= Tie-line endpoint covariance
{title2=$C_{ab}$}
\b[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> is
$$
C_{ab}=\mathbb E\bigl[(\mathbf a-\mathbb E\mathbf a)(\mathbf b-\mathbb E\mathbf b)^T\bigr],
\qquad C_{ba}=C_{ab}^T.
$$
An 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> gives
$$
u^2(\lambda)\approx\mathbf g\Bigl[
\Sigma_y+(1-\lambda)^2\Sigma_a+\lambda^2\Sigma_b
+\lambda(1-\lambda)(C_{ab}+C_{ab}^T)
\Bigr]\mathbf g^T.
$$
The 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equations13–16, correlated inputs]{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> yields
$$
u(\lambda)\approx\frac{s}{0.60}\sqrt{0.6^2+0.4^2+2(0.6)(0.4)\rho}
=0.01\sqrt{0.52+0.48\rho}.
$$
This gives \b[$0.002$ 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. Then
$$
\frac{(z+c)-(a+c)}{(b+c)-(a+c)}=\frac{z-a}{b-a}.
$$
The shared offset cancels exactly. More generally, a common <affine map> $x\mapsto mx+c$ with $m\ne0$ 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 $0.01\sqrt{1+0.6^2+0.4^2}\approx0.01233$ 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 $u^2(\lambda)\approx JCJ^T$ automatically includes correlations with the overall composition. The measurement guide specifically identifies common instruments, <measurement standards> and reference data as sources of such correlations.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.4, common influences]{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.
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