# The Relationship between Process Capability and Quality of Measurement System
## 1 Introduction
It was calculated using Equation (2) as the difference between the average value of the five repeated measurements of the end gauge and its actual value \(x_{s}\). \[u_{sys}=0.6\cdot\left(\left|\overline{x}_{g}-x_{s}\right|\right) \tag{2}\]  
The fourth source of calibration uncertainty was type A measurement uncertainty. The standard uncertainty \(u_{A}\) was calculated using Equation (3). \[u_{A}=\frac{SD}{\sqrt{5}} \tag{3}\]  
where:  
_SD_--standard deviation;  
5--number of trials. A simplified procedure according to Equation (4) was used to calculate the standard calibration uncertainty, which was based on the instructions given in [22] and took into account the above sources of uncertainty. \[u_{cal}=\sqrt{u_{micometer}^{2}\cdot u_{standard}^{2}\cdot u_{A}^{2}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Metrological Appraisers** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\ \hline Male/Female & M & F & M & M & M & M & F & F \\ \hline Age (years) & 48 & 38 & 41 & 60 & 61 & 32 & 25 & 37 \\ \hline \multirow{4}{*}{Optical power (D)} & \multirow{2}{*}{Left eye} & \(-0.0\)/ & \(-0\)/ & \(-0\)/ & \(-2.5\)/ & \(-0.5\)/ & \(-0\)/ & \(-0.75\)/ & \(-0\)/ \\  & & +1.0 & +0 & +0 & +0.25 & +1.5 & +0 & +0 & +0 \\ \cline{2-10}  & \multirow{2}{*}{Right eye} & \(-0.0\)/ & \(-0\)/ & \(-0\)/ & \(-4.0\)/ & \(-0.5\)/ & \(-0\)/ & \(-0.75\)/ & \(-0\)/ \\  & & +1.0 & +0 & +0 & +1.75 & +2.5 & +0 & +0 & +0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The age of the appraisers and condition of their vision (in diopters). whereas in the range of temperatures at which the calibration took place, the influence of ambient temperature on uncertainty was negligible (on average 0.069%), and it was not taken into account in the calculations. The calculated values of the standard uncertainty of the \(u_{\alpha l}\) calibration for all appraisers are given in the already mentioned Table 2. The standard uncertainty of the \(u_{\alpha lXY}\) calibration for the pair of appraisers was calculated as the geometric mean using Equation (5) and the uncertainties of the calibration are given in Table 2. As follows from Table 4, metrological appraiser **B** showed the highest uncertainty, and metrological appraiser \(F\) and the pairs \(CF\), \(EF\) and \(FG\) showed the lowest uncertainty. However, due to the fact that the uncertainty \(u_{\alpha lXY}\) of the calibration affected the results only indirectly, the values given in Table 4 were considered only as ancillary. \[u_{\alpha lXY}=\sqrt{u_{\alpha lX}^{2}\cdot u_{\alpha lY}^{2}} \tag{5}\]  
Metrological appraiser B appeared to be critical, and the pairs CF, EF and FG showed the lowest uncertainty. As can be seen from Figure 2, the value of the calibration uncertainty \(u_{\alpha l}\) of the metrological appraisers generally increased with age. For the two oldest metrological appraisers (D and E), the uncertainty was lower, despite their age. Their many years of practice and experience could have contributed to this fact. Using the Grubbs test with significance level \(\alpha\) = 0.05, it was found that no value of calibration uncertainty was an outlier. The calibration was followed by the measurement of the diameter of shafts no. 1 to no. 45. The arithmetical average of the measured dimensions was calculated from three measurements. The mean values \(\overline{x_{1}}\) to \(\overline{x_{45}}\) and the standard deviations SD\({}_{1}\) to SD\({}_{45}\) were calculated. Each appraiser obtained a set of 135 measured values. Table 1 shows the values for this set: average value \(\overline{x}\) (mean), standard deviation SD, normality (\(p\) value) and, in Table 2, average standard uncertainty of shaft diameter measurement \(u_{th}\). As shown in Table 1, we could assume that all sets had a normal distribution, because all p values were greater than 0.05. The Grubbs test found that there were no outliers in the sets. These would occur if the sets were not in a statistically controlled state.