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An unbalanced sampling-uncertainty design estimates collection and analytical contributions while performing different numbers of analyses on the collected samples. One useful duplicate design collects two samples independently at each target, analyses one twice and the other once. The Eurachem guide presents this three-result arrangement as a less costly alternative to analysing both samples twice.^[ref]

An original variance calculation shows why both components remain identifiable. At target $i$, letHere $j=1,2$ labels independently collected samples; only $j=1$ has analytical replicates $r=1,2$. Assume centered independent random effects, common variance components and the same analytical precision for each result. Define within-target contrastsThe target mean cancels. The first contrast also cancels the sample effect; the second compares independent collections with unequal analytical averaging. Consequently,The coefficient $3/2$ matters: one sample mean contains half the analytical variance, while the other result contains all of it. Applying the balanced-design coefficient would misallocate uncertainty.

For stipulated contrast variances $s_D^2=4$ and $s_Q^2=25$ in squared concentration units, the corresponding method-of-moments estimates areA single analysis of a single collected sample then has modeled standard uncertainty $\sqrt{11+2}\approx 3.61$ in concentration units. These are invented values illustrating the model, not measurements from the guide.

Across eight targets, this design requires sixteen collections and twenty-four analyses, compared with thirty-two analyses for two analytical replicates on each collection. The saving is in analytical runs; equal cost does not imply equal statistical precision. Repeat the nested design across targets to estimate the contrast variances. One triple of results does not reliably estimate the collection variance, and duplicate collection by one unchanged protocol does not reveal a bias shared by every sample. A negative estimated variance component signals finite-data or model issues requiring an appropriate estimation method, rather than a physically negative uncertainty.

For $S=A\cos (q\theta )$ with independent standard uncertainties, linear propagation gives ${\sigma }_S^2={\cos }^2(q\theta ){\sigma }_A^2+[qA\sin (q\theta ){]}^2{\sigma }_{\theta }^2$. Angular uncertainty must be in radians when using this derivative. Covariance adds cross terms for correlated inputs. Strict error bounds instead add absolute first-order contributions; they do not justify Gaussian significance tests.

The uncertainty of a gravimetric dilution describes how uncertain mass measurements propagate into the retained stock fraction $f=m_s/(m_s+m_d)$. Write $M=m_s+m_d$, where $m_s$ is stock mass and $m_d$ is added diluent mass. The first-order propagation of uncertainty isThe negative covariance coefficient follows from the opposite signs of $\partial f/\partial m_s=m_d/M^2$ and $\partial f/\partial m_d=-m_s/M^2$. This is the covariance form of the uncertainty law applied to this specific measurement model.^[ref]

For an original sequential-weighing example, let the empty vessel reading be $A=10.000\mathrm{g}$, the reading after adding stock $B=12.000\mathrm{g}$, and the final reading $C=30.000\mathrm{g}$. Then $m_s=B-A=2.000\mathrm{g}$, $m_d=C-B=18.000\mathrm{g}$ and $f=0.1$. Suppose the three readings have independent standard uncertainties of $0.001\mathrm{g}$, with other effects neglected. Both net masses have variance $2\times 10^{-6}{\mathrm{g}}^2$, butThus $u(f)=6.75\times 10^{-5}$, a relative standard uncertainty of about $0.0675\%$. Omitting this shared-reading covariance would give $6.40\times 10^{-5}$ and underestimate the uncertainty. Equivalently, propagate the three independent readings directly through $f=(B-A)/(C-A)$.

A common multiplicative balance scale error instead cancels exactly from $f$, since multiplying both masses by the same positive number leaves their ratio unchanged. Identify which errors are shared before assigning independent terms. If the desired result is final mass fraction $w_f=w_{s}f$, the stock mass fraction $w_s$ and its measurement uncertainty also belong in the measurement model.

