Reviving the inter-laboratory comparison measurement results

Abstract

Key comparison measurements serve as an ultimate tool of quality assurance of results. Whenever the inter-comparison results indicate inconsistency, the participating laboratory needs to take the corrective actions. Practically, the systematic errors involved in the measuring system confines the achievable accuracy. Therefore, the corrective action involves either empirically determine the influences afresh or intuitively reassigns these error values. Alternatively, an analytical method based on inter laboratory comparison results is proposed. The novelty of the proposal is considering task-specific errors in the model that is used for the analysis of interlaboratory comparison results. Without accounting the uncertainties of task-specific errors, the analysis grows complicated and even sometimes it is not feasible. To supplement the proposed method, task-specific errors due to the imperfect geometry of ring gauge, practical inability in implementing the measurement, and unattended environmental influences are explored. The proposed method is demonstrated using some internal diameter key comparison data. The systematic errors responsible for the outlier in the measurement comparison are clearly distinguished.

Keywords

Key comparison diameter measurement consistency dimensional metrology task-specific deviations corrective action

Introduction

The mutual recognition arrangement encourages free trade between countries by removing technical barriers. Mutual recognition establishes the basis for a “product tested once and accepted everywhere” principle (CIPM MRA documents:policy @ https://www.bipm.org). The international quality infrastructure aims to assure comparability of measurement results. Consensually, the designated laboratory called national metrology institute (NMI) of different economies participates in the interlaboratory measurement comparisons to demonstrate their equivalence of routine calibration services offered by to clients (CIPM MRA documents: guidance on comparison @ https://www.bipm.org). Participant NMIs measure the same artifact using the same apparatus and methods as routinely applied to client gauges. The measurement values reported by participants are compared with the agreed /assigned reference value ( $x_{KCRV}$ ) by considering reported measurement uncertainties and the uncertainty of the assigned value. The protocol of key comparison suggests the basic procedure of measurement. Usually, the statistical analysis adopted for the interlaboratory comparison is independent of the specific measurement process. The performance of the participants in an interlaboratory comparison is evaluated by analyzing the measurement data collected from all the participants applying an agreed statistical approach (Kacker et al., 2004). The criterion normalized error |E_n| < 1 provides objective evidence that the estimate of uncertainty and results are consistent. The statistical interpretation of key comparison is given in Kacker et al. (2003). Thus, the key comparisons complement the calibration and measurement capabilities (CMCs) claimed by the NMIs. Whenever an inconsistency (i.e. |E_n| > 1) is detected, the participating laboratory should take corrective actions. An inconsistency indicates a deficit of knowledge of systematic influences on the measurement results of one or several laboratories.

On the contrary, the analytical model required for the corrective action in response to the key comparison outcome has to be specific to the given measurement (Dobaczewski et al., 2014). Rukhin (2009) expressed that accounting the exact nature of all the participant laboratory measurement techniques needed in the formulation of a finite population sampling is difficult. Conceptually, the corrective action belongs to the jurisdiction of the participant. The work carried out (Chunovkina, 2008) for improving the estimation of bias of intercomparison assume that the probability density function (PDF) that encodes the size of its bias is available and results are consistent. Otherwise, the uncertainty about their biases cannot be reduced. Moreover, the consistency of data relies on the results provided by the participating laboratories including their quoted uncertainties. Often, we do not have the knowledge of PDF of each participant. Therefore, the benefit of intercomparison is not optimally extended for the corrective action using only statistical methods.

In this work, a model is devised to bridge the measurement model and the interlaboratory comparison analytical model to ease the above difficulties. Grabe (1987) also suggests the metrological statistics that account for the systematic bias in the model used for the analysis of intercomparison. His result advocates that the error budget can easily be established and conserved. Then, one can assess the reasons for a given uncertainty. It was also pointed out that these simplifications are intimately connected with a modest increase in measurement uncertainties. To fill these gaps, the concept of the task-specific error is applied here. The authors explored a common minimum task-specific uncertainty due to misalignment, form deviations of artifacts and constant environmental influence involved in the ring gauge measurement. After the evaluation of the degree of equivalence, a common minimum uncertainty is appended to all the results reported by participants and their respective systematic uncertainty components are adjusted accordingly. The authors also investigated deformation of a steel ring gauge of size 4 mm due to atmospheric pressure and gravity using finite element analysis. The proposed method is described using some artificial interlaboratory comparison data. Finally, the revised E_n values are calculated after readjusting the systematic errors. The revised E_n values highlighted nearly outliers also. During the redistribution of bias of participant laboratories, the misinterpreted errors and underestimated uncertainty components are revealed.

