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An affine pressure-tap coordinate calibration converts an instrument coordinate $u$ to a physical coordinate through an affine map, $x=a+bu$. The offset $a$ fixes the datum and the nonzero scale $b$ fixes the length conversion. Its adequacy is a question about the actual viewing or measuring geometry, rather than a consequence of having fitted two points.

For two distinct instrument readings $u_0,u_1$ at known physical references $x_0,x_1$,An original imaging example takes the leading edge at $u_0=100$ pixels and $x_0=0$, and the trailing edge at $u_1=500$ pixels and $x_1=0.20$ m. Hence $b=0.0005$ m/pixel and $a=-0.050$ m. A tap at $u=220$ pixels is at $x=0.060$ m, or dimensionless chord position $\xi =0.300$.

If all three features share the same affine map, the dimensionless coordinate is simplyChanging the displayed coordinate to $u^{\prime }=30+2u$ gives readings 230, 1030 and 470 pixels and leaves $\xi =0.300$. This is the exact pressure-tap datum cancellation; it does not assert that the original feature readings are known without uncertainty.

For example, with $D=u_{\mathrm{TE}}-u_{\mathrm{LE}}$, the sensitivity coefficients of $\xi$ with respect to the tap, leading-edge and trailing-edge readings areTheir sum is zero, as required for invariance to a common offset. At the example readings they are $0.0025,-0.00175,-0.00075$ per pixel. Use their full covariance matrix when propagating uncertainty, because independently attaching three origin errors would destroy the cancellation.^[ref]

An affine pressure-tap fiducial check uses an additional independently located mark. At $u=300$ pixels the fitted map predicts $x=0.100$ m. A measured location $0.102$ m leaves a 2 mm discrepancy to investigate. Two fitted reference points necessarily have zero regression residual; that fact cannot validate the map between them. Perspective, surface curvature, optical distortion and reference-location uncertainty can all matter. A proposed calibration records extra marks across the image and compares residual patterns before using the map for surface pressure integration for airfoil lift.

An analytical evaluation function converts an observed analytical response into an estimated analyte amount or concentration by inverting an analytical calibration function. If the forward function is $y=f(c)$ on a specified calibrated interval, the analytical evaluation function is $g=f^{-1}$ on the corresponding response interval. IUPAC defines the evaluation function as this inverse.^[ref]

The direction matters. A forward model describes an expected response at a given concentration; its inverse function is used with the response observed for an unknown sample. For an original dimensionless model $f(c)=c^2$ restricted to $c\ge 0$, the analytical evaluation function is $g(y)=\sqrt{y}$. Response4gives estimate2. Without the nonnegative restriction, the same response would leave the two possibilities $c=\pm 2$, so there would be no single-valued inverse function.

The inverse function rule gives a local sensitivity coefficient:For the example at $y=4$, $g^{\prime }(4)=1/4$, so a response standard uncertainty of0.2would contribute approximately0.05to the concentration standard uncertainty. Uncertainty in the fitted analytical calibration function adds further contributions and may introduce covariances.

Inversion also changes the effect of averaging. Equal-probability responses3and5have mean4, yetThus inverting the mean response and averaging individually inverted responses need not agree. This original calculation illustrates a nonlinear transformation, not a universal correction for analytical bias. The calibrated range, response noise and uncertainty in the forward fit must be specified before interpreting the resulting estimate.

The bearing thermal reaction slope is the local rate of change of a bearing assembly's thermal constraint force with temperature along a specified thermal path. For constant combined axial mechanical compliance $C=C_b+C_s$, relative free-expansion rate $\beta$ and an engaged end stop, the linear model givesIts units are force per temperature, such asN/K. It combines thermal mismatch with elastic restraint; it is not a stiffness measured inN/m. The need to accommodate differential expansion motivates locating/non-locating arrangements in the bearing manufacturer's design guidance.^[ref]

For an original one-stop model with initial relative position ${\delta }_0$, gap $g$, uniform temperature change $\Delta T$ and $\beta >0$,The slope is zero while the gap is open and $\beta /C$ after engagement. At the ideal switching point, the two one-sided slopes differ. The stated slope therefore needs the contact state; it should not be applied throughout the open-gap regime.

