## 2 Theoretical Background and Calculation
The occurrence of points outside the regulatory limits and the existence of non-random groupings of points are considered to be a signal of the action of definable causes. If non-accidental causes of variability occur, the process is considered to be statistically unmanageable/unstable/not under control. If no signals of the action of definable causes appear, the control chart analysis characterizing the production process's position, i.e., the \(\bar{x}\)-chart, is continued. Its evaluation is analogous to the \(R\)-chart. If signals of the action of identifiable causes are detected in practice, it is important to identify these causes and take corrective measures to eliminate them. Since the control chart for the arithmetic mean has already been calculated, it is possible to proceed with the control chart's calculation and construction for the standard deviation [46; 47; 48; 49]. The \(s\)-standard deviation control chart is the most accurate of the control charts by measurement. It is suitable for larger ranges of logical subgroups (\(2\leq n\leq 25\)) and where it is possible to process data on a computer. When selecting a subgroup range, it is recommended to select \(n>9\). The sample standard deviation is used to describe the variability of the technological process. The procedure for constructing a control chart \(s\) is similar to that for a control chart \(\bar{x}-R\). In each subgroup, the sample arithmetic mean \(\bar{x}_{i}\) and sample standard deviation \(s_{i}\) were calculated. It is applied that  
\[\bar{x}_{i}=\frac{1}{n}\sum_{j=1}^{n}x_{ij}, \tag{11}\]  
\[s_{i}=\sqrt{\frac{1}{n-1}\sum_{j=1}^{n}\big{(}x_{ij}-\bar{x}_{i}\big{)}^{2}}, \tag{12}\]for \(i=1,2,...,m,j=1,\ 2,...,n\), where \(m\) is the number of subgroups and \(n\) represents the range of the subgroup. The overall arithmetic mean \(\overline{\overline{x}}\) and average of sampling standard deviations \(\bar{s}\) were calculated according to equations  
\[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]0  
\[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]1  
The control limits for the \(s\)-chart can be calculated according to the following equations:  
\[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]2  
\[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]3  
\[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]4  
where \(A_{2}\), \(D_{3}\), \(D_{4}\), \(B_{3}\), and \(B_{4}\) are constants that depend on a range of subgroups and presented are in tables (see Table 5). When evaluating the process control via the \(s\)-chart, the procedure was the same as when evaluating the previous control charts. The following important control chart, which was constructed for the drilling technological process's needs, is the chart for the median \(\tilde{x}\). The chart for the median \(\tilde{x}\) is, on the one hand, easy to use and does not require more complex calculations; on the other hand, it is less sensitive than the chart for the sample average. The control chart for the median shows the range of the technological process and gives a continuous view of the process's measured variables' fluctuations. The procedure for constructing a control chart for the median \(\tilde{x}\) is similar to the other charts. It is commonly used for logical subgroups with a range of 10 or less than 10, with odd ranges of subgroups being more preferred. In each subgroup, sample characteristics were calculated, i.e., the sampling median \(\tilde{x}_{i}\) and the sampling range of variation \(R_{i}\), which has already been calculated.