Uncertainty of a gravimetric dilution ($u(f)$)

The uncertainty of a gravimetric dilution describes how uncertain mass measurements propagate into the retained stock fraction $f=m_s/(m_s+m_d)$. Write $M=m_s+m_d$, where $m_s$ is stock mass and $m_d$ is added diluent mass. The first-order propagation of uncertainty isThe negative covariance coefficient follows from the opposite signs of $\partial f/\partial m_s=m_d/M^2$ and $\partial f/\partial m_d=-m_s/M^2$. This is the covariance form of the uncertainty law applied to this specific measurement model.^[ref]

For an original sequential-weighing example, let the empty vessel reading be $A=10.000\mathrm{g}$, the reading after adding stock $B=12.000\mathrm{g}$, and the final reading $C=30.000\mathrm{g}$. Then $m_s=B-A=2.000\mathrm{g}$, $m_d=C-B=18.000\mathrm{g}$ and $f=0.1$. Suppose the three readings have independent standard uncertainties of $0.001\mathrm{g}$, with other effects neglected. Both net masses have variance $2\times 10^{-6}{\mathrm{g}}^2$, butThus $u(f)=6.75\times 10^{-5}$, a relative standard uncertainty of about $0.0675\%$. Omitting this shared-reading covariance would give $6.40\times 10^{-5}$ and underestimate the uncertainty. Equivalently, propagate the three independent readings directly through $f=(B-A)/(C-A)$.

A common multiplicative balance scale error instead cancels exactly from $f$, since multiplying both masses by the same positive number leaves their ratio unchanged. Identify which errors are shared before assigning independent terms. If the desired result is final mass fraction $w_f=w_{s}f$, the stock mass fraction $w_s$ and its measurement uncertainty also belong in the measurement model.