Take $\beta =2\mu m/K$ and $C=0.125\mu m/N$. The engaged slope is16N/K. If the bearing end-stop engagement temperature corresponds to a40K increase, increases of60K and70K give forces320N and480N. Conversely, those two force measurements would imply a16N/K slope, and a separately established $\beta$ would imply $C=\beta /(dF/dT)=0.125\mu m/N$. This is a possible bearing thermal stiffness identification measurement, not a reported manufacturer's test.

A bench assessment could measure temperature and axial force at successive equilibrated states after engagement. Measuring both shaft and housing temperatures helps distinguish their actual thermal path from an assumed common temperature. The inferred combined mechanical compliance does not separately identify $C_b$ and $C_s$. Temperature-dependent elasticity, friction, changing contact or temperature gradients can make the observed slope vary, while transient heating can produce a path different from the equilibrated one.

As a local sensitivity coefficient, the engaged slope also propagates a temperature contribution to force standard uncertainty: for a known16N/K slope and $u(T)=0.5\mathrm{K}$, that contribution is8N. Uncertainty in the thermal coefficient, compliance and gap requires its own terms, with covariance included when inputs are correlated.^[ref]

Capacitance uncertainty in heterodyne current recovery is the contribution of uncertain shunt capacitance to the inferred generated beat electric current. In the capacitive current division in heterodyne receivers model, a measured load-current amplitude $A$ is corrected towith load electrical resistance $R$, angular frequency $\omega$ and shunt capacitance $C$. These amplitudes must refer to the stated current-plane convention in heterodyne calibration and use the same peak or root mean square convention.

Differentiation gives the logarithmic sensitivity coefficientThus capacitance contributes little when $x\ll 1$, contributes half its fractional change at $x=1$, and contributes nearly its full fractional change when $x\gg 1$. High-frequency correction increases the required amplitude estimate; it does not make uncertainty in the correction disappear.

Let $y=(\log A,\log \omega ,\log R,\log C{)}^{\mathsf{T}}$ have covariance matrix ${\Sigma }_y$. First-order uncertainty propagation givesThis form retains correlations rather than silently assigning four independent inputs. It is the GUM's sensitivity-and-covariance law applied to this particular measurement model.^[ref]

For an original small-error example, set $x=1$, neglect frequency uncertainty, and assign fractional standard uncertainties of $2\%$ to $A$, $1\%$ to $R$ and $5\%$ to $C$. With independent inputs,If only the $R,C$ errors have correlation coefficient $+0.8$, the result becomes about $3.54\%$; correlation $-0.8$ instead gives about $2.92\%$. The same marginal uncertainties can therefore produce different output uncertainty.

Nonlinearity also shifts the mean. Holding $A,\omega ,R$ fixed and writing $C=C_0+ϵ$ with centered small error, a second-order Taylor series givesAt $x_0=1$ and $u(C)/C_0=0.05$, this is about $0.0313\%$, distinct from the first-order spread. Broad or asymmetric inputs require a fuller distribution model.

A proposed diagnostic characterizes the load and shunt response at the actual bias and relevant beat frequencies, then propagates the joint calibration record. NIST's primary apparatus separately calibrates electrical networks and mismatches as well as controlling optical beating.^[ref] A fitted capacitance is only one part of that response: a single-pole model should not absorb untested cable, transit-time or amplifier effects.

A centripetal speed-gain error is an error in a calculated centripetal correction caused by a multiplicative scale error in the measured angular velocity. For true centripetal magnitude $C=r{\omega }^2$ and indicated speed $\hat{\omega }=g\omega$, the computed magnitude with correct radius isWriting $g=1+ϵ$ gives exact relative error $2ϵ+{ϵ}^2$. The leading relative error is therefore twice the speed-scale error, rather than equal to it.

