Propagating Uncertainties Through Calculations

A final result in chemistry is rarely measured directly. A concentration comes from a mass, a volume and a molar mass; an enthalpy change from a mass, a temperature change and a number of moles; a Ka from a pH reading. Each measurement brings its own uncertainty, and those uncertainties propagate through the calculation into the final answer.

This guide covers the rules for propagating uncertainties, from the simple versions used in most school courses to the more rigorous methods used at university and in research.

Two sets of rules

There are two common approaches:

1. The simple (worst-case) rules, used in many school and college courses. They add uncertainties directly, which gives a slightly generous (larger) estimate.
2. The quadrature rules, used in university and professional work. They assume the uncertainties are independent and random, and combine them as a square root of the sum of squares, which gives a smaller, statistically more realistic estimate.

Use whichever your course requires. Both are shown below.

Rule 1: adding and subtracting

If R = A + B or R = A − B:

• Simple rule: ΔR = ΔA + ΔB (add absolute uncertainties)
• Quadrature: ΔR = √(ΔA² + ΔB²)

Example: a burette titre. Initial reading 0.40 ± 0.05 cm³, final reading 23.85 ± 0.05 cm³.

Titre = 23.45 cm³

• Simple: ±(0.05 + 0.05) = ±0.10 cm³
• Quadrature: ±√(0.05² + 0.05²) = ±0.071 cm³

Note that for subtraction, the uncertainties still combine; they never cancel.

Rule 2: multiplying and dividing

If R = A × B or R = A ÷ B:

• Simple rule: %ΔR = %ΔA + %ΔB (add percentage uncertainties)
• Quadrature: %ΔR = √((%ΔA)² + (%ΔB)²)

Example: concentration of a standard solution. 2.508 g (±0.010 g) of sodium carbonate (M = 105.99 g/mol, uncertainty negligible) is dissolved and made up to 250.0 cm³ (±0.30 cm³).

c = mass ÷ (M × V) = 2.508 ÷ (105.99 × 0.2500) = 0.09465 mol/dm³

Percentage uncertainties:

• mass: 0.010 ÷ 2.508 × 100 = 0.40%

• volume: 0.30 ÷ 250.0 × 100 = 0.12%

• Simple: 0.40 + 0.12 = 0.52% → ±0.00049 mol/dm³

• Quadrature: √(0.40² + 0.12²) = 0.42% → ±0.00040 mol/dm³

Result: c = 0.0947 ± 0.0005 mol/dm³ (simple) or ± 0.0004 mol/dm³ (quadrature).

Rule 3: powers

If R = Aⁿ, then %ΔR = n × %ΔA.

This rule is the same in both approaches, because the “errors” in each factor of A are the same error, not independent ones.

Example: a sphere’s volume V = (4/3)πr³. If r has an uncertainty of 1%, V has an uncertainty of 3%.

Chemistry example: Ka = [H⁺]² ÷ c for a weak acid, if [H⁺] has an uncertainty of 5%, the [H⁺]² term contributes 10%.

Rule 4: multiplying by a constant

If R = k × A, where k is an exact number (such as 2 in a mole ratio, or 1000 in a unit conversion):

• ΔR = k × ΔA, and the percentage uncertainty is unchanged.

Exact numbers carry no uncertainty. Constants such as molar masses and the gas constant have tiny uncertainties that are usually ignored.

Rule 5: logarithms and exponentials

This matters for pH, pKa and anything logarithmic.

If R = log₁₀ A, then ΔR ≈ 0.434 × (ΔA ÷ A).

If A = 10ᴿ, then ΔA ÷ A ≈ 2.303 × ΔR.

Example: from pH to [H⁺]. A pH meter reads 3.40 ± 0.02.

[H⁺] = 10⁻³·⁴⁰ = 3.98 × 10⁻⁴ mol/dm³

Relative uncertainty = 2.303 × 0.02 = 0.046 = 4.6%

So [H⁺] = (3.98 ± 0.18) × 10⁻⁴ mol/dm³.

This shows something important: a seemingly tiny pH uncertainty of ±0.02 means almost 5% uncertainty in [H⁺]. Calculating Ka from a single pH reading is therefore much less precise than it might appear.

Worked example: Ka from pH

A 0.1000 mol/dm³ (±0.5%) solution of a weak acid has pH 2.88 ± 0.02. Find Ka and its uncertainty.

