 Recovery-correction uncertainty

Recovery-correction uncertainty quantifies uncertainty in an amount inferred through a recovery correction. If $\hat{q}=y/r$ 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 $\partial \hat{q}/\partial y=1/r$ and $\partial \hat{q}/\partial r=-y/r^2$. Combining them givesThis is the linear propagation of uncertainty law with correlated inputs.^[ref] With positive nominal inputs and correlation coefficient $\rho$, the relative form isRecovery-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.