# A Bayesian Control Chart for Monitoring Process Variance
## 6 An Example for Demonstration  
A banking example from [PERSON] et al. [21] is applied to illustrate the proposed Bayesian EWMA variance chart. The banking industry regards service time as an important quality characteristic. Under the demand for standardization and uniformity of service time, a control chart will be constructed to monitor the service time and detect whether the variance is offset. Service time usually comes from a complex service process, whose distribution may not be normal. Sometimes, the distribution of the service time is time-varying even under the in-control condition. Because we know little about the nature of the service process, the proposed Bayesian EWMA variance is applicable to the banking case. In order to measure the efficiency in the service system of a bank branch, the in-control sampling service time was measured from 10 counters every day for 15 days. So, 15 samples of size \(2n=10\) were collected. A histogram shows that these samples are right-skewed. Table 8 shows the service time from 10 counters in a bank branch. The variance of service time is estimated with \(\hat{\sigma}^{2}=30.0969\) through Equation (20). The prior distribution of \(p\) is \(Beta(1,1)\). After incorporating the information of 15 preliminary samples, we get \(\hat{\texttt{a}}=1+\sum\limits_{t=1}^{T}m_{t}=23\) and \(\hat{\beta}=1+nT-\sum\limits_{t=1}^{T}m_{t}=54\) by Equation (21) as well as \(E_{0}(p)=23/77=0.2987\) by Equation (22). Past experience has confirmed that 30.0969 is very close to the true value of in-control process variance, \(\sigma_{0}^{2}\), and 0.2987 is also very close to the true value of the in-control mean of \(p\), \(E_{0}(p)\). When the smoothing parameter \(\lambda\) is 0.05, referring to the procedure described in Section 4.1, we obtain \(k_{1}=3.07\) and \(k_{2}=2.86\). Thus, the control limits are calculated as \(UCL_{EWMA}=2.0094\) and \(LCL_{EWMA}=1.0129\) through Equations (31) and (32). From Equation (30), we preset \(EWMA_{0}=1.4935\), then \(EWMA_{t}\) is computed in order by Equation (25) and presented in Table 8.  
Figure 2 shows the proposed Bayesian EWMA variance chart. The observed statistics \(EWMA_{t}\)s are plotted in order. Because all the \(EWMA_{t}\)s fall into the region between control limits, the chart reveals that the process is in control. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \(t\) & \(X_{1}\) & \(X_{2}\) & \(X_{3}\) & \(X_{4}\) & \(X_{5}\) & \(X_{6}\) & \(X_{7}\) & \(X_{8}\) & \(X_{9}\) & \(X_{10}\) & \(S^{2}\) & \(M\) & \(EWMA\) \\ \hline
1 & 0.88 & 0.78 & 5.06 & 5.45 & 2.93 & 6.11 & 11.59 & 1.2 & 0.89 & 3.21 & 11.59 & 1 & 1.4688 \\
2 & 3.82 & 13.4 & 5.16 & 3.2 & 32.27 & 3.68 & 3.14 & 1.58 & 2.72 & 7.71 & 86.35 & 2 & 1.4954 \\
3 & 1.4 & 3.89 & 10.88 & 30.85 & 0.54 & 8.4 & 5.1 & 2.63 & 9.17 & 3.94 & 77.86 & 2 & 1.5206 \\
4 & 16.8 & 8.77 & 8.36 & 3.55 & 7.76 & 1.81 & 1.11 & 5.91 & 8.26 & 7.19 & 19.77 & 1 & 1.4946 \\
5 & 0.24 & 9.57 & 0.66 & 1.15 & 2.34 & 0.57 & 8.94 & 5.54 & 11.69 & 6.58 & 18.47 & 1 & 1.4699 \\
6 & 4.21 & 8.73 & 11.44 & 2.89 & 19.49 & 1.2 & 8.01 & 6.19 & 7.48 & 0.07 & 31.88 & 2 & 1.4964 \\
7 & 15.08 & 7.43 & 4.31 & 6.14 & 10.37 & 2.33 & 1.97 & 1.08 & 4.27 & 14.08 & 24.85 & 2 & 1.5215 \\
8 & 13.89 & 0.3 & 3.21 & 11.32 & 9.9 & 4.39 & 10.5 & 1.7 & 10.74 & 1.46 & 25.00 & 4 & 1.6455 \\
9 & 0.03 & 12.76 & 2.41 & 7.41 & 1.67 & 3.7 & 4.31 & 2.45 & 3.57 & 3.33 & 12.78 & 1 & 1.6132 \\
10 & 12.89 & 17.96 & 2.78 & 3.21 & 1.12 & 12.61 & 4.23 & 6.18 & 2.33 & 6.92 & 31.47 & 1 & 1.5825 \\
11 & 7.71 & 1.05 & 1.11 & 0.22 & 3.53 & 0.81 & 0.41 & 3.73 & 0.08 & 2.55 & 5.62 & 0 & 1.5034 \\
12 & 5.81 & 6.29 & 3.46 & 2.66 & 4.02 & 10.95 & 1.59 & 5.58 & 0.55 & 4.1 & 8.51 & 0 & 1.4282 \\
13 & 2.89 & 1.61 & 1.3 & 2.58 & 18.65 & 10.77 & 18.23 & 3.13 & 3.38 & 6.34 & 44.71 & 2 & 1.4568 \\
14 & 1.36 & 1.92 & 0.12 & 11.08 & 8.85 & 3.99 & 4.32 & 1.71 & 1.77 & 1.94 & 12.63 & 1 & 1.4340 \\
15 & 21.52 & 0.63 & 8.54 & 3.37 & 6.94 & 3.44 & 3.37 & 6.37 & 1.28 & 12.83 & 39.96 & 2 & 1.4623 \\ \hline \end{tabular}
\end{table}