The uncertainty of a three-point baseline integral depends on both the sample uncertainties and their correlations. With a known fixed gate duration $T$, Simpson baseline integration for pulse-energy correction uses the linear combinationIf $C_b$ is the covariance matrix of the baseline estimates, thenFor fixed weights this variance identity is exact; it does not require Gaussian errors. Its use as a complete measurement uncertainty still requires an adequate baseline model. The general covariance propagation rule retains these cross terms.^[ref]

An original equal-precision model assigns each sample standard uncertainty ${\sigma }_b$ and common pairwise correlation coefficient $\rho$, with $-1/2\le \rho \le 1$. Substitution givesFor $T=10\mu \mathrm{s}$ and ${\sigma }_b=0.10\mathrm{mW}$, independent samples give $u(B)=0.707\mathrm{nJ}$. At $\rho =0.60$ this rises to $0.894\mathrm{nJ}$; a perfectly common additive error gives $1.00\mathrm{nJ}$. Three readings under one uncertain offset need not behave like independent measurements.

The final corrected pulse integral has another covariance. If $Y$ is the raw gate integral and $\hat{U}=Y-B$, thenThe covariance between a raw pulse integral and its baseline correction matters when both share a readout offset. If that offset is $c$, it contributes $Tc$ to both $Y$ and $B$ and cancels exactly in their difference. Treating these appearances as independent would invent uncertainty.

Use repeated matched blank acquisitions to estimate the relevant covariance structure, keeping the gate and readout sequence comparable. A contaminated midpoint, an inadequate baseline shape or uncertain timing adds effects outside the fixed-weight sample-noise calculation. Numerical quadrature exactness alone does not establish a sound pulse-energy uncertainty budget.

The vibroscope area estimator converts the fundamental frequency of a tensioned material fibre into a cross-sectional area through a mechanical model. For a uniform flexible strand of length $L$, positive tension $T$, linear mass density $\mu$ and mass density $\rho$, the fixed-end string relation givesThe second equation uses $\mu =\rho A$: the frequency first identifies mass per length under the stated tension, and a chosen mass density then converts it into area. Circular geometry is needed only to infer a diameter from that area.^[ref]

The experimental study by Bo Madsen, Rajnish Kumar and Hans Lilholt used a Favimat+ fibre testing machine. Acoustic waves excited the strand, and a frequency sweep plus a polynomial fit located the resonance peak. A load cell measured tension, while grip separation supplied $L$. The reported frequency range was 500–4000 Hz, with an estimated resolution of 0.1 Hz; the stated force and length readout resolutions were $1\mu \mathrm{N}$ and $0.1\mu \mathrm{m}$. Those readout resolutions do not themselves specify the estimator's full measurement uncertainty.^[ref]

The area and mass density must refer to the same loaded or reference configuration. The companion theory derives volume and area corrections for recovering an untensioned reference area from the tensioned measurement.^[ref]

The vibroscope area uncertainty follows locally by logarithmic differentiation:Treat the small input errors as independent random variables, and let each $u$ denote the corresponding standard uncertainty. First-order uncertainty propagation then givesCorrelated inputs require their covariances as well. Four independent 1% relative standard uncertainties give $\sqrt{10}\%\approx 3.16\%$ for the area. A systematic 1% overestimate of frequency alone instead multiplies the area estimate by $(1.01{)}^{-2}\approx 0.9803$: it produces about a 1.97% underestimate.

Bending resistance can also increase the observed frequency. The companion theory paper models fully clamped supports and derives the corresponding frequency equation and deformation corrections.^[ref] An independently derived, simply supported example isolates the bending-stiffness bias in the vibroscope area estimator. Let $B=EI$ be the filament bending modulus and impose zero deflection and zero bending moment at each endpoint. Substituting $y=\sin (n\pi x/L)\cos (2\pi f_{n}t)$ into the tensioned Euler-Bernoulli beam equation $\mu y_{tt}+B{y}_{xxxx}-T{y}_{xx}=0$ givesWith $r=b/a={\pi }^{2}B/(T{L}^2)$, the uncorrected flexible-string estimate satisfies $\hat{A}/A=1/(1+r)$. At $r=0.10$, it is 9.09% too small. The same model yields two-mode separation of tension and bending stiffness:For $T>0$ and $B\ge 0$, the predicted mode ratio is $f_2/f_1=2\sqrt{(1+4r)/(1+r)}$, ranging from 2 toward 4. These identities belong to the simply supported model; clamped grips require their own boundary conditions. Neither a single resonance peak nor an assumed diameter establishes the appropriate support, density or bending correction.