For an original example, let $r=0.020\mathrm{m}$ and $\omega =100rad/s$, giving $C=200m/{\mathrm{s}}^2$. A1% high speed indication gives $\hat{C}=204.02m/{\mathrm{s}}^2$, an overestimate4.02m/s². Subtracting that value from a radial reading would leave an error of the opposite sign if the true motion were otherwise modeled correctly. The example illustrates sensitivity; it is not a measured calibration error of a particular sensor.

Radius error combines with speed gain. If the indicated radius is $r+\delta r$, thenTo first order, relative error is $\delta r/r+2ϵ$. Apparent cancellation between these terms at one condition does not separately identify either calibration parameter. An independent radius measurement or speed reference helps distinguish their contributions.

An additive speed offset $b$ behaves differently:Measurements at several speeds can distinguish the quadratic scale contribution from the linear and constant offset contributions, subject to the remaining model and measurement errors. Near zero true speed, a relative-error description becomes unsuitable even though the absolute expression remains meaningful.

The same gain factor applies to the first-order ripple correction $2r{\omega }_0\epsilon$ when both mean speed and ripple amplitude share that speed scale. The resulting mismatch enters the centripetal amplitude-error budget, while relative channel timing enters the centripetal phase-error tolerance. A primary rotating MEMS-sensor experiment identifies rotating-frame components and reconstructs a rotation reference, making the physical need for consistent speed and timing information concrete.^[ref]

At a specified operating point, the sensitivity coefficient to speed is $\partial C/\partial \omega =2r\omega$. A speed standard uncertainty therefore contributes approximately $2r|\omega |u(\omega )$ to the correction's standard uncertainty, with radius uncertainty and any covariance included separately.^[ref]

Common-gain cancellation and flow-offset bias in expiratory ratios distinguishes an error that rescales both volumes equally from one that accumulates differently over time. Let true FEV1 be $A$, true FVC be $B$, and their ratio be $R=A/B$. A shared positive volume gain $g$ cancels exactly: $(gA)/(gB)=R$. The result requires the same gain for both quantities, not merely the same instrument name.

A constant error $b$ in measured volumetric flow rate has another effect. If the first-second interval lasts $t_1$ and the full recording lasts $T$, integration givesThe ratio alone identifies neither $b$ nor $g$ separately; only their relative combination enters this model. Nor does the positive duration-dependent error necessarily raise the ratio.

The exact difference isFor small $b$, its sensitivity coefficient at zero offset is $(t_{1}B-TA)/(g{B}^2)$. If average early volumetric flow rate $A/t_1$ exceeds average whole-record volumetric flow rate $B/T$, a positive volumetric flow rate offset lowers the ratio. If the averages agree, that offset leaves the ratio unchanged in this model.

An original example uses $A=2.4$ L, $B=4$ L, $t_1=1$ s and $T=6$ s. The true ratio is 0.60. At $g=1$, the zero-offset sensitivity is $-0.65\mathrm{s}{\mathrm{L}}^{-1}$. An offset $b=0.020\mathrm{L}{\mathrm{s}}^{-1}$ gives a first-order shift of $-0.013$; the exact ratio is $2.42/4.12\approx 0.58738$. A common 5% gain error without the offset would leave 0.60 unchanged.

Independent numerator and denominator gains would instead multiply $R$ by $g_A/g_B$. Clipping, early termination and different timing conventions are additional measurement effects, so gain cancellation is not a general guarantee of ratio accuracy. The GUM requires the full measurement model and shared-input effects to be retained.^[ref] The spirometry statement separately checks the acceptability of the constituent volumes.^[ref]

Differential thermal-expansion uncertainty is the uncertainty in the unequal thermal length changes of two compared objects. In a gauge-block thermal mismatch correction, shared temperature and coefficient information determine which contributions cancel and which remain.

