## 4 Materials and Method
### Application of Statistical Quality Control Methods
#### 4.1.2 Implementation of Control Charts for Process Monitoring  
Once the assumption of normality has been confirmed, the next step involves the application of control charts. These charts, similar in structure to time series or trend graphs, are designed to visualize the behavior of data over time. The primary objective of control charts is to detect the presence of \"special causes of variability\", which may indicate shifts or disturbances in the process. Control charts play a key role in monitoring process stability, assessing process improvement, and identifying process variability. Among the available methods, Shewhart control charts are particularly suitable for evaluating process stability in real time. These charts are updated at regular intervals, providing a continuous overview of the current process state and allowing for early detection of deviations from expected behavior [27; 28]. In the present study, Shewhart control charts were constructed for the sample arithmetic mean, with control limits calculated according to established statistical principles [29]. The test statistic for the control chart of the sample mean is defined as follows [29]  
Test criterion is  
\[\overline{x}_{j}=\frac{1}{n}.\sum_{i=1}^{n}x_{ij} \tag{2}\]  
\[CL=\overline{\overline{x}}=\frac{1}{k}\sum_{j=1}^{k}\overline{x}_{j} \tag{3}\]  
\[CL=\overline{\overline{x}}-A_{2}.\overline{R} \tag{4}\]  
\[UCL=\overline{\overline{x}}+A_{2}.\overline{R} \tag{5}\]  
where  
\(\overline{\overline{x}}_{j}\): sample mean value,  
\(A_{2}\): constant for a certain range of a subgroup [30],  
\(\overline{R}\): calculation of the average spread,  
_UCL_: Upper Control Line,  
_LCL_: Lower Control Line,  
_CL_: Central Line. In conjunction with control chart analysis, it is appropriate to evaluate process capability indices (PCIs) to further assess the quality and consistency of the manufacturing process. These indices offer quantitative insight into how effectively a process operates within predefined specification limits. Assessing process capability requires determining whether the process is properly centered within the tolerance range and quantifying its inherent variability. Such analysis is essential for verifying whether the process can consistently produce outputs that meet required specifications. The process capability indices include C\({}_{\text{p}}\), C\({}_{\text{pk}}\), P\({}_{\text{p}}\), and P\({}_{\text{pk}}\). The indices C\({}_{\text{p}}\) and C\({}_{\text{pk}}\) are used to quantify process capability, which reflects the degree to which a process meets the required quality specifications for a given product. Determining true process capability requires continuous and long-term monitoring under stable operating conditions. For capability indices to be valid, the process must be under statistical control, meaning that all systematic influences are identified and consistently managed. During capability assessment, the dominant factors influencing variability are typically those inherent to the production equipment and environment [28; 29]. In cases where the process is not statistically controlled, it is more appropriate to calculate process performance indices rather than capability indices. These performance indices reflect only the current behavior of the process, without assuming long-term stability. While the formulas remain the same, the indices are denoted by the letter P (for Performance) instead of C (for Capability), indicating that the values apply to a process that may still be influenced by uncontrolled or unquantified sources of variation. P\({}_{\text{p}}\) and P\({}_{\text{pk}}\) are indices based on samples collected in equally sized subgroups over a long-term time interval. These are indices of process potential and represent long-term process performance. These indices characterize past results; they cannot be used to predict future process behavior but only indicate the current state of the process [29].