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A caliper measures a dimension by relating the separation of contacting measuring faces to a scale or readout. A sliding-jaw caliper has a fixed jaw and a movable jaw; appropriate faces can measure outside dimensions, inside dimensions or steps, while a depth bar can supply a separate depth measurement. These modes use different contact geometries even when they share one slider.^[ref]

A Vernier scale, dial or electronic readout converts slider position into a measurement indication. It does not remove the need to align the contacts with the desired length. For an outside measurement, the opposed faces must meet the intended surfaces without dirt, excessive force or tilt. An inside measurement uses its designated faces and can require geometry-specific corrections; it is not merely the outside-jaw reading with a different label.

An original small-angle calculation shows why jaw geometry matters. Suppose the displacement reference and contact point are separated by $h=40\mathrm{mm}$, and a moving-jaw tilt produces angle $\theta =0.002\mathrm{rad}$. The resulting offset contribution has magnitude approximatelyIt exceeds a $0.05\mathrm{mm}$ reading increment. These are stipulated dimensions and tilt, not a measured error for a particular caliper. The primary guide gives the same lever-arm mechanism for caliper jaw tilt error and identifies contact force, wear and thermal differences as physical error sources.^[ref]

A useful check closes clean outside faces under consistent light force, then compares readings at several separations with suitable calibrated artefacts such as gauge blocks. Repeating the comparison near different contact positions can reveal a geometry-dependent error that a single zero check misses. Apply a Vernier zero correction when using that readout, while retaining uncertainty from the reference, alignment and repeatability.

The measurement result concerns the specified object dimension under stated conditions, not just slider displacement. Thermal expansion of either the object or instrument, deformation under contact force and the selected faces belong in the measurement model. More displayed digits or a finer scale cannot by themselves establish better measurement accuracy.

Caliper jaw tilt error is a change in the indicated dimension caused by rotation of a caliper's moving jaw relative to its reference geometry. If a contact lies distance $h$ from the displacement reference, a small rotation $\theta$ gives a signed offset contributionThe sign depends on the direction of rotation and contact geometry. The approximation requires small $|\theta |$; it describes this angular contribution rather than every possible measurement error.

Mitutoyo gives a concrete geometry: a jaw slope of $0.01\mathrm{mm}$ over $50\mathrm{mm}$ and a contact depth of $40\mathrm{mm}$ produce a tip error of $0.008\mathrm{mm}$. Excessive contact force and wear of the guide can cause the moving-jaw rotation.^[ref]

An original diagnostic repeats a gauge block comparison at contact depths $h_1=10\mathrm{mm}$ and $h_2=40\mathrm{mm}$. With stipulated constant $\theta =2\times 10^{-4}\mathrm{rad}$ and other effects unchanged, the angular contributions are approximately $2\mu \mathrm{m}$ and $8\mu \mathrm{m}$. Their $6\mu \mathrm{m}$ difference isolates a depth-dependent effect that a single closed-jaw zero reading could miss. Real contacts may also change force or deformation, so this comparison does not uniquely identify rotation without controls.

A Vernier zero correction removes an estimated constant offset; it cannot generally remove an error proportional to contact depth or varying force. Use consistent light contact, a clean reference and a stable temperature, and compare several separations and contact positions. A fine Vernier scale improves reading increments without guaranteeing that these mechanical contributions are small.

This lever-arm mechanism is distinct from merely projecting a fixed length onto a tilted measurement axis: the latter involves $\cos \theta$ and has a quadratic small-angle change, whereas the offset term above is linear in $\theta$. Keeping the geometry in the measurement model prevents applying the wrong angular correction.

Cluster-resampled wringing allocation uses pilot contacts as the resampling units when assessing how uncertain wringing and readout variability affect a planned gauge block study. Readings from one prepared contact stay grouped. Resampling them as unrelated observations would remove the dependence that motivates nested contact-study allocation.

Write the pilot as vectors $B_i=(y_{i1},\dots ,y_{i{r}_0})$ from $m_0$ independently prepared contacts. A cluster bootstrap draws $m_0$ of these vectors with replacement, copies every reading in each selected vector, recomputes the variance-component estimates and then reevaluates candidate allocations. For an original five-contact illustration, drawn indices $(2,2,5,1,4)$ copy contact2 twice and omit contact3. Each draw is a separate bootstrap occurrence; neither duplication nor omission creates a new physical contact. The empirical-with-replacement construction is the general bootstrap procedure, applied here to contact vectors.^[ref]

For a balanced random-intercept linear mixed model, each resample can yield planning estimatesWith preparation/read costs $c_p,c_r$, a finite candidate $(m,r)$ has cost $m(c_p+c_{r}r)$ and modeled mean variance ${\hat{\tau }}^2/m+{\hat{\sigma }}^2/(mr)$. Comparing candidates within each resample reveals how often the preferred allocation changes, instead of treating one pilot ratio as known exactly.