Proposed approach

The proposed model represents the measurement of internal diameter. The parameter (diameter of ring gauge) is expressed in terms of form imperfection ( $Δ Φ$ ) of artifact under test, probing deviation $Δ p$ and an effective environmental influence ( $Δ E$ ) in equation (1).

\emptyset = Φ + Δ Φ + Δ p + δ p + Δ E

(1)

The deviation $Δ p$ includes the form deviations of probe and imperfections in the travel of probe in the given plane of measurement, following the assigned direction for probing. It is noted that these deviations are exact in nature and they are specific to the artifact and probing strategy. Considering only such influences of measurement process leads to the estimate of diameter as in equation (2)

E [\emptyset] = E [Φ + Δ Φ] + E [Δ p + δ p] + E [Δ E]

(2)

\emptyset (θ_{a}) = Φ + Δ Φ (θ_{a}) + Δ p (θ_{a}) + Δ E + e_{i}

The probing direction $θ_{a}$ is being dictated by two lines made on the surface of ring gauge. Therefore, the common task-specific errors are applicable to all the laboratories who participate in the intercomparison. However, the estimate of these biases will vary for each laboratory during the interlaboratory measurement. Here $δ p = N (0, σ_{p})$ is randomness in probing within the laboratory. Further, the statistical representation (equation (3)) can be deduced from this measurement model. It is a generalized form that expresses the estimates of diameter reported by participant laboratories in intercomparison measurements. Let, the result $x_{i}$ reported by the i^th participant is equaled to sum of true value $μ = Φ$ (i.e. diameter of gauge), the innate biases $x_{μ i} = Δ Φ (θ_{a}) + Δ p (θ_{a}) + Δ E$ , and the biases ${\overset{\cdot}{B}}_{i}$ specific to the participant’s premises (Dobaczewski et al., 2014)

x_{i} = μ + x_{μ i} + {\overset{\cdot}{B}}_{i} + e_{i}

(3)

During inter laboratory measurements, the estimates of the task-specific error $x_{μ i}$ due to (i) geometrical imperfections of the artifact (ii) misalignments (iii) boundary coordinate detection anomalies vary for each laboratory. $e_{i}$ is finite for individual result $x_{i}$ , but the randomness over the intercomparison data will be $e = N (0, σ_{e})$ . Let us proceed with the conventional method of analysis of degrees of equivalence of each laboratory after eliminating the inconsistent set of reported results. Consider the conventional heteroscedastic one-way random effects model to determine the offset $d_{i}$ of individual participant (Wimmer, 2008). The evaluation of $x_{KCRV}$ is popularly discussed in the literature (Acko et al., 2014).

d_{i} = (x_{i} - x_{KCRV}) = \sum_{j = 1}^{m} b_{ij} + e_{i}

(4)

To reduce the offset $d_{i}$ for corrective action analysis, the individual (i^th participant) needs to reassess the incidental systematic errors. Without losing generality, the i^th participant can ramify the offset d_i into m number of systematic error components. The reported systematic uncertainties components ${u (B_{i 1}), u (B_{i 2}), \dots u (B_{ij}) . . u (B_{im})}$ of individual lab (i^th participant) can be bifurcated into systematic errors ${b_{i 1}, b_{i 2}, \dots b_{ij} . . b_{im}}$ and respective associated uncertainty components ${u (b_{i 1}), u (b_{i 2}), \dots u (b_{ij}) . . u (b_{im})}$ according to equation (5) [JCGM 100:2008]. The bifurcation is performed assuming all systematic errors behave as a rectangular probability distribution. Otherwise, the bifurcation can be done applying appropriate probability distribution $pdf (b_{ij})$ of each systematic error estimate $b_{ij}$

u^{2} (B_{ij}) = u^{2} (b_{ij}) + {b_{ij}}^{2} / 3

(5a)

\begin{matrix} u^{2} (B_{ij}) = u^{2} (b_{ij}) + E ({b_{ij}}^{2}) - {(E (b_{ij}))}^{2} \\ u^{2} (B_{ij}) = u^{2} (b_{ij}) + \int {b_{ij}}^{2} pdf (b_{ij}) \partial b_{ij} - {(\int b_{ij} pdf (b_{ij}) \partial b_{ij})}^{2} \end{matrix}

