Errors & Propagation of Uncertainty — Maths for Analytical Chemistry

How uncertainty travels

Every measured result carries a ±

No measurement is exact. A burette can be read to ±0.05 mL; an analytical balance to ±0.0001 g. When you combine several measurements to calculate a final result, each individual uncertainty propagates into the answer by well-defined rules. Understanding those rules lets you report a meaningful value ± uncertainty rather than a bare number with unspecified reliability.

The two fundamental cases are straightforward: for addition and subtraction you add the absolute uncertainties; for multiplication and division you add the relative (fractional) uncertainties. The dominant source of error — the one that contributes most to the final ± — immediately shows where instrumental effort is best spent.

z = x + y : δz = δ x + δy

z = x \cdot y : \frac{δz}{z} = \frac{δ x}{x} + \frac{δy}{y}

Converting between absolute and relative uncertainty is straightforward: $δ x_{rel} = δ x / x$ (express as a fraction or percentage). Once the relative uncertainty of the result is known, multiply by the result itself to recover the absolute form.

Key formulae at a glance

Absolute uncertainty

δ x

— has the same units as

x

. For a burette reading,

δ V = 0.05 mL

Relative uncertainty

δ x / x

— dimensionless; multiply by 100 for a percentage.

Sums / differences

δ (x \pm y) = δ x + δy

Products / quotients

\frac{δ ( x y )}{x y} = \frac{δ x}{x} + \frac{δy}{y}

Power rule

\frac{δ ( x ^{n} )}{x ^{n}} = ∣ n ∣ \frac{δ x}{x}

Reporting

Quote to the same decimal place as the uncertainty:

1.234 \pm 0.003

, not

1.234 \pm 0.00312

When a result depends on several measured quantities, always identify the dominant error term before spending effort improving any single measurement. For a product $z = x^{a} y^{b}$ , the relative uncertainty is:

\frac{δz}{z} = (a \frac{δ x}{x})^{2} + (b \frac{δy}{y})^{2}

(quadrature rule for random errors; simple addition gives the worst-case bound). The power law means that squaring a quantity doubles its relative contribution, while a logarithm propagates as $δ (ln x) = δ x / x$ . For pH specifically:

δ (pH) = δ (- lo g [H^{+}]) = \frac{δ [ H ^{+} ]}{[ H ^{+} ] ln 10} = \frac{δ [ H ^{+} ] / [ H ^{+} ]}{2.303}

A 5 % uncertainty in $[H^{+}]$ translates to only $0.05/2.303 \approx 0.022$ pH units — the logarithm compresses errors.

Log propagation

δ (ln x) = δ x / x

;

δ (lo g_{10} x) = δ x / (x ln 10)

Power propagation

δ (x^{n}) / x^{n} = ∣ n ∣ δ x / x

— squaring doubles relative error.

Quadrature (random errors)

δz / z = (δ x / x)^{2} + (δy / y)^{2}

— always smaller than simple addition.

Dominant term rule

If one relative error is twice another, it contributes four times as much to the quadrature sum. Improve the dominant term first.

Common pitfalls

Adding absolute errors for a product/quotient. For $z = x y$ you must add relative uncertainties ( $δz / z = δ x / x + δy / y$ ), not absolute ones. Adding $δ x + δy$ gives a result with wrong units and a grossly inflated error.
Mixing absolute and relative uncertainties in the same expression. Convert everything to the same form before combining.
Forgetting the power factor. For $z = x^{2}$ , $δz / z = 2 δ x / x$ , not $δ x / x$ . The coefficient matters.
Double-counting a shared measurement. If the same burette reading appears twice (e.g. titre = final − initial, and both readings carry the same instrument error), the errors genuinely add — do not treat them as independent.

Honest results

A density from mass ÷ volume, a concentration from a titration, a rate constant from a fit — each needs its uncertainty propagated so you can quote a meaningful ±. Open the propagation calculator next to see how each input error feeds through.

