= Uncertainty of a gravimetric dilution
{title2=$u(f)$}
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> is
$$
u^2(f)\simeq\frac{m_d^2u^2(m_s)+m_s^2u^2(m_d)-2m_sm_d\operatorname{Cov}(m_s,m_d)}{M^4}.
$$
The 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>.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{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> $2\times10^{-6}\,\mathrm{g^2}$, but
$$
\operatorname{Cov}(B-A,C-B)=-u^2(B)=-10^{-6}\,\mathrm{g^2}.
$$
Thus $u(f)=6.75\times10^{-5}$, a relative <standard uncertainty> of about $0.0675\%$. Omitting this shared-reading <covariance> would give $6.40\times10^{-5}$ 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> $w_f=w_sf$, the stock <mass fraction> $w_s$ and its <measurement uncertainty> also belong in the <measurement model>.
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