (±0.06 cm³) of NaOH is titrated with 0.1000 mol/dm³ HCl (assume negligible uncertainty). Titre = 22.40 cm³ (±0.10 cm³). Calculate [NaOH] and its uncertainty.

Calculate the result: n(HCl) = 0.1000 × 0.02240 = 2.240 × 10⁻³ mol [NaOH] = 2.240 × 10⁻³ ÷ 0.02500 = 0.08960 mol/dm³

Percentage uncertainties:

• Titre: 0.10 ÷ 22.40 × 100 = 0.45%
• Pipette: 0.06 ÷ 25.00 × 100 = 0.24%
• Total: 0.45 + 0.24 = 0.69%

Absolute uncertainty: 0.69% of 0.08960 = 0.00062 mol/dm³

Result: [NaOH] = 0.0896 ± 0.0006 mol/dm³

Worked example 5: enthalpy change

50.0 g of solution (assume no uncertainty in mass) rises by 6.5 °C (±0.2 °C with a digital thermometer). 0.0250 mol of water forms (uncertainty from the pipettes about 0.5%). Calculate ΔH and its uncertainty. (c = 4.18 J/g/°C)

• q = 50.0 × 4.18 × 6.5 = 1358.5 J
• ΔH = −1358.5 ÷ 0.0250 = −54,340 J/mol = −54.3 kJ/mol

Percentage uncertainties:

• ΔT: 0.2 ÷ 6.5 × 100 = 3.1%
• moles: 0.5%
• Total ≈ 3.6%

Absolute: 3.6% of 54.3 = 2.0 kJ/mol.

Result: ΔH = −54 ± 2 kJ/mol

Comparing with an accepted value

Once you have an uncertainty, you can judge whether your result agrees with the accepted value.

Percentage error (or percentage difference):

percentage error = |experimental − accepted| ÷ accepted × 100

Example: In example 5, the accepted value is −57.0 kJ/mol. Percentage error = |−54.3 − (−57.0)| ÷ 57.0 × 100 = 4.7%

The percentage uncertainty was 3.6%. Because the percentage error (4.7%) is bigger than the percentage uncertainty (3.6%), the measurement uncertainty alone can’t explain the difference. There must be a systematic error too, most likely heat lost to the surroundings, which would make the measured ΔT too small.

If the percentage error had been smaller than the percentage uncertainty, you could say the result agrees with the accepted value within experimental uncertainty.

How to reduce percentage uncertainty

1. Measure bigger quantities. A larger titre, mass or temperature change makes the fixed reading uncertainty a smaller percentage.
2. Use more precise instruments. A 0.001 g balance instead of 0.01 g; a digital thermometer instead of a 1 °C scale.
3. Use Class A glassware for critical volumes.
4. Repeat and average. Repeats reduce the effect of random errors (though not systematic ones).

In evaluation questions, identify the measurement with the largest percentage uncertainty and suggest improving that one: it’s the bottleneck.

Significant figures

Your final answer’s precision should match its uncertainty. Round the uncertainty to one (or sometimes two) significant figures, then round the result to the same decimal place:

• 0.089 60 ± 0.000 62 → 0.0896 ± 0.0006
• −54.34 ± 1.96 → −54 ± 2

See significant figures in chemistry for more.

Uncertainty vs error vs mistake

These three words are easy to mix up. Uncertainty is the range within which the true value probably lies, estimated before or during the experiment. Error is the difference between a measured result and the true or accepted value, which you can only find if the true value is known. A mistake (such as misreading a scale or spilling some solution) isn’t an uncertainty at all; the right response is to repeat the measurement, not to include it in the uncertainty.

Common mistakes

• Forgetting to double for two-reading measurements (titres, temperature changes).
• Adding absolute uncertainties when multiplying or dividing (use percentages).
• Quoting a result to more decimal places than the uncertainty allows.
• Confusing error with uncertainty. Uncertainty is the expected spread; error is the difference from the true value.
• Using stopwatch resolution (0.01 s) as the uncertainty, ignoring human reaction time.

Key takeaways

• Percentage uncertainty = absolute uncertainty ÷ measured value × 100.
• Differences between two readings have double the reading uncertainty.
• When multiplying or dividing, add percentage uncertainties; when adding or subtracting, add absolute uncertainties.
• If percentage error is bigger than percentage uncertainty, there’s a systematic error.
• Reduce uncertainty by measuring larger quantities with more precise instruments.