For approximately equal known nominal length $L$, coefficient difference $a={\alpha }_X-{\alpha }_R$ and common temperature offset $\theta =T-T_0$, the differential correction is $d=La\theta$. Its first-order sensitivity coefficients are $\partial d/\partial a=L\theta$ and $\partial d/\partial \theta =La$. HenceLength uncertainty and other inputs require further terms when they are material to the result. General propagation must retain input covariance.^[ref]

The coefficient difference itself hasAssuming two coefficient estimates independent merely because two blocks are present can therefore misstate the uncertainty. Similarly, a common thermometer contribution is not two unrelated temperature errors. NIST's handbook explicitly treats differences in block temperatures and expansion coefficients in comparisons.^[ref]

An original example takes $L=100\mathrm{mm}$, $a=6.5\times 10^{-6}{\mathrm{K}}^{-1}$, $\theta =2\mathrm{K}$, independent $u(a)=0.5\times 10^{-6}{\mathrm{K}}^{-1}$ and $u(\theta )=0.1\mathrm{K}$. The correction is $1.3\mu \mathrm{m}$. The two first-order standard-uncertainty contributions are 0.10 and $0.065\mu \mathrm{m}$, giving combined $u(d)\approx 0.119\mu \mathrm{m}$. These are stipulated inputs, not a certificate.

A useful boundary case exposes second-order thermal mismatch uncertainty. If both central values $a$ and $\theta$ are zero, the linear sensitivities vanish even though both inputs remain uncertain. For independent inputs with those means, their product has exact variance $L^2{u}^2(a)u^2(\theta )$. With $L=100\mathrm{mm}$, $u(a)=10^{-6}{\mathrm{K}}^{-1}$ and $u(\theta )=0.1\mathrm{K}$, this gives 10 nm standard uncertainty, while first-order propagation gives zero. The GUM explicitly requires attention to higher-order terms when nonlinearity matters.^[ref]

Specify the thermal model, shared inputs and approximation used. A small nominal correction is not automatically evidence that its uncertainty is negligible.

Finite-perturbation reference sensitivity measures the change in a comparison when one shared reference value is changed by a finite amount everywhere it occurs. It checks both local sensitivity coefficients and their usefulness over the actual reference-uncertainty range. The reference-sensitivity cancellation check is incomplete if it tests only a derivative at one point.

Let $F(R)$ be the final comparison, with all other independent inputs held fixed. A central finite difference and a second difference areFor sufficiently smooth $F$, these approach $F^{\prime }(R_0)$ and $F^{\prime \prime }(R_0)$, with errors of order $h^2$ when the required derivatives exist. Their units differ: comparison per reference unit and comparison per squared reference unit. The GUM defines sensitivity coefficients as partial derivatives and also discusses determining them by varying an input experimentally.^[ref]

An original model writes $x=R-R_0$ andIt gives $C_h=ϵ+\kappa h^2$ and $Q_h=2\beta$. Comparing $C_h$ with $C_{h/2}$ can expose cubic contamination of the local slope. But if $ϵ=\kappa =0$, every central slope is exactly zero although $F(R_0+h)-F(R_0)=\beta h^2$ is nonzero. A symmetric slope check alone misses an even-in-perturbation response.

For $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $h=0.001\mathrm{mm}$, the one-sided change is 1 nm, while $C_h=0$ and $Q_h=2\mathrm{m}{\mathrm{m}}^{-1}$. If the shared reference error instead has a zero-mean normal distribution with standard deviation $s=h$, the quadratic contribution has mean 1 nm and standard deviation approximately 1.414 nm. These are original model calculations. They show why exact cancellation, zero local slope and negligible finite-range influence are distinct claims.

Use step-size stability of a reference perturbation to select a sensible test range. Extremely small steps can magnify numerical subtraction errors; large steps probe a different region of a nonlinear response. Test several scales near the physical standard uncertainty and inspect one-sided changes as well as central slopes. If the model is stochastic, separate simulation noise from the response to the changed reference.

A computational perturbation changes the assigned shared reference in every dependent calculation while keeping its identity shared. Independently perturbing two uses of the same reference changes the covariance model. A physical experiment must additionally hold other influences stable, because changing a standard or its temperature may change contact or instrument behaviour. NIST separates master-block uncertainty from the comparison process in its uncertainty budget.^[ref] Retain those physical distinctions when interpreting a numerical sensitivity test.

