Uncertainty propagation

Result

Uncertainty propagation

T-108Home CU-308Threads equilibrium · structure

Statement

Combining measurement errors through a calculation.

Why it matters

gravimetric-analysis, chromatography-partition, and calibration-standard-addition all end in a single reported number — a mass, a concentration, a percentage composition — but that number is never exact: every measurement contributes its own uncertainty, and a calculation combining several measured quantities combines their uncertainties too. Error propagation is the systematic method for tracking exactly how each individual measurement's uncertainty carries through a calculation to determine the uncertainty in the final, reported result, without which a stated result of "\(25.4\,\text{g}\)" carries no real information about how much confidence to place in that last digit at all.

It is also the essential prerequisite for the significance testing and confidence-interval methods that follow it in this unit, since deciding whether two measured results genuinely differ, or whether a measured value is consistent with an expected one, is only meaningful once the uncertainty attached to each individual result has first been properly quantified.

Hypotheses

Individual measurement uncertainties are assumed random and statistically independent of one another.Independence is what justifies combining uncertainties in quadrature (via the sum of squares, Proof) rather than by simple addition: independent random errors are as likely to partially cancel as to reinforce, and the quadrature formula correctly captures this partial-cancellation behaviour, whereas simple addition would systematically overestimate the combined uncertainty by assuming every error reinforces every other in the worst possible way. Uncertainties are small enough, relative to the measured quantities themselves, that the calculation's result can be well approximated by a first-order (linear) Taylor expansion around the measured values.This linearisation is what makes the general partial-derivative propagation formula (Proof, Step 3) valid; for unusually large relative uncertainties, higher-order terms in the expansion can become non-negligible, and the simple propagation formulas given here become correspondingly less accurate approximations to the true propagated uncertainty. Only random uncertainty is addressed by this framework; systematic errors (a consistently biased instrument, for instance) are not reduced by any amount of repeated measurement or statistical combination.Because a systematic error shifts every measurement in the same direction by roughly the same amount, it does not average out or partially cancel the way independent random errors do; error propagation formulas correctly combine random uncertainties, but a systematic bias must instead be identified and corrected separately, often via calibration against a known standard (calibration-standard-addition).

Proof

z = x \pm y \ \Rightarrow\ \delta z = \sqrt{(\delta x)^2 + (\delta y)^2}

For addition or subtraction of independently measured quantities, the absolute uncertainties are combined in quadrature (the square root of the sum of squares), not by simple linear addition; this quadrature combination is the direct mathematical consequence of the independence assumption (Hypotheses), since the variance of a sum of independent random variables is the sum of their individual variances. A

z = xy \ \text{or}\ z=x/y \ \Rightarrow\ \frac{\delta z}{|z|} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}

For multiplication or division, it is the relative (fractional) uncertainties, not the absolute uncertainties, that combine in quadrature; this follows from taking the logarithm of the product/quotient relation and applying the same additive, quadrature-combination logic of Step 1 to the resulting sum or difference of logarithms, since \(d(\ln z) = dz/z\) converts a product relationship into an additive one on which Step 1's rule directly applies. B

z = f(x,y,\dots) \ \Rightarrow\ \delta z = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2(\delta x)^2 + \left(\frac{\partial f}{\partial y}\right)^2(\delta y)^2 + \dots}

The fully general propagation formula, of which Steps 1 and 2 are specific special cases, follows directly from the linear Taylor expansion of Hypotheses: each partial derivative measures how sensitively the result \(z\) responds to a small change in that particular input variable, and weighting each input's own uncertainty by its corresponding sensitivity before combining in quadrature correctly propagates the full set of independent input uncertainties through an arbitrary calculation. B

z = x^n \ \Rightarrow\ \frac{\delta z}{|z|} = |n|\,\frac{\delta x}{x}

Applying the general formula of Step 3 to a single power-law relationship gives a simple, frequently used special case: raising a measured quantity to a power \(n\) multiplies its relative uncertainty by \(|n|\), a direct consequence of the derivative \(dz/dx=nx^{n-1}\) evaluated within the general propagation formula; this is a standard source of surprisingly large propagated uncertainty whenever a squared or cubed measured quantity (e.g. a radius appearing as \(r^3\) in a volume calculation) enters a final result. A