Calculate Ka:

• [H⁺] = 10⁻²·⁸⁸ = 1.318 × 10⁻³ mol/dm³
• Ka ≈ [H⁺]² ÷ c = (1.318 × 10⁻³)² ÷ 0.1000 = 1.74 × 10⁻⁵

(ignoring the small correction to c for the ionised fraction)

Uncertainty:

• [H⁺]: 2.303 × 0.02 = 4.6%
• [H⁺]²: 2 × 4.6% = 9.2%
• c: 0.5%
• Simple total: 9.2 + 0.5 = 9.7%
• Quadrature: √(9.2² + 0.5²) = 9.2%

Ka = (1.74 ± 0.17) × 10⁻⁵ (simple rule)

The pH reading dominates the uncertainty completely. To measure Ka more precisely, use the half-equivalence point of a titration curve (where pH = pKa directly), which is less sensitive to concentration errors. See plotting a pH curve.

Worked example: enthalpy of combustion

Burning 0.52 g (±0.01 g, by difference) of ethanol heats 100.0 g (±0.5 g) of water by 31.0 °C (±0.2 °C). M(ethanol) = 46.07 g/mol. Find ΔcH and its uncertainty.

• q = 100.0 × 4.18 × 31.0 = 12,958 J
• n(ethanol) = 0.52 ÷ 46.07 = 0.01129 mol
• ΔcH = −12,958 ÷ 0.01129 = −1,148,000 J/mol = −1150 kJ/mol

Percentage uncertainties:

• mass of water: 0.5 ÷ 100.0 = 0.50%
• ΔT: 0.2 ÷ 31.0 = 0.65%
• mass of ethanol: 0.01 ÷ 0.52 = 1.9%
• Simple total: 0.50 + 0.65 + 1.9 = 3.1% → ±36 kJ/mol

Result: ΔcH = −1150 ± 40 kJ/mol.

The accepted value is −1367 kJ/mol, a difference of about 16%, far bigger than the 3.1% uncertainty. So measurement uncertainty can’t explain the gap: there are large systematic errors, mainly heat lost to the surroundings and incomplete combustion.

Uncertainty from repeated measurements

When you repeat a measurement several times, the spread of results gives a direct, experimental estimate of uncertainty that includes all random effects, including ones you didn’t think of.

Standard deviation (s) measures the spread:

s = √[Σ(xᵢ − x̄)² ÷ (n − 1)]

Standard error of the mean tells you how precisely you know the average:

SE = s ÷ √n

Example: four titres. 22.40, 22.45, 22.35, 22.40 cm³.

• Mean x̄ = 22.40 cm³
• Deviations: 0.00, +0.05, −0.05, 0.00
• Σ(deviations²) = 0.0050
• s = √(0.0050 ÷ 3) = 0.041 cm³
• SE = 0.041 ÷ √4 = 0.020 cm³

The mean titre is 22.40 ± 0.02 cm³ (as a standard error). That’s smaller than the single-titre reading uncertainty of ±0.10 cm³, which shows the value of repeating measurements. Note that repeats reduce random uncertainty only; a systematic error (like an unrinsed burette) affects every repeat equally.

Why quadrature gives a smaller answer

The simple rule assumes the worst case: that every measurement is off by its full uncertainty, in the same direction, at the same time. In reality, independent random uncertainties are just as likely to partly cancel as to add up. Squaring, adding and taking the square root reflects that statistically, which is why quadrature totals are always smaller than simple totals but never smaller than the largest single contribution. For school work, the simple rule is safer and easier; for research, quadrature is more honest.

Which rule should you use?

Situation	Suggested approach
Most school and college exams	simple rules: add absolute uncertainties for sums, percentages for products
University labs, research	quadrature, and statistical analysis of repeats
A quick check of which measurement matters most	either; compare the individual percentage uncertainties

Whichever rule you use, the practical lesson is the same: find the measurement with the largest percentage uncertainty. Improving that one gives the biggest gain. In the ethanol example, weighing a larger mass of fuel would help far more than a better thermometer.

Key takeaways

• Sums and differences: combine absolute uncertainties. Products and quotients: combine percentage uncertainties.
• Simple rules add directly; quadrature takes the square root of the sum of squares.
• Powers multiply the percentage uncertainty by the exponent; exact constants add no uncertainty.
• For pH, a ±0.02 uncertainty means about ±4.6% in [H⁺].
• Repeats let you calculate a standard deviation and standard error, reducing random uncertainty.
• Start with the basics in calculating percentage uncertainty.