# A New Approach to Production Process Capability Assessment for Non-Normal Data
## 6 Research Results
### Process Quality Capability Analysis  
According to the methodology outlined in item 4.3, the values of \(C_{p}\) and \(C_{pk}\) indices were determined for each production site analyzed for the data after transformation. Each process had a unilateral specification limit for the product treatment time--_USL_. Table 5 shows the results obtained: the statistical values for each machine, including the upper specification limit (_USL_) for the data before and after the transformation, the absolute values, the values of the distribution and the process performance indicators (\(C_{p}\), \(C_{pk}\)). It should be noted that the parameter indicators \(\lambda\) for the proposed power transform obtained negative values, which resulted in the specification limit for the transformed data being \"swapped\". Its value and location for the transformed data (red dotted line) are marked for each machine in Figures 2-4. For Machine 1, the \(C_{p}\) and \(C_{pk}\) indicators are 1, which means that the process is capable of producing products within tolerance. For Machine 2, the \(C_{p}\) indicator is 0.8 and \(C_{p}\) is 0.7, suggesting that the process is less stable and may lead to more defective products. For Machine 3, the \(C_{p}\) indicator is 0.9, and \(C_{pk}\) is 0.8, which means the process is less stable than Machine 1, but more stable than Machine 2. Analysis of these results can help identify machines that require additional investigation and improvements to the production process to improve process performance and production quality. The values of the distributions for all three machines are very high, which means that almost all products produced by these machines are within tolerance. At the analyzed stations, if the manufacturer wants to improve product quality, the focus should be placed on machines that have lower process performance indicators. For Machine 2, whose \(C_{pk}\) indicator is the lowest, the cause of the low performance should be identified and appropriate corrective action taken. It is also important to consider other factors, such as the costs associated with streamlining the process and making changes. Depending on the situation, it may be more \begin{table}
\begin{tabular}{c c c} \hline \hline
**Statistics** & **Treatment Time** & **Treatment Time Transformation** \\ \hline mean & 0.0184 & 4.0170 \\ st. deviation & 0.0123 & 0.7640 \\ min & 0.0057 & 2.3738 \\
1Q & 0.0090 & 3.4294 \\
3Q & 0.0233 & 4.6852 \\ max & 0.0716 & 5.4588 \\ skewness & 1.6794 & \(-\)0.0189 \\ kurtosis & 3.2309 & \(-\)0.9942 \\ JB statistic & 178.2859 & 8.1258 \\ \(p\)-value & \(<\)0.0001 & 0.0172 \\ USL & 0.1000 & 2.1279 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Based statistics and result of Jarque–Bera test. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process** & **Machine 1** & **Machine 2** & **Machine 3** \\ \hline USL before transformation & 0.10 & 0.12 & 0.10 \\ USL after transformation & 1.5050 & 11.3771 & 2.1279 \\ \(\lambda\) & \(-\)0.1775. & \(-\)1.1468 & \(-\)0.3279 \\ Absolute value & 2.87 & 1.91 & 2.47 \\ Distribution function value & 0.997882 & 0.971930 & 0.993053 \\ \(C_{p}\) & 1.0 & 0.8 & 0.9 \\ \(C_{pk}\) & 1.0 & 0.7 & 0.8 \\ Condition of the process & Capable & Inadequate & Inadequate \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical values and \(C_{p}\), \(C_{pk}\) for the stations analyzed. cost-effective to focus on machines with higher process performance indicators that also have a low deviation from the specification limit. Identifying machines that need to be improved in process performance helps ensure higher production quality and increase customer satisfaction, but it is equally important to consider costs and choose corrective actions that are most cost-effective for the company. Therefore, the next step after analyzing the results for individual machines should be to conduct a more detailed study of the production process for each machine.