Propagation calculator

Set x ± δx and y ± δy, choose addition or multiplication, and watch the combined uncertainty and dominant error term respond in real time.

Operation:

\frac{δz}{z} = \frac{δ x}{x} + \frac{δy}{y}

x2.5

δx0.01

δy0.2

result12.500 ± 0.550

relative error of x0.4%

relative error of y4.0%

dominant sourcey

Improve the result by attacking the dominant uncertainty — there's no point measuring x to four decimals if y is only known to 4%.

Worked example 1Propagating through a titration volume difference

A burette is read as $V_{final} = 24.35 \pm 0.05 mL$ and $V_{initial} = 0.15 \pm 0.05 mL$ . Find the titre volume and its absolute uncertainty.

Worked example 2Uncertainty in a molarity calculation

A standard solution is prepared by dissolving $n = 0.02500 \pm 0.00005 mol$ of NaCl in a volumetric flask of capacity $V = 250.0 \pm 0.3 mL$ . Find the concentration and its uncertainty.

Worked examples

End-to-end propagation problems — try each before revealing the solution.

Worked example 3Density from mass and volume

A liquid has mass $m = 12.46 \pm 0.01 g$ and occupies volume $V = 14.8 \pm 0.2 mL$ . Calculate the density and its uncertainty.

Worked example 4Propagating through a serial dilution

A stock solution at $c_{0} = 0.500 \pm 0.002 mol L^{- 1}$ is diluted by transferring $V_{1} = 10.00 \pm 0.02 mL$ into a flask and making up to $V_{2} = 100.0 \pm 0.1 mL$ . Find the final concentration and its uncertainty.

Worked example 5Identifying the dominant error term

A Beer–Lambert absorbance experiment gives $A = ε c l$ with $ε = 1500 \pm 30 M^{- 1} c m^{- 1}$ , $c = 2.00 \times 1 0^{- 4} \pm 3 \times 1 0^{- 6} M$ , and $l = 1.000 \pm 0.002 cm$ . Which term dominates the relative error in $A$ ?

Worked example 6Propagation through a log — pH uncertainty

A pH meter is calibrated so that the absolute uncertainty in $[H^{+}]$ is $δ [H^{+}] = 0.40 \times 1 0^{- 7} mol L^{- 1}$ at a measured value of $[H^{+}] = 2.00 \times 1 0^{- 7} mol L^{- 1}$ (i.e. pH ≈ 6.70). Find the uncertainty in the pH reading.

Worked example 7Multi-step molarity: mass, molar mass, volume

Molarity is calculated as $c = m / (M V)$ where $m$ is mass, $M$ the molar mass, and $V$ the volume. For potassium hydrogen phthalate (KHP, $M = 204.22 g mo l^{- 1}$ ):

$m = 0.5124 \pm 0.0002 g$
$M = 204.22 \pm 0.10 g mo l^{- 1}$ (from atomic masses)
$V = 250.0 \pm 0.2 mL$

Find c and its uncertainty. Which source of error dominates?

ChallengeChallenge — full quadrature propagation to a molarity with dominant-source analysis

A titration determines NaOH concentration from a KHP primary standard. Three measurements are made:

KHP mass: $m = 0.4085 \pm 0.0001 g$
Molar mass of KHP: $M = 204.22 \pm 0.05 g mo l^{- 1}$
Titre volume: $V_{titre} = 20.05 \pm 0.10 mL$

The NaOH molarity is $c_{NaOH} = m / (M V)$ (1:1 stoichiometry). Calculate $c_{NaOH}$ and its absolute uncertainty in quadrature, and state which measurement dominates. (Assume all errors are random and independent.)

Check yourself

Four quick questions on absolute vs relative uncertainty and error propagation.

Question 1 of 4 · Score 0

When you ADD or SUBTRACT two measurements, you combine their:

Choose an answer.