Lesson 4 · 25 min

Propagation of uncertainty

The burette, the pipette and the standard solution each carry an uncertainty — how do they combine in the final concentration? Absolute uncertainties for sums, relative ones for products, and special rules for logs and powers.

In this lesson

Combine absolute uncertainties for sums and differences, and relative uncertainties for products and quotients
Handle logarithms and powers: from the uncertainty of a pH to that of [H⁺], and back
Turn glassware tolerances into standard uncertainties and build the uncertainty budget of a titration
Find the dominant source of uncertainty and report a result with its expanded uncertainty

A single acid–base titration uses at least three things with errors: the standardised NaOH solution, the burette and the pipette. A class A 50 mL burette has a tolerance of ±0.05 mL, a 25 mL pipette ±0.03 mL, and the NaOH concentration is known only to ±0.2 %. How reliable is the final hydrochloric-acid concentration? Do you add all the errors, or keep only the largest? The answer lies in between: independent errors partly cancel, and they combine as the square root of the sum of squares.

Predict first

c(HCl) = c(NaOH)·V(NaOH)/V(HCl). The three inputs have relative uncertainties of 0.20 %, 0.23 % and 0.12 %. What is the relative uncertainty of the result, roughly?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

0.55 % (added up) 0.33 % (root sum of squares) 0.23 % (the largest one wins) 0.12 % (the smallest one)

The rules of propagation

\begin{aligned} y = a + b - c:&\quad u_y = \sqrt{u_a^2 + u_b^2 + u_c^2}\\ y = \frac{a\,b}{c}:&\quad \frac{u_y}{y} = \sqrt{\left(\frac{u_a}{a}\right)^2 + \left(\frac{u_b}{b}\right)^2 + \left(\frac{u_c}{c}\right)^2}\\ y = a^n:&\quad \frac{u_y}{y} = |n|\,\frac{u_a}{a}\end{aligned}

Provided the errors of the inputs are independent. Sums and differences use absolute uncertainties; products and quotients use relative ones.

$u_a$: standard uncertainty of a (same units as a)
$u_a/a$: relative uncertainty (dimensionless, often in %)
$n$: power: squaring doubles the relative uncertainty

Logarithms and powers have their own rules; in chemistry the pH is the classic case:

y = \lg a:\ u_y = \frac{u_a}{a\ln 10} = 0.434\,\frac{u_a}{a},\qquad y = 10^a:\ \frac{u_y}{y} = \ln 10 \cdot u_a = 2.303\,u_a

The uncertainty of a pH is absolute; that of [H⁺] is relative. pH = 4.00 ± 0.02 means $[\ce{H+}] = 1.00\times10^{-4}$ M with a relative uncertainty of $2.303 \times 0.02 = 4.6\ \%$ — a pH reading 0.02 off puts [H⁺] out by nearly 5 %.

A general function $y = f(x_1, x_2, \ldots)$ uses partial derivatives: $u_y^2 = \sum (\partial f/\partial x_i)^2 u_{x_i}^2$ . This is the internationally agreed GUM method; the share of the total variance from each term $(\partial f/\partial x_i)^2 u_{x_i}^2$ makes up the uncertainty budget, which tells you which step to improve first.

Worked example

The uncertainty budget of a titration

0.1000 ± 0.0002 M NaOH titrates 25.00 mL of hydrochloric acid (class A pipette, ±0.03 mL); the burette reads 0.52 mL at the start and 25.02 mL at the end (each reading ±0.02 mL, class A 50 mL burette tolerance ±0.05 mL). Taking the tolerances directly as standard uncertainties, find c(HCl) and its uncertainty.

First-order propagation (GUM) checked by Monte Carlo

Propagation-of-uncertainty calculator

[H⁺] from pH: [H⁺] = 10^(−pH)

Standard uncertainty of the pH reading · 0.020

Input	Value	Standard uncertainty u	Sensitivity	Share of variance
pH reading	4.00	0.02	−2.30 × 10⁻⁴	100 %

Combined standard uncertainty: 4.6 × 10⁻⁶ M
Relative uncertainty: 4.61 %
Expanded uncertainty U (k = 2, about 95 %): 9.2 × 10⁻⁶ M
Report: [H⁺] = 1.00 × 10⁻⁴ ± 9.2 × 10⁻⁶ M (k = 2)

The largest contribution comes from: pH reading

Monte Carlo (20 000 draws)

Monte Carlo (20 000 draws)
Normal curve from first-order propagation

Standard deviation from Monte Carlo: 4.57 × 10⁻⁶ M

Raise the uncertainty of the pH reading from 0.02 to 0.10: the first-order normal curve (orange) and the Monte Carlo histogram start to part — the power of ten turns a symmetric pH error into a lopsided [H⁺] distribution. Then switch back to the titration and compare the three ways of treating tolerances.

Concept check

In the titration example above, which input contributes most to the uncertainty of the result?

The NaOH concentration (0.20 %) The volume delivered by the burette (0.23 %) The pipette (0.12 %) All three equally

Try the calculation

A pH meter reads 3.45 ± 0.03. What is the relative uncertainty of [H⁺], in per cent?

Key takeaways

Sums and differences: absolute uncertainties in quadrature; products and quotients: relative uncertainties in quadrature; powers: multiply the relative uncertainty by the power.
u(pH) is absolute; the relative uncertainty of [H⁺] is 2.303·u(pH).
The uncertainty budget points to the dominant source; start improving with the largest term.
Report the expanded uncertainty U = k·u (usually k = 2) and say how tolerances were treated.

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Data and references

JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), BIPM.
JCGM 101:2008, Evaluation of measurement data — Supplement 1: Propagation of distributions using a Monte Carlo method, BIPM.
S. L. R. Ellison, A. Williams (eds.), EURACHEM/CITAC Guide CG 4: Quantifying Uncertainty in Analytical Measurement, 3rd ed. (2012).
D. C. Harris, Quantitative Chemical Analysis, 9th ed., W. H. Freeman (2016), chapters 2 and 3 (glassware tolerances, propagation of uncertainty).