## 2 Theoretical Background and Calculation
In the next step, the number of logical subgroups \(m\) (the recommended minimum number is 20 to 25 subgroups), and the time interval was determined. Since the process of rock disintegration by rotary drilling is fast and short-lived, a seconds time interval of measuring variables was chosen, from which samples were selected for the logical subgroup. The sample arithmetic mean \(\bar{x}_{i}\) and the sample range of variation \(R_{i}\) were calculated for the \(i\)-th subgroup from the measured values. These parameters are calculated as the following:  
\[\bar{x}_{i}=\frac{1}{n}\sum_{j=1}^{n}x_{ij}, \tag{1}\]  
\[R_{i}=x_{i,\max}-x_{i,\min}, \tag{2}\]  
for \(i=1,2,...,m\) and \(j=1\), \(2,...,n\), where \(n\) is the range of logical subgroup, \(m\) is the number of subgroups, \(x_{i,\max}\) is maximal, and \(x_{i,\min}\) is the minimum measured value in the \(i\)-th subgroup. Finally, the overall arithmetic mean \(\overline{\overline{x}}\) and the average range \(\bar{R}\) to which it applies  
\[\overline{\overline{x}}=\frac{1}{m}\sum_{i=1}^{m}\bar{x}_{i}, \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rock** & **Class** & **Degree of Rock Strength** & **Rock Type** & **Strength Factor \(f\)** \\ \hline Andesite & IIIa. & strong & strong andesites & 8 \\ Limestone & III. & strong & very strong limestones & 10 \\ Granite & II. & very strong & strong granites & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Detailed technological properties of the rocks. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Physical Properties of Rocks**} & \multicolumn{3}{c}{**Mechanical Properties of Rocks**} \\ \hline \multirow{2}{*}{**Rock**} & **Specific Mass** & **Volume Mass** & **Porosity** & **Absorbability** & **Resistance to Simple Compression** & **Resistance to Crossfeed Tension** \\ \cline{2-7}  & \(\mathbf{r}\) (**logm\({}^{-3}\))** & \(\mathbf{r}\) (**logm\({}^{-3}\))** & \(\mathbf{p}\) (**\%**) & \(\mathbf{n}\) (**\%**) & **(MPa**)** & **(MPa**)** \\ \hline Andesite & 2720 & 2700 & 0.07 & 0.67 & 282 & 16 \\ Limestone & 3000 & 2960 & 1.33 & 0.1 & 225 & 13 \\ Granite & 2680 & 2650 & 1.12 & 0.14 & 196 & 72 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical and mechanical properties of the rocks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rock** & **Sekant Reshaping Modulus \(\mathbf{M_{i}}\) (MPa)** & **Tangent Reshaping Modulus \(\mathbf{M_{i}}\) (MPa)** & **Elastic Modulus \(\mathbf{E}\) (MPa)** & **Poisson Number \(\mathbf{p}\)** \\ \hline Andesite & 84,607 & 84,314 & 84,608 & 0.21 \\ Limestone & 43,725 & 70,294 & 56,471 & 0.13 \\ Granite & 54,020 & 61,275 & 57,059 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Detailed mechanical properties of the rocks. \[\bar{R}=\frac{1}{m}\sum_{i=1}^{m}R_{i}, \tag{4}\]  
were calculated. The control chart contains the upper (UCL), lower control limit (LCL) and center line (CL). The regulatory limits for the \(\bar{x}\)-chart were calculated according to the following equations:  
\[\text{CL}_{\bar{x}}=\overline{\bar{x}}, \tag{5}\]  
\[\text{UCL}_{\bar{x}}=\overline{\bar{x}}+A_{2}\overline{R}, \tag{6}\]  
\[\text{LCL}_{\bar{x}}=\overline{\bar{x}}-A_{2}\overline{R}. \tag{7}\]  
The regulatory limits for the \(R\)-chart were calculated according to the equations:  
\[\text{CL}_{R}=\overline{R}, \tag{8}\]  
\[\text{UCL}_{R}=D_{4}\overline{R}, \tag{9}\]  
\[\text{LCL}_{R}=D_{3}\overline{R}. \tag{10}\]  
Since the value of the average range of variation evaluated in the \(R\)-chart is used to calculate the control limits of the \(\bar{x}\)-chart, the analysis of the control chart is first performed, which characterizes the variability of the process, i.e., the \(R\)-chart. This chart was obtained by plotting the values \(R_{i}\) of the selective variation ranges of the individual subgroups and the control limits. From the chart analysis, it can be monitored whether the regulated variable does not signal the effect of definable causes of variability.