= Shared-reference floor in averaged isotope stiffness ratios
{title2=$\operatorname{Var}(\overline\ell)=4(s^2/m+s_r^2)$}
A \b[shared-reference floor in averaged isotope stiffness ratios] is the calibration uncertainty that remains when many sample measurements reuse one uncertain reference. In the first-order gain model of <covariance of differential spectral-calibration gains>, let the error in log stiffness-ratio estimate $i$ be $2(e_i-e_r)$. Assume independent sample errors with <variance> $s^2$ and one independent reference error with <variance> $s_r^2$. Averaging $m$ estimates gives
$$
\operatorname{Var}(\overline\ell)=4\left(\frac{s^2}{m}+s_r^2\right).
$$
The sample contribution decreases with replication, while the shared reference contribution is unchanged. This is the <covariance> propagation rule applied to the actual reuse of the reference, rather than treating every ratio as independent.<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 example, choose sample calibration standard uncertainty $s=0.2\%$ and reference uncertainty $s_r=0.1\%$. One ratio has approximately $0.447\%$ standard uncertainty on the log-ratio scale. Sixteen sample measurements reduce it to about $0.224\%$, but the limiting floor is $2s_r=0.200\%$. Reaching within ten percent of that floor requires
$$
2\sqrt{s^2/m+s_r^2}\leq1.1(2s_r),
$$
so $m\geq20$ for these chosen numbers. Replication beyond that point has progressively less effect unless the reference uncertainty is reduced.
If a new independent reference is measured for each sample, the model changes: both error contributions can average down, giving $4(s^2+s_r^2)/m$ under the corresponding independence assumptions. Alternatively, averaging several independent reference measurements reduces their random component, while a systematic component shared by all references still remains. A <reference-versus-sample allocation in repeated isotope-ratio measurements> must distinguish those resources and common influences.
The floor is conditional on the measurement model. Additive spectral calibration errors, reference drift, isotope-composition uncertainty and mode-assignment error are separate contributions. A useful acquisition protocol records which sample spectra share which reference observations; numerical averaging alone cannot remove an uncertainty component reused unchanged in every estimate.
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