Result

\delta z = \sqrt{\textstyle\sum_i \left(\frac{\partial f}{\partial x_i}\right)^2(\delta x_i)^2}

Reading. Uncertainty in a calculated result is found by weighting each independent input's own uncertainty by how sensitively the result depends on that input (its partial derivative), then combining all these weighted contributions in quadrature, correctly accounting for the partial statistical cancellation expected among independent random errors.

Scope. Assumes independent, random uncertainties and a sufficiently linear response of the calculation to small changes in each input (Hypotheses); correlated uncertainties (e.g. the same instrument's calibration error entering two different measured quantities in the same calculation) require a more careful treatment that explicitly accounts for that correlation, beyond this simple, independent-error formula.

Corollaries & converses

gravimetric-analysis's final reported analyte mass or percentage composition, calculated from several individually weighed masses, requires exactly this propagation framework to report a properly justified uncertainty on the final result, rather than an unsupported single number.
confidence-interval and significance-test methods take a properly propagated uncertainty (or an experimentally measured standard deviation serving the same role) as their essential starting input; without first applying error propagation correctly, any subsequent statistical significance claim rests on an unjustified foundation.
calibration-standard-addition, which corrects for certain systematic matrix effects, addresses a genuinely different problem from error propagation (Hypotheses' third assumption): propagation combines random uncertainties correctly, but does nothing to detect or correct a systematic bias, which requires separate calibration methodology entirely.

Fails without

Apply the quadrature-combination formula (Steps 1–3) to two inputs whose errors are actually correlated (e.g. both measured on the same imperfectly calibrated instrument), violating the independence hypothesis: the simple formula, which omits any covariance term, systematically over- or under-estimates the true propagated uncertainty relative to a treatment that correctly accounts for the shared, correlated error source.
Assume a well-propagated random uncertainty also accounts for systematic bias (violating the third hypothesis): the reported result can appear precise (a small propagated \(\delta z\)) while still being substantially inaccurate, since propagation says nothing about a consistent instrumental or methodological bias shifting every measurement in the same direction.

Common errors

Combining uncertainties by simple linear addition (\(\delta z = \delta x + \delta y\)) rather than in quadrature (Step 1), which overestimates the true propagated uncertainty for independent random errors by ignoring their partial statistical cancellation.
Applying the addition/subtraction rule (absolute uncertainties combine, Step 1) to a multiplication or division calculation, where it is instead the relative uncertainties that combine (Step 2), a frequent source of significant numerical error in propagated results.
Forgetting the amplifying effect of a power-law relationship (Step 4): a squared or cubed measured quantity propagates a relative uncertainty two or three times larger than the original measurement's own relative uncertainty, easily overlooked if the power-law rule is not applied explicitly.
Assuming error propagation, correctly applied, can account for or reduce systematic error; it cannot (Hypotheses' third assumption), and a well-propagated random uncertainty says nothing about whether an unaddressed systematic bias is also present in the result.

Discussion

The formal statistical treatment of error propagation traces back to Carl Friedrich Gauss's early nineteenth-century work on the theory of errors and the method of least squares, developed originally in the context of astronomical and geodetic measurement, where combining many individually imperfect observations into a single best estimate, with a properly quantified uncertainty, was already a central practical concern.

A subtlety often glossed over in introductory treatments is that the quadrature-combination rule (Steps 1–3) specifically requires the input uncertainties to be statistically independent; when two measured quantities entering the same calculation share a correlated source of error (for instance, both were measured using the identical, imperfectly calibrated instrument), the simple independent-error formula can meaningfully underestimate or overestimate the true propagated uncertainty, and a more general formula including an explicit covariance term between the correlated inputs is required instead.