Vibroscope area uncertainty quantifies uncertainty in an area inferred from a vibration model and its measured inputs. For the flexible-string vibroscope area estimator, length and frequency enter with inverse-square dependence, whereas tension and density enter to the first power. The parent article gives the full first-order propagation formula. A measurement uncertainty budget records the input standard uncertainties, their sensitivities and any correlations.^[ref]

For an original independent-input example, stipulate relative standard uncertainties of $1\%$ in tension, $2\%$ in density, $0.5\%$ in length and $0.1\%$ in frequency. After applying the sensitivity coefficients, their relative area contributions are $1\%,2\%,1\%,0.2\%$. Combining their squares givesFor a stipulated estimate $\hat{A}=80\mu {\mathrm{m}}^2$, this is $u_A\approx 1.97\mu {\mathrm{m}}^2$. Density contributes $4/6.04\approx 66.2\%$ of the combined variance. Improving the frequency readout alone would therefore make little difference to this budget. Reducing the density contribution from $2\%$ to $1\%$ instead gives $\sqrt{3.04}\%\approx 1.74\%$ overall.

These input uncertainties are assigned model values, not the readout resolutions quoted for the Favimat+ fibre testing machine. A resolution increment does not determine a standard uncertainty without a model for the associated error. Correlated inputs require covariance terms; their signed sensitivities can either reinforce or cancel contributions.

The budget also presumes an adequate measurement model. Bending-stiffness bias in the vibroscope area estimator, uncertain supports or deformation conventions may require a correction and additional uncertainty terms. Increasing the numerical precision of the frequency does not repair a wrong physical model.

Wringing-film uncertainty is the uncertainty contributed by differences or changes in the effective contact layers formed during gauge block wringing. Because the defined length of a gauge block includes a wringing-layer convention, the issue is the difference between calibration and use, not automatically an extra layer correction added to every certificate value.

For a stack of $n$ blocks wrung to an auxiliary plane, let $w_i$ denote each calibrated effective layer contribution and $w_j^{\prime }$ the contributions from the $n-1$ internal contacts plus the base contact in use. The residual contact correction isIts standard uncertainty requires a model for these differences. NIST's handbook describes dependence on surface condition, fluid and elapsed time, and notes residual uncertainty when an unknown block's layer is transferred through comparison with a master.^[ref]

An original model for $n$ effective contact differences writes ${\delta }_j=b+{ϵ}_j$, with one shared zero-mean contribution $b$ of variance $u_b^2$ and independent contact contributions ${ϵ}_j$ of variance $u_{ϵ}^2$. ThenThe common term grows quadratically in variance because every contact contains the same contribution. Treating all contacts as independent would omit their covariance.

For three contacts with stipulated $u_b=3\mathrm{nm}$ and $u_{ϵ}=2\mathrm{nm}$, the combined standard uncertainty is $\sqrt{9\cdot 9+3\cdot 4}=\sqrt{93}\approx 9.64\mathrm{nm}$. Assuming independence for three marginal uncertainties $\sqrt{13}\mathrm{nm}$ instead gives $\sqrt{39}\approx 6.24\mathrm{nm}$. The numerical difference comes entirely from the assumed shared effect; these values are illustrative, not a general wringing specification.

Repeatedly separating, preparing and rewringing contacts can characterize a repeatability contribution. Repeated detector readings of an unchanged stack do not sample new wrings. Averaging independent rewrings may reduce a random contact term while leaving a shared calibration or preparation contribution. Film evolution and thermal expansion during handling must also be distinguished from readout noise.

Report surface preparation, reference geometry, elapsed time and the treatment of shared effects. A stack that adheres firmly can still have a length contribution relevant to a small uncertainty target. Wringing-film covariance model assumptions therefore belong with the result, rather than being hidden inside an unspecified repeatability number.