= Recovery-estimate covariance
<Recovery-estimate covariance> describes joint variation between an estimated <extraction recovery> and another input used with it, such as the measured recovered amount. It matters when a <recovery correction> divides one estimate by the other: shared data can make their uncertainties correlated.
For corrected amount $q=y/r$, first-order <uncertainty propagation> gives
$$
u_q^2\approx\frac{u_y^2}{r^2}+\frac{y^2u_r^2}{r^4}-\frac{2y\operatorname{Cov}(y,r)}{r^3}.
$$
The sign of the covariance contribution follows from the opposite signs of the quotient's sensitivity coefficients. A positive <covariance> reduces the propagated variance relative to a calculation assuming independence; a negative <covariance> increases it. Neither sign is a discretionary choice.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated input quantities and covariance terms.]{ref}
An original shared-data example sets $r=y/z$, where independent measured amounts have nominal values $y=6\,\mathrm{nmol}$ and $z=10\,\mathrm{nmol}$, with <standard uncertainties> $u_y=0.30\,\mathrm{nmol}$ and $u_z=0.40\,\mathrm{nmol}$. Then $r=0.60$ and
$$
u_r^2\approx\frac{u_y^2}{z^2}+\frac{y^2u_z^2}{z^4}=0.001476,\qquad
\operatorname{Cov}(y,r)\approx\frac{u_y^2}{z}=0.009\,\mathrm{nmol}.
$$
Substitution gives $u_q^2=0.25+0.41-0.50=0.16\,\mathrm{nmol^2}$, hence $u_q=0.40\,\mathrm{nmol}$. This agrees with the exact algebraic cancellation $y/(y/z)=z$. Ignoring the shared numerator would instead give $\sqrt{0.66}\approx0.81\,\mathrm{nmol}$.
The cancellation also reveals a validation trap: using the same recovered amount both to estimate its recovery and to correct itself does not independently demonstrate accurate recovery of an unknown sample. Build the joint <measurement model> from the underlying observations. Shared calibration standards, preparations or instrument drift may create other covariance structures; matched replicate data or an explicit common-input model can estimate them. Do not assume independence merely because a table lists two separate estimates.
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