Common misconception: that reporting more decimal places in a calculated result automatically makes that result more precise or more trustworthy. A calculated value's genuinely justified precision is set entirely by its properly propagated uncertainty (the Result), not by how many digits a calculator happens to display; reporting digits beyond what the propagated uncertainty actually supports misrepresents the true precision of the underlying measurement.

Worked examples

\text{Mass of analyte} = m_{\text{crucible+ppt}} - m_{\text{crucible}} = (24.716\pm0.002)\,\text{g} - (23.884\pm0.002)\,\text{g}

Both masses are measured independently on the same balance with the identical absolute uncertainty \(0.002\,\text{g}\); applying Step 1's quadrature rule for subtraction: \(\delta z=\sqrt{(0.002)^2+(0.002)^2}=\sqrt{0.000008}\approx0.0028\,\text{g}\), giving a mass of analyte of \(0.832\pm0.003\,\text{g}\) (rounded appropriately to match the propagated uncertainty's own precision). A

m_{\text{analyte}} = 0.832 \pm 0.003\,\text{g}

Reading. Subtracting two independently measured masses combines their individual uncertainties in quadrature, giving a combined uncertainty only modestly larger than either individual measurement's own uncertainty, not their simple sum.

Scope. The identical quadrature-combination procedure applies to any subtraction of two independent measured quantities, regardless of the specific instrument or measurement technique involved.

Problems

A rectangular sample's length is measured as \((5.20\pm0.02)\,\text{cm}\) and its width as \((3.10\pm0.02)\,\text{cm}\). Compute the area and its propagated uncertainty.
Solution
Area \(=5.20\times3.10=16.12\,\text{cm}^2\). Applying Step 2 (multiplication, relative uncertainties combine in quadrature): relative uncertainty in length \(=0.02/5.20=0.00385\); in width \(=0.02/3.10=0.00645\); combined relative uncertainty \(=\sqrt{(0.00385)^2+(0.00645)^2}=\sqrt{0.0000148+0.0000416}=\sqrt{0.0000564}\approx0.00751\). Absolute uncertainty in area \(=0.00751\times16.12\approx0.12\,\text{cm}^2\). Reported result: Area \(=16.12\pm0.12\,\text{cm}^2\).
The volume of a sphere, \(V=\tfrac43\pi r^3\), is calculated from a measured radius \(r=(2.00\pm0.03)\,\text{cm}\). Compute the relative uncertainty in the calculated volume, using Step 4.
Solution
Since \(V\propto r^3\), Step 4 gives the relative uncertainty in volume as \(|n|\) times the relative uncertainty in radius, with \(n=3\): relative uncertainty in \(r\) is \(0.03/2.00=0.015\), so relative uncertainty in \(V\) is \(3\times0.015=0.045\), i.e. \(4.5\%\) — three times larger than the radius's own \(1.5\%\) relative uncertainty, a direct, concrete illustration of the amplifying effect a cubed measured quantity has on propagated uncertainty.
Two students measure the same physical quantity by entirely different methods (different instruments, different underlying physical principles) and report results that agree within their combined propagated uncertainty. A third student argues this agreement proves neither result contains any systematic error. Explain why this argument is flawed, referencing the Hypotheses.
Solution
Error propagation, as developed here, addresses only random uncertainty (Hypotheses' third assumption); agreement between two results within their propagated random uncertainties shows the two measurements are statistically consistent with each other given their respective random errors, but says nothing directly about systematic error, which by definition shifts a result consistently in one direction and would not necessarily be revealed by comparing two measurements against each other alone, particularly if the two methods happened to share a similar, unrecognised systematic bias, or if neither student's propagated uncertainty accounted for a systematic effect present in their own method specifically. Ruling out systematic error genuinely requires independent calibration against a known standard (calibration-standard-addition), not simply agreement between two otherwise independent measurements.