For an original180-minute budget with $c_p=9$ and $c_r=1$ minute, consider $(m,r)=(15,3),(12,6),(10,9)$. With ${\sigma }^2=16\mathrm{n}{\mathrm{m}}^2$ and ${\tau }^2=4\mathrm{n}{\mathrm{m}}^2$, their mean variances are approximately0.6222,0.5556 and0.5778nm², favoring12 contacts with6 readings. Changing only the planning between-contact variance to1nm² gives0.4222,0.3056 and0.2778nm², favoring10 contacts with9 readings. A resampling distribution of planning inputs can therefore matter to the design decision.

The cost of flattening the data is visible analytically. Under the same model, the true mean variance is $({\tau }^2+{\sigma }^2/r)/m$, while an independence calculation over all $mr$ readings gives $({\tau }^2+{\sigma }^2)/(mr)$. Their ratio isusing intraclass correlation $\rho$. For $r=4$ and $\rho =1/2$, the variance ratio is2.5, a design effect that an independence-based analysis misses.

This nonparametric analysis inherits the limited empirical contact distribution and any boundary estimate $\hat{\tau }=0$. More bootstrap draws do not provide more independent wrings. Drift, shared standards or an unbalanced pilot need an appropriate model or resampling design. NIST documents physical wringing films and their dependence on preparation and elapsed time, which is why contact preparation belongs in the experimental record.^[ref]

Gauge-block thermal mismatch is the unequal change in length of a reference and an unknown gauge block caused by different temperatures or thermal expansion coefficients. Bringing both blocks to the same temperature does not cancel the effect when their expansion coefficients differ and that common temperature differs from the reference temperature.

Let the lengths at $T_0=20{}^{\circ }\mathrm{C}$ be $L_{X,0}$ and $L_{R,0}$, and let their linear expansion coefficients be ${\alpha }_X,{\alpha }_R$. To first order,Thus subtract the final two terms from the observed difference to estimate the difference at $T_0$. NIST's handbook explicitly treats both temperature and coefficient differences in gauge-block comparisons.^[ref]

For approximately equal nominal length $L$ at common temperature $T$, the differential thermal contribution becomesIn an original example with $L=100\mathrm{mm}$, ${\alpha }_X=11.5\times 10^{-6}{\mathrm{K}}^{-1}$, ${\alpha }_R=5.0\times 10^{-6}{\mathrm{K}}^{-1}$ and $T-T_0=2\mathrm{K}$, the contribution is $1.3\mu \mathrm{m}$. Identical temperatures have not made the comparator reading a reference-temperature difference. These are stipulated material coefficients, not certificates for particular blocks.

Even equal coefficients leave an effect if the temperatures differ: at $11.5\times 10^{-6}{\mathrm{K}}^{-1}$ and 100 mm, a temperature difference 0.2 K contributes $0.23\mu \mathrm{m}$. Handling and incomplete equilibration can therefore matter before the readout is considered.

Differential thermal-expansion uncertainty must include the temperature measurement, the coefficients and their dependence. A common uncertain temperature is one shared input, not two unrelated thermometer errors. In the common-temperature model its sensitivity is $L({\alpha }_X-{\alpha }_R)$; if the coefficients agree, that particular common-temperature term cancels. Independent block-temperature differences need separate terms. General propagation retains the appropriate covariance.^[ref]

A controlled comparison measures or bounds block temperatures, allows equilibration, and records the coefficient assumptions. Comparing unlike materials without that information can turn a thermal contribution into an apparent calibration error.

Gauge block wringing joins suitably finished measuring faces by bringing them into intimate contact with a controlled sliding or twisting motion. The resulting adhesion lets gauge blocks form a composite length standard or contact an auxiliary plane for interferometry. The contact layer is part of the length convention: a calibrated block length includes one wringing-layer contribution.^[ref]

Adhesion is not explained adequately by saying that air was removed and atmospheric pressure alone holds the blocks together. NIST's handbook describes multiple surface- and fluid-dependent contributions, including surface tension, and warns that film thickness varies with surface condition and time. It does not supply a universal predictive film model.^[ref]

An original bookkeeping model explains the length convention. Let $g_i$ be a block's geometric face separation and let $w_i$ be the effective layer contribution in its calibration, so its defined length is $l_i=g_i+w_i$. If a stack of $n$ blocks is wrung to an auxiliary plane, it has $n-1$ internal contacts and one base contact. Its top height can be writtenThus compatible contacts can make defined lengths additive without treating every interface as an additional correction beyond the certificates. Differences between calibration and use remain a wringing-film uncertainty.