Nonuniqueness of bounded uncertainty laws with a fixed covariance means that specifying finite bounds, expected values and a covariance matrix can still leave several different joint probability distributions. They agree on the variance of every linear combination of the errors, but can disagree on threshold probabilities and nonlinear outputs. A particular construction in bounded errors with prescribed correlation therefore requires a distributional assumption beyond its second moments.

Here is an original example with a common allowed box $[-\sqrt{2},\sqrt{2}{]}^2$. In model A, $X$ and $Y$ are independent fair signs: the four points $(1,1),(1,-1),(-1,1),(-1,-1)$ each have probability $1/4$. In model B, the four points $(\sqrt{2},0),(-\sqrt{2},0),(0,\sqrt{2}),(0,-\sqrt{2})$ each have probability $1/4$. Both are bounded, have zero expected value, and satisfyFor model B, each squared coordinate is two at two of the four points and zero otherwise, proving its unit variance directly. Its zero covariance does not make the coordinates independent random variables: when $X$ is nonzero, $Y$ must vanish. The common box is an allowed deterministic bound; its boundary need not be attained by every model.

The linear output $U=X+Y$ has expected value zero and variance two under either model. Yet under model A it takes $-2,0,2$ with probabilities $1/4,1/2,1/4$, while model B gives only $-\sqrt{2},+\sqrt{2}$ with equal probabilities. Consequently,Even a linear output's threshold risk is not fixed by its variance. For the nonlinear output $F=X^2{Y}^2$, model A gives $F=1$ always and model B gives $F=0$ always. Thus the complete output law, and even a nonlinear output's expected value, can differ.

The law of propagation of uncertainty computes first-order output variance from sensitivity coefficients and input covariance; it does not identify all probabilities. NIST correspondingly requires distributional justification to assign a confidence level to an uncertainty interval.^[ref] The finite counterexample above is an original exact calculation. For guaranteed linear limits, robust linear constraints for bounded covariance models must use the chosen support of a probability distribution, rather than treating a covariance matrix as a support description.

Raman band-area extrapolation estimates the complete area of a spectral band from a measured finite-window area and a model for the fraction outside that window. It uses the Raman captured-area fraction $f>0$:The extra area is inferred rather than observed. Its validity depends on the assumed line shape, width, center, background and separation from neighboring bands. Keeping these assumptions explicit prevents a corrected Raman band-area map from concealing the model dependence of its values.

For an original uncertainty example, let $A_{\mathrm{win}}=10.0$ in arbitrary area units with standard uncertainty0.20, and let $f=0.950$ with standard uncertainty0.010. The extrapolated area is about10.526. For independent inputs, the first-order combined standard uncertainty isin the same area units. Treating $f$ as exact would retain only the first term and understate the uncertainty in this example.

The fitted fraction and window area may be inferred from the same spectrum, so independence needs justification. The general first-order expression retains their covariance:The signs follow from the sensitivity coefficients $\partial A_{\mathrm{full}}/\partial A_{\mathrm{win}}=1/f$ and $\partial A_{\mathrm{full}}/\partial f=-A_{\mathrm{win}}/f^2$. The GUM's correlated-input propagation is the applicable measurement framework.^[ref]

A deterministic bound answers a different question. If $A_{\mathrm{win}}\in [9.8,10.2]$ and $f\in [0.94,0.97]$, positivity givesThis interval follows from allowed ranges and is not automatically a probabilistic confidence interval. Small $f$ makes both sensitivity coefficients large, so extrapolating from a small central portion can be unstable even when that portion is measured precisely.

Model inadequacy needs attention alongside parameter uncertainty. A background mistaken for a band tail, or an unresolved neighboring band, changes what is being extrapolated. Retaining the local Raman spectrum allows the fitted band parameters to be checked instead of relying solely on an intensity summary.^[ref]

A reference-sensitivity cancellation check determines whether a proposed comparison really removes the influence of a shared calibration reference. It checks the measurand, the reference's sensitivity coefficients and the range over which the claimed cancellation applies.