(5b)

Chunovkina (2008) has discussed the cases that reduce the uncertainty $u (b_{ij})$ associated with the resulting estimates $b_{ij}$ of the biases. The uncertainty $u (b_{ij})$ can be significantly smaller than those associated with the estimates of the single laboratories $u (B_{ij})$ . Thus, the laboratories can improve the degree of equivalence using the state of knowledge of about their biases. With this inspiration, the proposed approach suggests the laboratory to intuitively anticipate its m systematic errors. Instead of arbitrarily assigning a value to $b_{ij}$ , the magnitude should be chosen within the plausible range. The reassigned j^th systematic error ( $b_{ij}$ ) of i^th participant can be used to revise their respective offset ( $d_{i}$ ). The revised offset ${\hat{d}}_{i}$ can be attained by reducing the sum of systematic errors from $x_{i} - x_{KCRV}$ as given in equation (6). Further, the asymmetry of distribution of estimate of systematic error infers definite, it can be separated by splitting, that is, simply subtracted from the bias $\sum_{j = 1}^{m} b_{ij}$ (Grabe, 1987)

{\hat{d}}_{i} = x_{i} - e_{i} - x_{KCRV} - \sum_{j = 1}^{m} b_{ij}

(6)

According to Kacker et al. (2007), the revised error is parsed into lab specific bias and task-specific bias $(x_{KCRV} - μ)$ as in equation 7. The task-specific bias of individual laboratory will be equals to $x_{μ i} = x_{μ} + δ x_{μ i}$ . Usually, the mean value $x_{KCRV}$ evaluated during intercomparison represents the true value plus the task-specific bias ( $x_{μ}$ )

{\hat{d}}_{i} = (x_{i} - μ - \sum_{j = 1}^{m} b_{ij} - e_{i}) - (x_{KCRV} - μ)

(7)

{\hat{d}}_{i} = (x_{i} - μ - \sum_{j = 1}^{m} b_{ij} - e_{i}) - x_{μ i}

The variation $δ x_{μ i} = N (0, σ_{x_{μ}})$ is the randomness involved in intercomparison in estimating task specific errors. The common minimum systematic error $x_{μ}$ can be determined experimentally by individual laboratory. In the weighted mean analysis, the $x_{KCRV}$ will be influenced by the lowest uncertainty reported by the participant laboratories. Therefore, the offset $d_{i}$ of the given i^th lab depends on the reported competitive results. The uncertainty of the revised offset ${\hat{d}}_{i}$ of the i^th participant lab can be analytically evaluated as described in Rukhin (2009). As a matter of fact, the proposed approach does not alter the mathematical framework of the GUM; accordingly, the uncertainty of refined offset can be evaluated based on equation (8)

{\hat{d}}_{i} = x_{i} - (μ + δ x_{μ} + e_{i}) - \sum_{j = 1}^{m} b_{ij} - x_{μ}

(8)

Obviously, the uncertainty $u_{int}$ of weighted mean $x_{KCRV}$ has a correlation with reported uncertainty $u_{i}$ of i^th participant laboratory. The uncertainty $u ({\hat{d}}_{i})$ given in equation (9) comprises of a common minimum task specific error uncertainty $u (x_{μ})$ , and square sum of uncertainties of compensated biases used in the above analysis

u^{2} ({\hat{d}}_{i}) = u_{i}^{2} - u_{int}^{2} - \sum_{j = 1}^{j = m} u^{2} (b_{ij}) + u^{2} (x_{μ})

(9)

Since the proposed analysis should be specific to the individual laboratory; the uncertainty of offset ${\hat{d}}_{i}$ should be independent of the uncertainty of key reference value $(x_{KCRV})$ . The common minimum uncertainty should be task specific to the given strategy of measurement and gage under test (Lira, 2007). One needs to investigate the intricacies of the principle of measurement, common environmental influences, and form deviation to determine the task-specific uncertainty (IS 15393-5 (2003)). The reported uncertainty $u_{i}^{2}$ also imbibed the randomness due to task-specific errors. Now, the deviation ( ${\hat{d}}_{i}$ ) became independent of dominating uncertainty reported by the few laboratories (Phillips et al., 1998). The normalized error $E_{ni_adj}$ of the analysis can be evaluated using equation (10) to validate the reestablished the error values of the measurement setup (Kacker et al., 2010).

