 Nested wringing-repeatability study

A nested wringing-repeatability study separates variation between independently prepared gauge block wringings from readout scatter within one unchanged wring. Repeating a detector reading many times does not sample the contact variability introduced by separating and rejoining the surfaces.

For $m$ wrings with $r$ readings each, an original balanced random effect model iswhere independent wring effects have variance ${\tau }^2$ and independent readout errors have variance ${\sigma }^2$. Assume stable conditions and independence between those levels. One-way analysis of variance givesThus $M{S}_W$ estimates readout variance and $(M{S}_B-M{S}_W)/r$ estimates the between-wring variance in this model. A negative variance estimate reflects sampling variability or model limitations, not a physically negative contact variance.

For a small invented data set, duplicate readings in nm relative to an arbitrary reference are $(-3,-1)$, $(1,3)$ and $(5,7)$. Wring means are $-2,2,6$; $M{S}_W=2$ and $M{S}_B=32$, so the method-of-moments between-wring variance is $15\mathrm{n}{\mathrm{m}}^2$. These are arithmetic examples, not a recommended sample size or a NIST calibration result.

For the overall mean of the balanced study,With stipulated $\tau =6\mathrm{nm}$, $\sigma =2\mathrm{nm}$, five wrings and four readings per wring, its standard deviation is $\sqrt{7.4}\approx 2.72\mathrm{nm}$. Increasing to forty readings on the same five wrings lowers it only to about 2.69 nm. Fifty wrings with four readings each would instead give about 0.86 nm, if the same model and stable preparation remain valid.

A shared reference or persistent preparation contribution does not disappear under either averaging rule. An independent common uncertainty $u_0$ adds $u_0^2$ to the displayed variance. This connects the nested study with the wringing-film covariance model and the gauge-block reference uncertainty floor.

NIST's handbook documents contact-layer dependence on preparation and time, including evolution after thermal equilibration.^[ref] Record rewringing events, timestamps and surface preparation; use appropriate controls for drift and damage. A nested contact-study allocation should balance new wrings against repeat readings, rather than treating every reading as an independent contact experiment.