For estimates $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$, a contrast $D=c_{X}X+c_{Y}Y$ has local reference sensitivityIts first-order reference variance is $s_R^2{u}^2(R)$. Checking the contrast directly and checking the full covariance propagation should agree. The GUM defines the sensitivities and retains correlations between inputs.^[ref]

An original linear model is $X=R+d_X$ and $Y=1.02R+d_Y$. For $D=X-Y$, the sensitivity is $-0.02$. Recomputing the unchanged comparison with the assigned reference value increased by $1\mu \mathrm{m}$ changes $D$ by $-0.02\mu \mathrm{m}$. With reference standard uncertainty 20 nm, its contribution to this contrast is 0.4 nm. Treating the two uses of $R$ as independent would miss this suppression.

A useful computational check perturbs one shared input consistently in every place it occurs. Perturbing its occurrences independently tests a different model. Finite-perturbation reference sensitivity also reveals whether a derivative evaluated at one point adequately describes the actual uncertainty range.

Zero first derivative is weaker than exact invariance. For an illustrative nonlinear model, take $X=R+d_X$ and $Y=R+\beta (R-R_0{)}^2+d_Y$. The derivative of $X-Y$ vanishes at $R_0$, but a reference perturbation $\delta R$ leaves a term $-\beta \delta R^2$. If $\delta R$ is zero-mean normal with standard deviation $s$, that term has mean $-\beta s^2$ and standard deviation $\sqrt{2}|\beta |s^2$. The first-order cancellation has not removed the nonlinear contribution.

With stipulated $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $s=0.001\mathrm{mm}$, these are $-1$ nm and about 1.41 nm. These are model coefficients, not measured comparator behaviour.

Finally, verify that the chosen contrast is the quantity the experiment intends to determine. Replacing $X-Y$ by another combination solely to cancel a reference changes the question. Retain calibration lineage and independent checks on comparator response; NIST's physical uncertainty budget distinguishes the master history from comparison-process variability.^[ref]

A sensitivity coefficient describes how an estimated output of a measurement model changes when one input changes slightly. For $y=f(x_1,\dots ,x_n)$, the first-order coefficients, evaluated at the input estimates, areA sensitivity coefficient can have a sign and a unit of measurement: for an output in ohms and an input in volts, its unit is ohms per volt. The GUM also describes estimating sensitivity coefficients experimentally by varying inputs.^[ref]

In a measurement uncertainty budget, propagation of uncertainty uses these coefficients to transform input variances and covariances:Here $u_i$ denotes an input standard uncertainty. The derivative's sign disappears from its independent variance term but still matters in a covariance term.^[ref]

For an original electrical resistance example, $R=V/I$, $V=10\mathrm{V}$ and $I=2\mathrm{A}$ give $R=5\Omega$ andIndependent input standard uncertainties of0.1volt and0.02ampere each contribute0.05ohm on the output scale, giving $u_R=\sqrt{0.05^2+0.05^2}\Omega \approx 0.0707\Omega$. A larger sensitivity coefficient alone does not identify the dominant uncertainty: the input standard uncertainty and any correlations also matter.

The derivative is a local approximation. If the input spread spans a strongly nonlinear part of the measurement model, first-order propagation of uncertainty can be inadequate even when every derivative has been computed correctly.

Tetrazolium contrast uncertainty quantifies uncertainty in the four-reading cell-free tetrazolium interaction contrast, including correlations between its inputs. In the order reagent/test, no-reagent/test, reagent/vehicle and no-reagent/vehicle, writeFor finite input variances and known covariance matrix $\Sigma$, the variance is exactlyThe exactness here follows from linearity of the contrast. JCGM's general uncertainty propagation framework uses sensitivity coefficients and correlated input quantities; this contrast has coefficients $w_i$ independent of the readings.^[ref]

An original example gives each reading an independent residual with standard uncertainty $0.010$ and all four readings the same additive offset with standard uncertainty $0.030$. ThenSince $w^{\mathsf{T}}1=0$, the common offset cancels, giving $u_I=0.020$. Treating those four appearances of one shared offset as independent would instead give $\sqrt{4(0.010^2+0.030^2)}\simeq 0.0632$. The distinction concerns the physical source of variation, not the number of numbers in the subtraction.