E_{ni_adj} = \frac{{\hat{d}}_{i}}{2 u ({\hat{d}}_{i})}

(10)

Whenever a participant laboratory has mistaken the systematic errors and/or their associated uncertainty components, it would not be possible to redistribute systematic errors as explained above. The algorithm of the proposed method is summarized here:

Assign systematic errors and their associated uncertainty according to equation (5).

Investigate the given measurement to determine the task-specific error components.

Evaluate the common minimum uncertainty due to task-specific errors.

Reduce the reassigned errors from the bias $d_{i}$ .

Evaluate the uncertainty of refined offset ${\hat{d}}_{i}$ using equation (6).

Evaluate normalized error after readjustment of biases using equation (10).

Analysis of corrective action

Different NMIs have different machines for internal diameter measurement. Even then, the basic principle of internal diameter measurement remains the same for all. Majority of NMIs use single axis length measuring machine (LMM) as follows. The LMM holds the ring gauge on its tip-tilt table. A carriage equipped with a single (two) spherical anvil(s) moves on the guide-ways (Lee et al., 2013). The detector on the carrier reads a reference linear scale to announce displacement of anvils inside the cylinder. Initially, the internal diameter of a master ring gauge is probed. A probing correction is calculated by subtracting the measured displacement inside the hole from the diameter value provided in the calibration certificate of the master ring gauge. The probing correction accounts the probe tip diameter, indentation of a probe and other biases due to imperfections in the probing system. Then, the probe is displaced inside the ring gauge under test. The measured internal diameter will be the sum of displacement of the probe inside the ring gauge and the probing correction. Hence, the uncertainty component due to probing correction (Pahk and Kim, 1995) hikes the overall uncertainty of measurement.

NMIs publish their CMCs on the web portal of the International Bureau of Weights and Measures (BIPM). Ad infinitum, metrologist works to improve CMC by reducing the respective systematic errors (Neugebauer et al., 1997). Experimental quantification of the error components is tedious and impracticable. Neugebauer (1998) comprehensively narrated the various uncertainty components involved in internal diameter measurement. Such knowledge of each influence is required to estimate the undisputable uncertainty of measurement.

Figure 1 shows the strategy to achieve desirable uncertainty of measurement. Scientists can validate error component values in their measuring machine based on their performance in the key comparison. Further, designers can refine the measuring system to reduce the systematic errors that are conceived in the key comparison. The flowchart describes the process of improving the measurement system. An intercomparison data of internal diameter inspired from some key comparison are taken to discuss the proposed method [CCL-K4 Key Comparison, 2000]. The results reported by participants are analyzed by the weighted mean method. The uncertainty components reported by each participant are also given.

The key comparison data is analyzed to determine the weighted mean method ( $x_{KCRV}$ ). The mean equals to 78.6 nm with an associated uncertainty ( $u_{int}$ ) 12 nm is obtained. After rejecting the inconsistent results, the E_n values of thirteen laboratories are taken into Table 1. Often, some participant passes the intercomparison despite inconsistent results, merely due to the larger value of associated uncertainty. The proposed method is revealed in such cases here. From the analysis, it was found that the result of Lab 5 is an outlier with E_n = 1.02 and results of Lab 2, 11, 13 are about to outlier. The proposed analysis is applied to the reported results of lab 5, 11. The uncertainty component due to form deviations of ring gauge of lab 5 (11) is 31nm (17nm), respectively. According to equation (5), the magnitude of form error of lab 5 (11) can be reassigned as 22nm (-12nm), respectively. Their associated uncertainty is evaluated as equation (11).

{(31)}^{2} = {(28)}^{2} + {(- 22)}^{2} / 3 Lab 5

(11a)

{(17)}^{2} = {(16)}^{2} + {(- 12)}^{2} / 3 Lab 11

(11b)

Table 1.

