# On the Subrange and Its Application to the R-Chart
## 3 The Construction of Control Charts  
In this section, we provide the method for incorporating the proposed methods into constructing the control charts. We use the proposed subrange as an alternative of the range and standard deviation when there are outliers in the data of Phase I. We provided the \(\overline{X}\)-chart first, and then the \(R\)-chart. For the case of the \(\overline{X}\)-chart, it is immediately from the central limit theorem that we have  
\[\frac{\overline{X}-E(\overline{X})}{\text{SE}(\overline{X})}\ \sim N(0,1),\]  
where \(\overline{X}\)-is the sample mean from a sample of size \(n\), and \(\text{SE}(\cdot)\) is the standard deviation from a sample of size \(n\). Solving the below for \(\overline{X}\)  
\[\big{(}\overline{X}-E(\overline{X})\big{)}/\text{SE}(\overline{X})\ =\ \pm 3,\]  
we can construct the \(\text{CL}\pm 3\cdot\text{SE}\) control limits. \[E\big{(}\overline{X}\big{)}\pm 3\cdot\text{SE}(\overline{X})\ =\ \mu\pm\frac{3 \sigma}{\sqrt{n}}.\]Then we have \(\text{UCL}\ =\ \mu+\frac{3\sigma}{\sqrt{n}}\), \(\text{CL}\ =\ \mu\), and \(\text{LCL}\ =\ \mu-\frac{3\sigma}{\sqrt{n}}\). Since \(\mu\) and \(\sigma\) are unknown in practice, we need to estimate them. Suppose that there are \(m\) samples of size \(n\). The most widely used estimator of \(\mu\) is \(\overline{\overline{X}}\) where  
\[\overline{\overline{X}}\ =\ \frac{\overline{X}_{1}+\overline{X}_{2}+\cdots+ \overline{X}_{m}}{m}.\]  
The scale \(\sigma\) can be estimated by using the subrange. Let \(R_{[k],i}\) be the subrange with \(k\) trims from the \(i\)th sample. Then, as shown in (4), \(R_{[k],i}/d_{2}(n,k)\) is an unbiased scale estimator of \(\sigma\). Thus, it is quite reasonable to use \(\overline{R}_{[k]}/d_{2}(n,k)\) for the scale estimator, where  
\[\overline{R}_{[k]}\ =\ \frac{R_{[k],1}+R_{[k],2}+\cdots+R_{[k],m}}{m}.\]  
With the estimators of \(\mu\) and \(\sigma\), we have the following control limits:  
\[\text{UCL}\ =\ \overline{\overline{X}}+\frac{3\overline{R}_{[k]}}{ d_{2}(n,k)\sqrt{n}}\] \[\text{CL}\ =\ \overline{\overline{X}}-\frac{3\overline{R}_{[k]}}{ d_{2}(n,k)\sqrt{n}}.\]  
It should be noted that when \(k\ =\ 0\), we have \(\overline{R}_{[k]}\ =\ \overline{R}\) and \(d_{2}(n,k)\ =\ d_{2}\). The above control limits then become  
\[\text{UCL}\ =\ \overline{\overline{X}}+\frac{3\overline{R}}{d_{2}\sqrt{n}}\] \[\text{CL}\ =\ \overline{\overline{X}}\] \[\text{LCL}\ =\ \overline{\overline{X}}-\frac{3\overline{R}}{d_{2} \sqrt{n}}.\]  
which is the traditional \(\overline{X}\)-chart provided in the quality control literature. For the case of the \(R\)-chart, we briefly reviewed the conventional \(R\)-chart and proposed a new \(R_{[l]}\)-chart. Solving the below  
\[\big{(}\overline{X}-E(\overline{X})\big{)}/\text{SE}(\overline{X})\ =\ \pm 3,\]0  
for \(R\), we can construct the \(\text{CL}\pm 3\cdot\text{SE}\) control limits given by  
\[\big{(}\overline{X}-E(\overline{X})\big{)}/\text{SE}(\overline{X})\ =\ \pm 3,\]1  
Then we have \(\text{UCL}\ =\ d_{2}\sigma+3d_{3}\sigma\), \(\text{CL}\ =\ d_{2}\sigma\) and \(\text{LCL}\ =\ d_{2}\sigma-3d_{3}\sigma\). Analogous to the above construction, we can calculate the \(R_{[l]}\)-chart with the limits given by  
\[\big{(}\overline{X}-E(\overline{X})\big{)}/\text{SE}(\overline{X})\ =\ \pm 3,\]2  
In practice, the scale \(\sigma\) is unknown. Thus, we need to estimate \(\sigma\) using \(\overline{R}_{[k]}/d_{2}(n,k)\), as above.