= Feasibility threshold for averaging under shared uncertainty
A \b[feasibility threshold for averaging under shared uncertainty] is the smallest uncertainty that repetition can approach while a common uncertain contribution remains unchanged. In the <measurement model> $X_i=S+D+\epsilon_i$, suppose the shared term has <variance> $\tau^2$ and independent repeat noise has <variance> $\sigma^2$. Then
$$
u_n=\sqrt{\tau^2+\sigma^2/n}.
$$
For $\sigma>0$, every finite repeat count has $u_n>\tau$, although $u_n$ tends to $\tau$ as $n$ grows. A target below the floor is infeasible within this model, and a target exactly at it needs an infinite count. The GUM's correlated-input propagation explains why a common contribution is retained rather than divided by the repeat count.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13, and §5.2.4, common inputs]{ref}
Approaching the floor has a quantifiable cost. To obtain $u_n\le\tau(1+\delta)$ with $\tau>0$, solve
$$
n\ge\frac{\sigma^2}{\tau^2(2\delta+\delta^2)}.
$$
For small positive $\delta$, the required count grows approximately as $1/\delta$. With stipulated $\sigma=0.04\,\mathrm{m^2}$ and $\tau=0.08\,\mathrm{m^2}$, getting within 1% of the floor needs 13 repeats, within 0.1% needs 125, and within 0.01% needs 1,250. These are planning examples for <body surface area> prediction, not observed clinical performance limits.
<Precision limits of weighting correlated repeats> show why a clever average cannot remove the common term for free. A linear estimate $\widehat S=\sum_i w_iX_i$ that preserves the unknown $S$ requires $\sum_iw_i=1$. The shared discrepancy therefore enters with coefficient one as well, giving
$$
\operatorname{Var}(\widehat S)=\tau^2+\sigma^2\sum_iw_i^2\ge\tau^2+\frac{\sigma^2}{n}.
$$
The inequality follows from $\sum_iw_i^2\ge1/n$; equal weights attain it under the stipulated equal independent repeat-noise variances. Negative weights do not evade the floor while preserving the signal coefficient.
Reaching a lower uncertainty requires additional information or a changed measurement model that reduces the common contribution. Repeating a validation method with genuinely independent reference errors can supply different information, whereas repeating one person's height and <mass> measurements does not redraw that person's model discrepancy. A known bias calls for a correction, not a claim that averaging will make it disappear.<Guide to the expression of uncertainty in measurement by JCGM>[§4.1.2, corrections and incomplete knowledge in a measurement model]{ref}
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