 Feasibility threshold for averaging under shared uncertainty

A 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 $X_i=S+D+{ϵ}_i$, suppose the shared term has variance ${\tau }^2$ and independent repeat noise has variance ${\sigma }^2$. ThenFor $\sigma >0$, 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.^[ref]

Approaching the floor has a quantifiable cost. To obtain $u_n\le \tau (1+\delta )$ with $\tau >0$, solveFor 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 $\hat{S}={\sum }_i{w}_i{X}_i$ that preserves the unknown $S$ requires ${\sum }_i{w}_i=1$. The shared discrepancy therefore enters with coefficient one as well, givingThe inequality follows from ${\sum }_i{w}_i^2\ge 1/n$; 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.^[ref]