Propagation of Uncertainty

Introduction

An analytical result often depends on several measured inputs: sample mass, flask volume, titrant concentration and delivered titre. Each has uncertainty. Propagation carries those input uncertainties through the calculation to the final result. The familiar square-root-of-squares rules are useful for independent small effects, but correlated inputs and unrecognised bias cannot be ignored merely because a formula is available.

Core explanation

Suppose y = x₁ + x₂ for independent inputs with standard uncertainties u₁ and u₂. Then u(y) ≈ √(u₁² + u₂²). For subtraction, the same quadrature applies to independent random uncertainties: subtracting values does not subtract their uncertainty. For a product or quotient y = x₁x₂/x₃ with independent small relative uncertainties, [u(y)/ y ]² ≈ [u₁/x₁]² + [u₂/x₂]² + [u₃/x₃]². These rules arise from a first-order approximation to how small input changes affect the output.

More generally, for y = f(x₁, …, xₙ), the independent-input approximation is u(y)² ≈ Σ(∂f/∂xᵢ)²u(xᵢ)². Each derivative is a sensitivity coefficient: an input that strongly changes y contributes more. Correlated inputs add covariance terms. For example, two masses measured with the same balance calibration may share a scale error, so treating their calibration uncertainties as independent can overestimate or underestimate the uncertainty of a difference or ratio. A complete model distinguishes shared and independent effects.

Consider a concentration c = m/V. If m and V have relative standard uncertainties of 1.0% and 0.5% independently, relative combined uncertainty is √(1.0² + 0.5²)% ≈ 1.12%. Simply adding gives 1.5%, a conservative bound-like rule but not the usual combined standard uncertainty. The result should also include other relevant contributions, such as extraction recovery, purity and calibration; otherwise the computed number is too optimistic.

Expanded uncertainty U = k u c may be reported for a chosen coverage approach, with k stated. A common approximate k ≈ 2 is not a universal guarantee of exact 95% coverage, especially for few degrees of freedom or non-normal distributions.

Uncertainty evaluation begins by defining the measurand and formula. If a factor is missing entirely, propagation of the listed inputs cannot fix the wrong model. Similarly, a known blank should be corrected with its uncertainty; an unknown contamination source demands investigation rather than an arbitrary large uncertainty label.

Step-by-step reasoning

1. Write the result as a function of measured inputs and constants. 2. List standard uncertainty for each input and identify shared sources. 3. Apply sensitivity coefficients or suitable sum/product approximations. 4. Add covariance terms when inputs are correlated, and include major method effects. 5. Report the combined or expanded uncertainty with unit and interpretation.

Visual explanation

Draw input boxes for mass, volume, recovery and calibration, each with an uncertainty bar, feeding a calculation box marked c = m/V. Arrows have different thicknesses to represent sensitivity. Link two inputs with a curved line for shared balance calibration, showing why the arrows cannot always be treated independently.

Real-world analogy

A recipe's final salt concentration depends on how much salt was added and how much soup was made. Uncertainty in either quantity affects the final concentration, and a shared measuring-cup bias may affect several volume steps together. Simply counting uncertain numbers does not reveal their influence; the calculation's structure does.

Real-world example

A gravimetric chloride result uses precipitate mass and original water volume. A balance certificate, drying variability and sampling-volume calibration contribute. If drying variability is much larger than balance resolution, buying a finer balance will barely improve total uncertainty; improving conditioning has greater value.

Why?

Why do independent standard uncertainties combine in quadrature rather than by direct addition? Positive and negative random fluctuations can partly offset, and variance contributions add under independence. The square root returns the result to the original unit. Direct addition can describe a worst-case bound under different assumptions, but not the same statistical quantity.

Common misconception

“The last digit of the balance reading is the total result uncertainty” ignores chemical recovery, volume and calibration. Another error is subtracting uncertainties when measured quantities are subtracted. Even if two readings nearly cancel, their uncertainties can make the difference relatively uncertain.

Worked example

A result is c = m/V with m = 10.00 ± 0.10 mg and V = 100.0 ± 0.5 mL, where both ± values are independent standard uncertainties. Nominal c = 0.1000 mg mL⁻¹. Relative uncertainties are 0.10/10.00 = 1.0% and 0.5/100.0 = 0.5%. Combined relative uncertainty ≈ √(0.010² + 0.005²) = 0.01118, so u(c) ≈ 0.00112 mg mL⁻¹. A sensible standard-uncertainty report is about (0.1000 ± 0.0011) mg mL⁻¹, excluding other method effects.

Quick check

1. If two independent masses are subtracted, should their standard uncertainties be subtracted too? Answer: No. Their variances add, so the standard uncertainty of the difference is approximately the square root of the sum of their squared standard uncertainties.

Exam focus

Distinguish absolute from relative uncertainty rules. Use relative contributions for products and quotients, absolute for sums and differences under independence. State the correlation assumption and identify important chemical sources outside instrument specifications. Give k when reporting expanded uncertainty.

Advanced insight

Monte Carlo propagation can be useful when input distributions are non-normal, functions strongly nonlinear or uncertainties large enough that first-order derivatives are inadequate. It samples plausible inputs through the measurement equation and examines the output distribution. This is still only as trustworthy as the input distributions and model chosen.

Summary

Uncertainty propagation turns input uncertainties into a result uncertainty through the measurement equation. Independent small contributions often combine by quadrature, while shared effects require covariance. The numerical calculation must include relevant chemistry, calibration and sampling assumptions, not just visible instrument digits.

Practice questions

1. Two independent volumes have standard uncertainties 0.10 and 0.20 mL. What is the standard uncertainty of their sum? Answer: √(0.10² + 0.20²) ≈ 0.224 mL.

2. Why can a tiny difference between two large readings have large relative uncertainty? Answer: Each reading retains absolute uncertainty, which combines even though their nominal values nearly cancel. Dividing that uncertainty by a small difference yields a large relative fraction.

3. What extra information is needed if mass and volume errors are correlated? Answer: Their covariance or correlation structure is needed to add the appropriate cross term; an independent-input quadrature may be wrong.