= Repeat-count targets under shared model error
\b[Repeat-count targets under shared model error] choose how many independent readings are worth averaging when part of the uncertainty is common to every repeat. In the local <measurement model> $X_i=S+D+\epsilon_i$, let the shared discrepancy have <variance> $\tau^2$, and the independent repeat noise have <variance> $\sigma^2$. Assuming the two components are independent,
$$
u_n^2=\operatorname{Var}(\overline X)=\tau^2+\frac{\sigma^2}{n}.
$$
The common term produces <covariance> between readings and survives averaging. The GUM's correlated-input rule retains precisely such shared contributions.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13; §5.2.4, common instruments, standards and reference data]{ref}
For a desired total <standard uncertainty> $u_*>\tau$, solve the inequality before choosing the integer repeat count:
$$
n\ge\frac{\sigma^2}{u_*^2-\tau^2},\qquad
n_{\min}=\max\left\{1,\left\lceil\frac{\sigma^2}{u_*^2-\tau^2}\right\rceil\right\}.
$$
If $\sigma>0$, a target equal to $\tau$ cannot be attained with any finite repeat count, and a target below $\tau$ is unattainable within this model. This is the <feasibility threshold for averaging under shared uncertainty>.
In an original <body surface area> example, take $\sigma=0.04\,\mathrm{m^2}$ and $\tau=0.08\,\mathrm{m^2}$. A target $u_*=0.081\,\mathrm{m^2}$ needs ten repeats; $u_*=0.0801\,\mathrm{m^2}$ needs one hundred. A target of $0.079\,\mathrm{m^2}$ instead requires reducing the shared component or changing the model. Merely collecting more readings is insufficient.
Two relative criteria illustrate why the target must be explicit. Requiring repeat-noise <variance> to be at most $\eta\tau^2$ gives $n\ge\sigma^2/(\eta\tau^2)$: at $\eta=0.01$, 25 repeats suffice. Requiring total <standard uncertainty> within 1% of its floor instead gives
$$
n\ge\frac{\sigma^2}{\tau^2[(1.01)^2-1]},
$$
which needs 13 repeats with the same stipulated numbers. A 1% variance criterion and a 1% standard-uncertainty criterion are different requirements.
The calculation assumes independent repeat noise and a justified shared-error model. Repeated observations alone cannot estimate an individual fixed discrepancy; independent reference measurements or a suitable validation population are needed to constrain it.
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