For three stipulated blocks each carrying an effective $10\mathrm{nm}$ calibration contribution, three equivalent contacts contribute the same $30\mathrm{nm}$ already counted in their defined lengths. If one actual contact contributes $25\mathrm{nm}$ instead, the height in this simplified model differs by $15\mathrm{nm}$. These chosen values illustrate the convention; they are not a measured stack or a universal film specification.

Experimentally, NIST describes viewing a block wrung to a transparent optical flat: a uniform contact appearance supports a satisfactory wring, while coloured or bright regions indicate poor contact. Rewringing and checking the optical result tests an interface contribution that merely adding nominal lengths would miss.^[ref] Cleanliness, surface damage and thermal expansion during handling still affect the comparison. A visibly adhering stack is not, by itself, proof of the required dimensional accuracy.

John S. Beers is a NIST dimensional metrology researcher and coauthor, with Ted Doiron, of The Gauge Block Handbook by Ted Doiron and John S. Beers. Its preface explicitly attributes the handbook's single-wavelength interferometry discussion to Beers's work. The official publication record retains the name John S. Beers.^[ref]

A gauge block provides a physical length standard, but its calibrated length depends on a stated convention. The handbook defines length using a wrung auxiliary plane and discusses mechanical comparison as well as interferometry.^[ref] This makes contact geometry and gauge block wringing part of the measurement, rather than treating the artifact as an abstract number printed on a block.

An original comparison example uses a reference and unknown of nominal length $50\mathrm{mm}$. Let the reference's certified deviation from nominal be $+0.120\mu \mathrm{m}$, and let the corrected unknown-minus-reference difference be $-0.035\mu \mathrm{m}$. The inferred unknown deviation is then $+0.085\mu \mathrm{m}$.

For independent reference and comparison standard uncertainties $u_r=0.025\mu \mathrm{m}$ and $u_d=0.015\mu \mathrm{m}$,If five independent repeat comparisons reduce only the second contribution, the result becomesThe common reference contribution does not disappear by repeating the comparison. This gauge-block reference uncertainty floor illustrates why high measurement repeatability alone cannot establish the same numerical measurement uncertainty as a reference-free measurement. Correlated inputs require the corresponding covariance terms.^[ref]

These chosen numbers explain a calibration principle; they are not reported performance values for Beers's apparatus. His documented contribution concerns the measurement methods and their controlled realization, including the link between an optical observation and a defined physical length.

Ted Doiron, identified as Theodore D. Doiron in NIST's official publication record, is a dimensional metrology researcher and coauthor with John S. Beers of The Gauge Block Handbook by Ted Doiron and John S. Beers. The preface uses Ted Doiron and attributes substantial changes to the handbook's measurement-assurance program to the author abbreviated TD.^[ref]

The scientific challenge is to obtain a reproducible length from a material artifact under changing environmental and measurement conditions. Thermal expansion is a particularly clear example. For a small temperature change,where $\alpha$ is the linear expansion coefficient. An original example with $L(T_0)=100\mathrm{mm}$, stipulated $\alpha =11.5\times 10^{-6}{\mathrm{K}}^{-1}$ and $T-T_0=0.1\mathrm{K}$ gives a length increase $0.115\mu \mathrm{m}$. A small temperature difference can therefore matter when comparing precision gauge blocks.

Even a shared temperature does not cancel every thermal effect. For two equal nominal lengths with different expansion coefficients, the first-order differential change isUsing $L=100\mathrm{mm}$, coefficient difference $1\times 10^{-6}{\mathrm{K}}^{-1}$ and common offset $0.2\mathrm{K}$ gives $20\mathrm{nm}$. This gauge-block thermal mismatch survives a simple subtraction even though both blocks have reached the same temperature. A temperature difference between the blocks introduces another contribution.

The handbook treats temperature corrections and their control alongside comparison procedures and measurement assurance.^[ref] A useful experiment compares the corrected result over controlled environmental changes and checks an independent measurement standard, rather than equating a stable display with an accurate physical length. The examples above are illustrative calculations, not an attributed material dataset or a claim about current NIST instrument performance.

