= Recovery-correction uncertainty
\b[Recovery-correction uncertainty] quantifies uncertainty in an amount inferred through a <recovery correction>. If $\widehat q=y/r$ uses measured recovered amount $y$ and recovery estimate $r>0$, both inputs contribute. The uncertainty of the corrected result is not obtained merely by copying the uncertainty attached to $y$.
The first-order sensitivity coefficients are $\partial\widehat q/\partial y=1/r$ and $\partial\widehat q/\partial r=-y/r^2$. Combining them gives
$$
u^2(\widehat q)\approx\frac{u_y^2}{r^2}+\frac{y^2u_r^2}{r^4}-\frac{2y\operatorname{Cov}(y,r)}{r^3}.
$$
This is the linear <propagation of uncertainty> law with correlated inputs.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, independent and correlated input quantities.]{ref} With positive nominal inputs and correlation coefficient $\rho$, the relative form is
$$
\frac{u^2(\widehat q)}{\widehat q^2}\approx
\left(\frac{u_y}{y}\right)^2+\left(\frac{u_r}{r}\right)^2
-2\rho\frac{u_y}{y}\frac{u_r}{r}.
$$
<Recovery-estimate covariance> can matter because an amount and its estimated recovery may share measurements or common response changes.
For original values $y=6\,\mathrm{nmol}$, $r=0.60$, $u_y=0.30\,\mathrm{nmol}$ and $u_r=0.06$, the corrected amount is $10\,\mathrm{nmol}$. Independence gives relative variance $0.05^2+0.10^2=0.0125$ and standard uncertainty about $1.12\,\mathrm{nmol}$. With $\rho=0.8$, the variance becomes $0.0045$ and uncertainty about $0.67\,\mathrm{nmol}$. Correlation cannot be chosen to improve the answer; it must describe the actual joint measurements.
The formula is an approximation near the nominal values. A recovery near zero or with large relative uncertainty can make a quotient strongly asymmetric and poorly represented by a symmetric standard uncertainty. Model inadequacy—such as applying a recovery from an unrepresentative <sample matrix>—is a separate issue that careful numerical propagation alone does not resolve.
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