Variable Control Chart for any quality characteristic is based on a methodological sequential order, namely, specifying a probability model for the quality variable, collecting observations on measurements on the quality varies in a production process, formulating the rational subgroups of equal size preferably for the quality observations, considering a statistic and calculating its value for observations in each subgroup called subgroup statistics, say t_n where, n is the common subgroup size, finding the percentiles of t_n from its sampling distribution. Parallel to Shewart principle, we generally construct the 0.9973 two-tailed, equitailed probability limits of t_n .

The median or mean of t_n shall be chosen as criterion for central target value. Shifts in the process quality characteristic is indicated by deviations of the value of t_n from their median or mean. The larger is the deviation, the poorer is the quality. The limits of deviation are generally called UCL & LCL. These sequences of steps are generally followed for deciding the in control of a process on the basis of control chart as in [1] -[5] and [7]-[10]. The name of the control shall be generally the name of the statistic t_n like - chart if t_n is , range chart if t_n is range, the S-chart if t_n is standard deviation,……etc. If the specified probability model and the statistic t_n are not convenient to derive the sampling distribution of t_n and hence its 0.9973 probability limits, the theory of extreme order statistics can be explored in developing a control chart. The present paper is based on this principle and hence named as extreme value control charts. The basic model is Burr Type XII Distribution (BTXIID). The suggested theory of extreme order statistics for control charts with reference to BTXIID is presented in Section 1. The developed control limits and the control chart constants along with the associated control limits are given in Section 2. The results are explained in Section 3.

In the introduction, the authors said that if the statistic t_n of a subgroup falls within the probability limits of its sampling distribution, the process is said to be in control. Alternatively, this can be viewed as follows. Since a statistic is a function of all sample observations, they may conclude, that in control of the process quality, all the subgroup observations of each subgroup fall between a reasonably wide probability limits. In other words, they propose two limits; in such a way that the maximum of each subgroup is not above an upper limit and the minimum of each subgroup is not below a lower limit. That is, they need the percentiles of extreme order statistics in a subgroup of size n, that would serve as control limits. With this back drop they can use 0.99865 percentile of the highest order statistic and 0.00135 percentile of the least order statistic of a sample of size n assumed to follow a specified probability model as 0.9973 control limits for the subgroup as above. Accordingly, this paper specifies Burr XII with parameters c and k as probability model and subgroups of size n from it as a statistical quality control data for process control. In terms of c and k, the percentiles of extreme order statistics are,

The median of BTXII is taken as the central line. These three would develop a variable control chart for rational subgroups, which are assumed to follow BTXII Distribution.

It is well known that a Burr Type XII Distribution variable with parameters c=4.874 and k=6.158 follows a normal distribution with mean 0.83576 and standard deviation 0.40093.

2.1 Methodology and Objectives

Using this result, if is the mean of a subgroup of size n from a BTXII Distribution, then is a standard normal variate. From this, the authors can find the probability limits of the subgroup is . If this has l such subgroups, the extreme value control limits for subgroup mean are given by the probability identity with the help of areas of standard normal tables and the specified subgroup size and the number of subgroups that is fixing n and l, they get the probability limits +U and –U in the above identity. These are evaluated for different values of l and n and are given in Table 1. The in control of the process is verified by finding the minimum and maximum of each subgroup and if both of them fall within the control limits chosen for the given subgroup size and number of subgroups. The authors say that the process is in control over the process out of control. Using the same principle and relation between BTXII and normal distributions, they can develop the control charts for the mean of BTXII distribution also without going on to the number of subgroups. These limits are given in Table 2. If one wants to use extreme value to control limits, they have to go for Table 1 corresponding a given subgroup size, and the number of subgroups. On the other hand, the control chart for mean that is chart one can use Table 2.

Table 1. Extreme Value Control Chart Limits

Table 2. Extreme Value Control Limits

This procedure of using Table 3 is given by illustration borrowed from as in [6].

Table 3. Inside Diameter Measurements (mm) for Automobile Engine Piston Rings

In this example (Table 3), number of subgroups l=25, subgroup size n=5. These are given in Table 3, to find the individual observations in each subgroup, their mean and standard deviation also.

3.1 Conclusion of Extreme Value Control Chart

From Table 1 corresponding to n=5, and l=25, the pair of control limits is (1.01506, 1.07286). Find the mean of minimum of each subgroup say , similarly, find the mean of the highest (maximum) observation of each subgroup let it be . The process will be in control if the least of all subgroup observations fall above and the maximum of all subgroup observations fall below . It will be out of control otherwise.

If it uses only subgroup means of a common subgroup size say n limits of Table 2 are to be multiplied by the grand mean of the data . The LCL and UCL are . C_L , . C_U . where C_L , C_U is the table values in Table 2 corresponding to a given subgroup size n.

3.2 Calculations

Let n=5, l=25 ; Readings of Table 1 are (1.01506,1.07286).

From the given data, mean of all minima of the 25 subgroups = =73.98964.

Mean of all maxima of the 25 subgroups = = 74.05096.

The control limits of Extreme Value Control Limits are (1.01506 , 1.07286 ) = (75.10392, 79.44631). All the 25 size subgroups of Table 3 has to show with the extreme value limits of minimum and maximum subgroups to be identified with the conditions of in control and out of control as presented in Table 4.

Table 4. Control Limits of Extreme Value Control Limits

This paper concludes the construction of extreme value limits, other than a normal distribution approximation, which is used in Burr Type XII Distribution through Analysis of Means (ANOM) by this suggested way.