Thermal-product dependence beyond covariance is the influence of higher joint moments on uncertainty in a nonlinear thermal expansion correction. Two input models can have identical means, variances and covariance yet predict different uncertainty in a product. A thermal-product input-distribution check must therefore examine how the temperature offset and expansion-coefficient error occur together.

Let $a={\mu }_a+A$ and $\theta ={\mu }_{\theta }+T$, where the centered variables have expected values zero, variances $s_a^2,s_{\theta }^2$ and covariance $c=\mathbb{E}[AT]$. Direct expansion givesThe third joint moments and the mixed fourth moment of thermal inputs are extra information. For centered inputs, only the last two terms remain. A first-order uncertainty propagation rule at zero input means would miss the product entirely; the GUM explicitly requires higher-order treatment when nonlinearity matters.^[ref]

An original counterexample keeps even the marginal shapes fixed. Let $Z$ have a standard normal distribution and let $S$ independently take values $+1$ and $-1$ with equal probabilities. Define $A=s_{a}Z$ and $T=s_{\theta }SZ$. Both have centered normal distributions, the quoted variances and zero covariance. They are not independent: their absolute standardized values are identical.

Their product is $s_a{s}_{\theta }S{Z}^2$. Its expected value is zero, while its variance is $3s_a^2{s}_{\theta }^2$, because $\mathbb{E}[Z^4]=3$. Independent centered normal inputs with the same individual variances instead give product variance $s_a^2{s}_{\theta }^2$. The joint model, rather than a marginal histogram or correlation coefficient alone, determines the difference.

For an illustrative length correction $d=La\theta$ with zero means, $L=100\mathrm{mm}$, $s_a=10^{-6}{\mathrm{K}}^{-1}$ and $s_{\theta }=0.1\mathrm{K}$, these models give standard deviations $\sqrt{3}\times 10\approx 17.32$ nm and 10 nm. This constructed dependence is a mathematical test case, not a proposed universal thermometer model.

In a physical gauge block study, track paired temperature and coefficient information, shared thermometer corrections and any procedure that estimates one input using the other. NIST's thermal correction explicitly combines block length, expansion coefficient and temperature difference.^[ref] Preserve that dependence when generating joint draws. JCGM 101 distinguishes model formulation from propagation and gives a multivariate Gaussian input model as a specific choice, not a consequence of normal marginals alone.^[ref] More simulation trials reduce numerical error conditional on the chosen model; they cannot identify unmeasured joint moments.

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.

Wringing-film uncertainty is the uncertainty contributed by differences or changes in the effective contact layers formed during gauge block wringing. Because the defined length of a gauge block includes a wringing-layer convention, the issue is the difference between calibration and use, not automatically an extra layer correction added to every certificate value.

For a stack of $n$ blocks wrung to an auxiliary plane, let $w_i$ denote each calibrated effective layer contribution and $w_j^{\prime }$ the contributions from the $n-1$ internal contacts plus the base contact in use. The residual contact correction isIts standard uncertainty requires a model for these differences. NIST's handbook describes dependence on surface condition, fluid and elapsed time, and notes residual uncertainty when an unknown block's layer is transferred through comparison with a master.^[ref]

An original model for $n$ effective contact differences writes ${\delta }_j=b+{ϵ}_j$, with one shared zero-mean contribution $b$ of variance $u_b^2$ and independent contact contributions ${ϵ}_j$ of variance $u_{ϵ}^2$. ThenThe common term grows quadratically in variance because every contact contains the same contribution. Treating all contacts as independent would omit their covariance.

For three contacts with stipulated $u_b=3\mathrm{nm}$ and $u_{ϵ}=2\mathrm{nm}$, the combined standard uncertainty is $\sqrt{9\cdot 9+3\cdot 4}=\sqrt{93}\approx 9.64\mathrm{nm}$. Assuming independence for three marginal uncertainties $\sqrt{13}\mathrm{nm}$ instead gives $\sqrt{39}\approx 6.24\mathrm{nm}$. The numerical difference comes entirely from the assumed shared effect; these values are illustrative, not a general wringing specification.

Repeatedly separating, preparing and rewringing contacts can characterize a repeatability contribution. Repeated detector readings of an unchanged stack do not sample new wrings. Averaging independent rewrings may reduce a random contact term while leaving a shared calibration or preparation contribution. Film evolution and thermal expansion during handling must also be distinguished from readout noise.

Report surface preparation, reference geometry, elapsed time and the treatment of shared effects. A stack that adheres firmly can still have a length contribution relevant to a small uncertainty target. Wringing-film covariance model assumptions therefore belong with the result, rather than being hidden inside an unspecified repeatability number.