Data take from some key comparison (All numerical are in nm).

i	$x_{i}$	$u_{i}$	$u (B_{ij})$ uncertainty component of different categories
Lab	Result	Associateduncertainty	Alignment	Probing correction	Contact Deformation	Environment	Tempeffects	Artifactform	CTE	Repeatability
1	60	67	12.3	36	5	3.5	4.4	54.3	5	6
2	0	48	32.2	26	0.4	5	3	9	7	21
3	78	30	11.8	20.2	3.2	2.4	14.8	7.5	4	5.1
4	80	18	0.8	15.3	2.5	1.5	3.1	8.4	0	1.1
5	156	40	6.2	21.3	2	4	5.7	31	0.2	11.5
6	68	51	12.5	45	2.5	4	12.5	5	4	14
7	51	50	4	30	2	3.1	2.5	4	2.5	42
8	20	130	109	14.6	20	3	13	18	4	65
9	30	90	7	38.1	74.8	3	32.7	14.6	6.4	6
10	30	160	155	26	17	4	22	18	4	19
11	−100	100	60	14.8	47	3	50	17	5.5	34.7
12	140	130	50	25	24	3	32.9	18	4.3	6
13	210	70	50	25	17	3	32.9	18	4.3	6

Similarly, all the error components and their associated uncertainties can be split for all the laboratories. We (as a third party) cannot have a precise value of systematic error component of each participant. Therefore, we briefly estimated the magnitude of errors. The statistics of systematic errors reported by different NMIs during key comparisons (Banreti, 2010; CCL-K4 Key Comparison, 2000; Chin et al., 2014, Picotto, 2010) are compiled in Table 2. Usually, these perturbations giving rise to unknown offset from the true values have been formally randomized, that is, they are treated as if they were of random origin. Biased estimators thus again become fictitiously unbiased.

Table 2.

Typical systematic errors (nm) reported in some intercomparisons.

Systematic error	Minimum value	Maximum value	Average value	Standard deviation
Artifact form	4	54	17	18
Probe calibration	50	100	29	29
Cosine error	2	58	25	27
Abbe’s error	10	84	35	20

The individual laboratory can adjust the systematic error components following some thumb rules as follows. Theoretically, the form deviations of artifacts cause uncertainty equally to all the laboratories participants of key comparison. One can argue that the sign of form deviations will be same for all the labs. But, the individual labs might have assumed different values. The anticipated error components of the chosen inter-comparison results are listed in Table 3.

Table 3.

Adjusted systematic error components (nm).

Lab	Alignment	Probingcorrection	Contact Deformation	Environment	Tempeffects	Artifactform	CTE	$\sum_{j = 1}^{m} b_{ij}$
1	−9	25	4	2	3	−38	−7	−20
2	−23	−18	0	−4	−2	−6	−10	−63
3	−8	14	2	2	10	−5	−6	9
4	−1	10	5	1	2	−6	−10	2
5	−4	15	1	3	4	−22	0	−3
6	−9	−32	2	3	9	−4	−6	−36
7	−3	−21	1	2	2	−3	−4	−25
8	−77	10	14	2	9	−13	−6	−60
9	−5	−27	−53	2	23	−10	−9	−79
10	−110	18	12	3	16	−13	−6	−79
11	−42	10	−33	2	−35	−12	−8	−118
12	−35	18	17	2	23	−13	−6	6
13	−35	18	12	2	23	−13	−6	1

The associated uncertainty with each error components is taken apart from the respective reported uncertainty components. It should be noted that for each lab, the square root of the sum squared uncertainty component will be less than the reported uncertainty as given in Table 4.

Table 4.

Adjusted uncertainties of systematic error components (nm).

Lab	Alignment	Probing correction	Contact Deformation	Environment	Temp effects	Artifact form	CTE	$\sqrt{\sum_{j = 1}^{j = m} u^{2} (b_{ij})}$
1	11	33	5	3	4	50	3	61.0
2	29	24	0	5	3	8	4	39.2
3	11	18	3	2	14	7	2	26.5
4	1	14	0	1	3	8	0	16.4
5	6	19	2	4	5	28	0	35.4
6	11	41	2	4	11	5	2	44.6
7	4	27	2	3	2	4	1	28.2
8	100	13	18	3	12	16	2	104.1
9	6	35	68	3	30	13	4	83.7
10	141	24	16	4	20	16	2	146.7
11	55	14	43	3	46	16	3	85.8
12	46	23	22	3	30	16	2	65.3
13	46	23	16	3	30	16	2	63.5

Task-specific systematic errors

The measuring machines should have desirable design characteristics for specific motion, geometry that, if not perfect, will lead to systematic errors in the measurement (Mian et al., 2014). Probing as a means to detect the boundary of an object imposes practical limits on accuracy attainable in the internal diameter measurements. Prominently, the reproducibility of the instrument fall into broad categories viz. reference surface geometry, alignment, and undesirable motion. Reference surface geometry includes the flatness and parallelism of the anvils, the roundness of artifact and the sphericity of the probe balls (Kim et al., 2010).