A shared multiplicative gain does not cancel. If $A_i=(1+g){\mu }_i+b+e_i$, with independent zero-mean gain error $g$, common offset $b$ and independent residuals $e_i$, thenFor chosen $I_0=0.17$, $u_g=0.05$ and $u_e=0.010$, the combined standard uncertainty is approximately $0.0217$. The gain contribution is a relative uncertainty, whereas the residual contribution is in absorbance units.

The input model should distinguish repeated readings of one preparation from independently prepared controls. A small propagated uncertainty does not test whether controls reproduce the sample's chemistry or whether a nonzero optical contrast identifies a particular formazan. Those are separate questions about the measurement model.

Tie-line endpoint covariance records how the measurement errors of the two measured liquid-liquid tie line endpoints vary together. Using independent mole fraction coordinates, represent the two endpoint estimates by column vectors $\mathbf{a},\mathbf{b}$. Their cross-covariance matrix isAn individual off-diagonal block need not be symmetric; the full joint covariance matrix must be a symmetric positive semidefinite matrix. The uncertainties of the two endpoint vectors alone do not determine their cross-covariance matrix.

Near an exactly balanced, interior tie-line phase-fraction fit, let $\mathbf{g}$ be the row-vector sensitivity coefficient defined in tie-line phase-fraction sensitivity. If the overall composition measurement $\mathbf{y}$ is independent of both endpoints, first-order uncertainty propagation givesThe cross term has a positive coefficient because the two endpoint perturbations enter the phase fraction with the same sign. Correlated endpoints can therefore increase the result. The general rule is to propagate the full joint covariance matrix, rather than assume that correlation always increases or always reduces uncertainty.^[ref]

For a stipulated binary mixture example, use $a=0.20$, $b=0.80$, $z=0.44$, so $\lambda =0.40$. Treat $z$ as exact and give each endpoint standard uncertainty $s=0.006$ with correlation coefficient $\rho$. The lever rule yieldsThis gives $0.002$ for $\rho =-1$, about $0.00721$ for $\rho =0$, and $0.010$ for $\rho =1$. These are first-order standard uncertainties under the stipulated joint error model, not bounds on all possible errors.

A shared calibration error affecting the overall composition as well changes the calculation. Suppose the same additive error $c$ affects $a,b,z$ identically. ThenThe shared offset cancels exactly. More generally, a common affine map $x\mapsto mx+c$ with $m\ne 0$ also cancels in this binary ratio. This common affine calibration cancellation in the lever rule concerns only a truly common contribution; separate sample errors remain. If the shared offset has standard uncertainty $0.006$, incorrectly treating its three appearances as independent would assign a spurious contribution $0.01\sqrt{1+0.6^2+0.4^2}\approx 0.01233$ instead of zero.

In general, stack the inputs as $\mathbf{x}=(\mathbf{y},\mathbf{a},\mathbf{b})$ with joint covariance matrix $C$. Let $J$ be the corresponding Jacobian matrix row from tie-line phase-fraction sensitivity, including its residual correction when needed. Then $u^2(\lambda )\approx JC{J}^T$ automatically includes correlations with the overall composition. The measurement guide specifically identifies common instruments, measurement standards and reference data as sources of such correlations.^[ref]

An instructive measurement design analyzes the two equilibrium phases and the overall mixture using one instrument calibration, then repeats the entire procedure with independent instrument calibrations. The within-run contrasts and between-run changes probe different error contributions. Repeated readings under one unchanged instrument calibration cannot by themselves establish the uncertainty of its shared offset. The measurement model must identify which influences are common before the joint covariance matrix can represent them.

Tie-line phase-fraction sensitivity measures how an inferred phase fraction changes when the overall composition or the measured liquid-liquid tie line endpoints change. A sensitivity coefficient is a local partial derivative of the inference formula, rather than an uncertainty by itself. It becomes an uncertainty contribution only after combination with the input covariance matrix.^[ref]

For one composition coordinate of a binary mixture, write the lever rule aswhere $a,b$ are endpoint mole fractions, $z$ is the overall mole fraction and $\lambda$ is the mole-based amount of phase $\beta$. Differentiating the mass balance givesThus a measurement error in the overall composition and an equal measurement error in the phase-$\alpha$ endpoint have different effects. The conditioning of tie-line phase-fraction inference deteriorates as the endpoint separation becomes small: the same composition measurement error then produces a larger change in the inferred phase fraction.

