# The Relationship between Process Capability and Quality of Measurement System
## 1 Introduction
The measuring system includes the measured object (parts), metrological appraisers, measuring equipment, measuring method, environment (temperature, humidity, lighting), applicable standards and regulations used, software, etc. It is assumed that, if the system is capable, the process will also be capable. The capability of the measurement system can be evaluated and subsequently quantified by various methods, for example, using Measurement Systems Analysis (MSA) [20]. Their outputs are the indices that characterize the system as a whole as well as its constituent parts. A repeatability and reproducibility (R&R) assessment approach was used in the analysis. Analysis of variance (ANOVA) offered more indices, but through more complicated calculations. To assure the efficiency of resource spending, it was appropriate to first eliminate the undesirable influence of appraisers on the measurement system, provided that the measurement instrument is accurate and reliable. Dimension d\({}_{2}\) was measured again on 45 shafts, the same ones measured by appraiser A. Eight metrological appraisers (A-H) were involved in the measurement using the already presented values measured by appraiser A. The age of the appraisers and condition of their vision are shown in Table 3. The measurement was repeated three times in random order of the appraisers and shafts, and \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Metrological** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\
**Appraisers** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\ \hline \(\overline{X}_{g}\) (mm) & 10.0040 & 10.0066 & 10.0026 & 10.0028 & 9.9972 & 10.0016 & 10.0028 & 10.0044 \\ \hline SD (mm) & 0.0007 & 0.0005 & 0.0005 & 0.0011 & 0.0016 & 0.0005 & 0.0004 & 0.0015 \\ \hline \(u_{\mathrm{cal}}\) (mm) & 0.0026 & 0.0041 & 0.0017 & 0.0019 & 0.0018 & 0.0012 & 0.0018 & 0.0028 \\ \hline \(u_{h}\) (mm) & 0.0029 & 0.0041 & 0.0024 & 0.0019 & 0.0019 & 0.0013 & 0.0020 & 0.0030 \\ \hline \end{tabular}
\end{table} 
Table 2: Mean value \(\overline{X}_{g}\) and standard deviation SD measured during calibration, standard uncertainty of calibration \(u_{\mathrm{cal}}\) and average value of standard uncertainty of measured dimension d\({}_{2}\)\(u_{h}\). the results are given in the already mentioned Table 1. The sets of measured values of all appraisers had a normal distribution and were without outliers. The use of a precise measurement instrument did not guarantee accurate and precise measurement results. The measurement could be significantly affected by the incompetence of the metrological appraiser, noncompliance with the correct measurement procedure, incorrectly selected measurement method, noncompliance with the correct pressing force, zeroing, etc. To determine the \"individuality\" of the metrological appraiser, each performed a calibration using the end gauge (Johansson's gauges) as a standard of the second-class precision and the fourth grade according to the traceability scheme of length gauges [21] with nominal value (standard length) \(x_{n}\) = 10 mm. The calibration took into account the true value of the end gauge \(x_{s}\), which was calculated according to Equation (1). \[x_{s}=x_{n}-\epsilon_{standard} \tag{1}\]  
where \(\epsilon_{standard}\) = 0.0002 mm, and the end gauge error was detected during its calibration (bias of the standard). The first source of uncertainty of the calibration was the uncertainty of the used measurement instrument micrometer found during its calibration (\(u_{micometer}\) = 0.0005 mm). The second source of calibration uncertainty was the uncertainty of the true value of the end gauge \(u_{standard}\) = 0.0001 mm. From the five repeated measurements of the end gauge performed by each of the metrological appraisers, the mean and the standard deviation SD were calculated and are given in Table 2.  
The third source of uncertainty was the systematic error of the micrometer (bias) detected during calibration.