Misalignment in a vertical plane

According to analytical geometry, the intersection of a plane with a sphere will be a circle. However, the intersection of a plane and cylinder will be a circle provided the plane is normal to the extracted axis of the cylinder. Therefore, the trajectory of the probe should confine to the given measuring plane. Moreover, the extracted datum of the bore of ring gauge must be normal to this plane.

B_{v} = M [1 - \frac{1}{2} {(\frac{Δ_{Local}}{D_{N} - d})}^{2} - \frac{1}{2} {(\frac{Δ_{tilt}}{D_{N} - d})}^{2}]

(12)

Equation (12) gives the bias $B_{v}$ due to misalignments in a vertical plane. $Δ_{Tilt}$ ( $Δ_{local}$ ) represents tilt offset (the local squareness deviations inside the bore), respectively. $D_{N}$ is the nominal diameter value of ring under test. The machine measures a displacement (M) with the tactile probe of diameter d inside the ring under test.

Misalignment in a horizontal plane

Often, two lines are marked on the ring gauge to indicate the direction of measurement. Reproducible results can be ensured by specifying the direction and location of measurement on the face of ring gauge. The translation of probes in this direction should be maintained during the measurement of ring gauge. A microscope can be used to locate these marks when aligning the ring parallel to the translation of the probe. Precisely, the lines do not always pass through the centre of the ring, as shown in Figure 2.

The lateral offset ( $Δ_{L_offset}$ ) due to lateral misalignment and angular offset ( $Δ_{A_offset}$ ) results in an additional systematic error. The resultant bias $B_{H}$ occurs in the horizontal axis is given by equation (13)

B_{H} = M [1 - \frac{1}{2} {(\frac{Δ_{A_offset}}{D_{N} - d})}^{2} - \frac{1}{2} {(\frac{Δ_{L_offset}}{D_{N} - d})}^{2}]

(13)

Effect of force fields

The force distribution due to atmospheric air pressure (ATP), gravitational pull deforms the ring under test. The location-dependent deviation of gravity and ATP are studied by simulation. A ring gauge of size 4mm is chosen for finite element analysis. The fixed support is set as a reactive force at the base of ring gauge. The ATP is distributed around the ring except for the bottom of the ring gauge. The gravity acts throughout the entire thickness of the material of ring gauge. During finite element analysis, the gravity is acting on each element. The maximum internal diameter variation due to deformation under 10% variation of pressure from reference air pressure (101330 Pa) is about 4 nm. The outer edge of the ring gauge does not change appreciably at the bottom. It was found the increased wall thickness of ring gauge reduces the effect of ATP. Further, the gravity is applied to the entire body of ring gauge keeping the force field due to ATP. The gravity pulls the material layer by layer.

The deformation due to gravity is few tenths of a nanometre for the given sizes of master ring gauges. The combined effect due to gravity, ATP is plotted in Figure 3. The external diameter of the cylinder indenture longitudinally. Similarly, the internal diameter of the cylinder expands by 6 nm.

Results

Often, the results with the lowest uncertainty dominate the key reference value in the weighted mean method. The task-specific errors in the ring gauge measurement are summarized in Table 5. Often, each participant laboratory reports the different magnitude of systematic error ingested with these errors in the key comparison.

Table 5.

Typical value of task-specific systematic errors.

Category	Systematic error	Typical values of state-of-the-art machine (nm)	Range(nm)
Artifact alignment	Diameter plane tilt	5	5 to 25
	Eccentricity(Peak Searching)	10	5 to 20
	Out of plane probe travel	5	5 to 25
Force fields	Atmospheric pressure variation	4	4 to 15
	Gravity variation	4	4 to 15
	Non-linearity of indentation forceto deformation (Puttock et al., 1969)	10	10 to 50

The task-specific uncertainties due to these task-specific errors are given in Table 6. The combined uncertainty is evaluated as 14 nm. Usually, the uncertainty of the weighted mean value will be less than the smallest uncertainty reported among the participant labs. The systematic error value cannot be exclusively known during a given measurement experiment.

Table 6.

Task-specific uncertainty component of internal diameter measurement.