For an original example, take $a=0.20$, $b=0.80$, $z=0.44$, giving $\lambda =0.40$. Increasing only $a$ by $0.003$ gives a first-order change $\delta \lambda =-0.003$. Direct substitution gives $\lambda =0.237/0.597\approx 0.396985$, a change of $-0.003015$. For the closer endpoints $a=0.20$, $b=0.26$ and balanced composition $z=0.224$, the local sensitivity to the same endpoint error is ten times larger. This is a property of the inference geometry; it does not establish whether either hypothetical pair is in liquid-liquid equilibrium.

The multidimensional tie-line phase-fraction fit has an additional detail away from an exactly balanced composition. On independent mole fraction coordinates, hold a symmetric positive-definite matrix $W$ fixed and definewith distinct endpoints. Define row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$ and $\mathbf{h}={\mathbf{r}}^{T}W/q$. Differentiating the generalized least squares normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$ givesThe residual correction in tie-line sensitivity vanishes when $\mathbf{r}=0$. Otherwise, moving an endpoint also rotates the fitted line relative to the nonzero ternary tie-line consistency residual. Omitting that contribution is an additional approximation. If $W$ itself changes with the measured inputs, its derivative must also enter the calculation.

These formulas describe the unconstrained fit locally. Imposing $0\le \hat{\lambda }\le 1$ introduces boundary effects, and coincident endpoints make this inference undefined. A practical sensitivity check recomputes the fit after small positive and negative perturbations of one measured coordinate, keeping composition closure and the stated weight matrix. Agreement as the perturbation shrinks tests the derivative calculation; it does not validate the underlying equilibrium model.

Unequal gauge-block reference sensitivities occur when two length estimates respond differently to the same uncertain reference value. A reference contribution cancels from their difference only when its coefficients agree. Sharing a calibration source is necessary information, but does not by itself guarantee cancellation.

Let estimates have measurement models $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$. At the stated input values define sensitivity coefficients $a_X=\partial f_X/\partial R$ and $a_Y=\partial f_Y/\partial R$. The reference contribution to $D=X-Y$ is, to first order,This follows either by differentiating the contrast directly or by retaining shared gauge-block reference covariance in the two-output propagation. The general law includes both sensitivities and covariance.^[ref]

An original ratio-transfer model illustrates the difference: $X=q_{X}R+d_X$, $Y=q_{Y}R+d_Y$, with fixed known $q_X,q_Y$. If $R=100\mathrm{mm}$, $u(R)=20\mathrm{nm}$, $q_X=1$ and $q_Y=2$, their length difference retains a 20 nm reference contribution. The reference error is shared, but appears twice as strongly in $Y$. Treating the reference contributions as independent would instead give $\sqrt{20^2+40^2}\approx 44.72\mathrm{nm}$.

The contrast $2X-Y$ cancels this model's reference term because $2q_X-q_Y=0$. It is a different measurand from $X-Y$; cancellation is not permission to replace the quantity the experiment intended to determine. More generally, $c_{X}X+c_{Y}Y$ retains reference variance $(c_X{a}_X+c_Y{a}_Y{)}^2{u}^2(R)$.

Nearly matched sensitivities can provide strong suppression. With $a_X=1$, $a_Y=0.98$ and the same 20 nm reference uncertainty, the residual reference contribution is 0.4 nm. That does not imply the total uncertainty is 0.4 nm: comparison scatter, coefficient uncertainty, temperature and wringing contributions remain.

If the transfer ratios are themselves uncertain, their derivatives introduce terms proportional to the reference length, along with any shared covariance. Determine them from the actual calibration model rather than assigning ideal coefficients solely to obtain cancellation. NIST separates reference-master history from comparator-process variability in its uncertainty calculation.^[ref] A reference-sensitivity cancellation check should accompany a claimed low-uncertainty contrast.