 Wringing-allocation pilot uncertainty

Wringing-allocation pilot uncertainty is uncertainty in the noise components used to choose how many gauge block contacts to prepare and how many readings to take per contact. The optimum in nested contact-study allocation depends on estimated between-contact and within-contact variability; a short pilot does not determine their ratio exactly.

For a balanced pilot with $m_0$ independent wrings and $r_0$ readings per wring, use the random-intercept linear mixed modelwith all random effects independent. The within-wring mean square $M{S}_W$ has expected value ${\sigma }^2$; the between-wring mean square $M{S}_B$ has expected value ${\sigma }^2+r_0{\tau }^2$. Thus method-of-moments planning estimates areTruncating a negative estimate is a boundary convention, not proof that true between-wring variance vanishes. The expectations follow by decomposing the independent within-wring residuals and wring means in this stated model.

An original pilot has $r_0=2$, $M{S}_W=36\mathrm{n}{\mathrm{m}}^2$ and $M{S}_B=38\mathrm{n}{\mathrm{m}}^2$. The estimates are $\hat{\sigma }=6$ nm and $\hat{\tau }=1$ nm. If another short pilot gives $M{S}_B=34\mathrm{n}{\mathrm{m}}^2$ with the same $M{S}_W$, truncation instead gives $\hat{\tau }=0$. Substituting that boundary value into the continuous planning rule drives the preferred readings per wring toward infinity, revealing a fragile design decision rather than an experimentally established absence of contact variability.

With independent normal residuals, $M{S}_W$ has relative standard deviation $\sqrt{2/{\nu }_W}$, where ${\nu }_W=m_0(r_0-1)$. Five wrings with two readings each give ${\nu }_W=5$ and approximately 63% relative scatter in this variance estimate. The between-component estimate subtracts two noisy mean squares and can be still less stable. More readings of the same contact do not substitute for observing more independent contacts.

A useful planning sensitivity analysis holds readout standard deviation at a stipulated 6 nm while testing plausible $\tau$ from 1 to 4 nm. With preparation cost 9 minutes and reading cost 1 minute, the continuous optimumranges from 18 to 4.5 readings per wring. Under a 180-minute budget, feasible allocations $(m,r)=(15,3),(12,6),(10,9)$ give mean variances approximately $(0.867,0.583,0.500)\mathrm{n}{\mathrm{m}}^2$ when $\tau =1$ nm, but $(1.867,1.833,2.000)\mathrm{n}{\mathrm{m}}^2$ when $\tau =4$ nm. The best choice among these candidates changes with the pilot assumption; no one set of counts is universally optimal.

Cluster-resampled wringing allocation can retain the pilot's contact grouping when assessing planning sensitivity; treating all readings as independent destroys the nesting. Any resampling still inherits the pilot's limited contact population. Physically, record fresh preparation, waiting time and handling: NIST documents changes in wringing films with conditions and elapsed time.^[ref] Drift and common reference errors require their own model terms rather than being silently absorbed into independent readout noise.