Source of uncertainty	Estimate (nm)	Limits (nm)	Type/Probability distribution	Uncertainty (nm)
u ( $Δ_{A - offset}$ ) probe travel	−10	10	Type B, rectangular	5.8
u ( $Δ_{L - offset}$ ) Marking offset	−5	10	Type B, rectangular	5.8
u ( $Δ_{tilt}$ ) Plane tilt of artifact	−5	15	Type B, rectangular	8.7
u ( $Δ_{local}$ ) Bore straight	–	6	Type B, rectangular	3.5
u (P) Air pressure variation	–	6	Type B, rectangular	3.5
u (g) gravity variation	–	10	Type B, rectangular	5.8
u (I) Nonlinearity	–	10	Type B, rectangular	5.8

Combined task-specific uncertainty $u (x_{μ})$ (k=1) =14 nm.

One needs to perform additional experiments to find the number of systematic errors in each experiment (Stone et al., 2011). The participant laboratories enlarge the uncertainties above their CMC so that the measurement results can be consistent with the KCRV during the draft-A stage of key comparison. Instead, being task specific error are unknown experimentally with respect to magnitude and sign, this component is expected to lie within intervals symmetric to zero. The E_{ni_adj} value of each participant is given in Table 7. It was found that the results of lab 5 are considerably outlier now. It is evident that the lab results are qualified in the inter-comparison due to the large uncertainty claim. Surprisingly, the adverse effect of large uncertainty vanished for lab 13. If we assign positive signed misalignment error value (i.e., 35 nm), lab 13 will qualify with E_{n_adj} = 0.96. Theoretically, misalignment results in a lesser value than the actual (nominal) value of the internal diameter of the ring. Therefore, the systematic error component of all labs was set with a negative sign. Individual labs get insight into competitive values of error components. Thus, lab 13 can decide upon the suitable error values in the root-cause analysis. The same is in the case of lab 11; its result will be an outlier with the positively valued temperature compensation.

Table 7.

Results of participant labs.

Lab	Key comparison results			Root cause results
Lab	$d_{i}$ /nm	$u (d_{i})$ /nm	$E_{ni}$	${\hat{d}}_{i}$ /nm	$u ({\hat{d}}_{i})$ /nm	$E_{ni_adj}$
1	−19	66	−0.14	1	29	0.02
2	−79	46	−0.85	−16	29	−0.28
3	−1	27	−0.01	−10	16	−0.32
4	1	13	0.05	0	15	−0.01
5	77	38	1.02	81	20	2.02
6	−11	50	−0.11	26	26	0.50
7	−28	49	−0.28	−3	42	−0.03
8	−59	129	−0.23	1	78	0.01
9	−49	89	−0.27	30	34	0.45
10	−49	160	−0.15	31	64	0.24
11	−179	99	−0.90	−60	52	−0.58
12	61	129	0.24	56	113	0.25
13	131	69	0.95	130	30	2.15

Conclusion

The proposed approach appears statistically intuitive due to the lack of comprehensive information of each and every participant laboratory. Without the consideration of task-specific uncertainty, the analysis remains intuitive. Since the individual participants have their measurement system at hand; the results of this analysis precisely confirm the compensatable systematic error values and the respectively associated uncertainty. Therefore, the novelty of the analysis involves recognizing inevitable systematic errors. Then, appending the corresponding uncertainty to all participants eliminates the biasing of reference value due to dominant reported uncertainty.

Using this approach, the uncertainty associated with the estimates of the biases is significantly reduced. Thereby, the state of knowledge of the laboratories about their biases is reaffirmed. These benefits of the proposed approach are illustrated by the analysis of internal diameter key comparison data. A common minimum uncertainty is concocted. The deformation due to the variation of atmospheric pressure and gravity are estimated. Task-specific uncertainty component of internal diameter measurement are evaluated. It is also found that the standard uncertainty due to task-specific errors ranges from 11 nm to 25 nm. The results of the proposed analysis discriminated the merely consistent laboratory results. For example, labs 5 and 13 are still outlier despite increasing the uncertainty of their results. The proposed method is easy to practice and fetches the benefits of the intercomparison.

The author(s) declared no potential conflict of interests with respect to the research, authorship and/or publication of this article.

Footnotes

Appendix

Acknowledgements

The authors pay thanks to the Director at National Physical Laboratory, New Delhi for his encouragement.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mahammed Arif Sanjid

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