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Konscp untuk bngan l<cndali X dan hugan licndirli X trita ul<urln sullgru;lnva sanrir

Bagan kendali X dan bagan kcrrdali J binr.tryo clipandang scbagai clua teknik l'cngendalian l)roscs Statistis lang nrcnrrlrki konsep yang berbeda. Dalam tulisan ini kami usulkan dua buah proposisi yang lneniuugkinkan mcmandang kcdua bagan kcndali itu dari satu konsep yang salna. Proposisi pertama adalah uutuk tahap as'al pernbuatan bagan kendali (start-up stage) dan lang lain untuk pengendalian proses nrelalui pengamatan bcrikutnya. Keuutungan lain dari konsep itu tcrlctak pada batas kendali lang tidlk lagi harus ditentukan rnelalui n.retode pendckatan scpcrti yang biasa dikcnrukakan dalanr bcrbagai pustaka lcntiurg pcugcrrdaliarr proses. Metode eksak akan diperkenalkan.

Katakunci:bagankendali.\.,bagctttlicnda!ir'tnl,,,paltttlpcntbtntanbagrnkettdaIi(stot,t.l|P.s|ttge),pcttgt:t batas kendali.

1 Introduction

X Chart and X-Bar Chart'are usually sccn as tl\'o techniques inSPC rvith different concepts. XChart or individual chart is usuallv used in conjunction u'ilh rnoving range and X-Bar Chart is oflcn uscd togethcr rvith standard deviation. Fuflherntorc, in thosc tu'o charts, it is customary lo use limitirrg clistribution to detennine the controlimits either in start-u1l stag,e or in process control for fu(ure obsen'ations.

In this paper rve try to develop a urrilf irrg conccpt for constructing:

• l. X Chart based on the standard dcvialion .r of individuals.
• 2. X-Bar Chart in conjunction rvith slandardo'ialion of subgroup means for spccial casc: u,hen strbgroup sizes are equal.

With this concept, the exact distribtrtions for dctccting controlinrits rvill be obtaincd. For thrs putposc. itt

scclion 2 u'c prcscnt a unificd point of vicu, of individrral charl or X Chart and X-Bar Chart. This includcs thc control lirnits citlrcr in stult-rrp stlrgc or in proocss control for fulurc obscn,alions in X Chart. Scction 3 rr ill be dcvotcd to lhc dclcrnrinlliorr ol'sinrilar conlrol lirnits in X-Bar Clurrl. bv using nrathcnratical rcason dcvclopcd in thc prcviorrs scction.

2 X Chart

I ncliviclual cha rl or X Chart is usuallv trsed irr conjunction uillr rnoving range. Tlris can bc sccn rn un1' litcraturc. o'cn irt thc urosl rcccnt oncs. srrclr its Badavas (1993). Doty (1991) and Srtrith (1995). In this cirsc. its controlirnits arc

LCL = \[\overline{X} - k\]. \(\overline{MR}\) and UCL = \(\overline{X} + k\). \(\overline{MR}\) where k is given by an approximation method.

In this section we try to develop X Chart based on the standard deviation s of individuals which will provide us with exact control limits. Another advantage is that this unify the concept of X Chart and X-Bar Chart. First we introduce the following proposition.

Proposition 1. Let \(X_1\), \(X_2\), ..., \(X_m\) be random sample from normal distribution \(N(\mu, \sigma^2)\). If \(\overline{X}\) and \(s^2\) represent respectively sample mean and sample variance, then we have

\[\frac{m}{(m-1)^2} \frac{(X_i - \overline{X})^2}{s^2} \sim \text{Beta}\left(\frac{1}{2}, \frac{m-2}{2}\right)\] for all i = 1, 2, ..., m.

Proof.

We know that

\[\frac{(m-1)s^{2}}{\sigma^{2}} = \sum_{k=1}^{m} \frac{(X_{k} - \overline{X})^{2}}{\sigma^{2}} \sim \chi^{2}_{(m-1)}\] which means that it follows chi-square distribution with (m-1) degrees of freedom.

On the other hand, for all i = 1, 2, ..., m we have

\[(X_i - \overline{X}) \sim N(0, \frac{m-1}{m} \sigma^2)\]

Consequently,

\[\frac{m}{m-1} \frac{(X_i - \overline{X})^2}{\sigma^2} \sim \chi^2_{(1)}\]

Now we write the statistic \(\frac{(X_i - \overline{X})^2}{s^2}\) as follows

\[\frac{(X_i - \overline{X})^2}{s^2} = (m-1)\frac{\frac{(X_i - \overline{X})^2}{\sigma^2}}{\sum_{k=2}^{m} \frac{(X_k - \overline{X})^2}{\sigma^2}}\] or,

\[\frac{(X_{i} - \bar{X})^{2}}{s^{2}} = \frac{(m-1)^{2}}{m} \frac{\frac{m}{(M-1)} \frac{(X_{i} - \bar{X})^{2}}{\sigma^{2}}}{\sum_{k=1}^{m} \frac{(X_{k} - \bar{X})^{2}}{\sigma^{2}}}\]

But the denominator equals

\[\frac{m}{m-1} \frac{(X_i - \overline{X})^2}{s^2} + \sum_{k=1}^m A_k \frac{(X_k - \overline{X})^2}{\sigma^2}\] where \(A_k = -\frac{1}{m-1}\) for k = i and \(A_k = 1\) otherwise. Now we obtain the following expression

\[\frac{(X_{i} - \overline{X})^{2}}{s^{2}} = \frac{(m-1)^{2}}{m} = \frac{\frac{m}{(m-1)} \frac{(X_{i} - \overline{X})^{2}}{\sigma^{2}}}{\frac{m}{m-1} \frac{(X_{i} - \overline{X})^{2}}{\sigma^{2}} + \sum_{k=1}^{m} A_{k} \frac{(X_{k} - \overline{X})^{2}}{\sigma^{2}}}\] or,

\[\frac{m}{(m-1)^2} \frac{(X_i - \bar{X})^2}{s^2} = \frac{\frac{m}{(m-1)} \frac{(X_i - \bar{X})^2}{\sigma^2}}{\frac{m}{m-1} \frac{(X_i - \bar{X})^2}{\sigma^2} + \sum_{k=1}^{m} \frac{(X_k - \bar{X})^2}{\sigma^2}}\]

We recognize that the numerator is distributed as \(\chi^2_{(1)}\). Consequently, the second term of denominator is distributed as \(\chi^2_{(m-2)}\). If the numerator and the denominator are divided by 2, then now the numerator is distributed as Gamma \((\frac{1}{2}, 1)\) and the second term of denominator is distributed as Gamma \((\frac{m-2}{2}, 1)\).

Hence the proof is done.

In practice \(X_1\), \(X_2\), ..., \(X_m\) represent individual random observations. Its realizations are used as historical data in start-up stage. Hence, in this stage \(X_i\) and \(\overline{X}\) are not independent. The two statistics \(X_i\) and \(s^2\) are also not independent. But if \(X_f\) represents future observation, then \(X_f\), \(\overline{X}\) and \(s^2\) are independent. This means that control chart in start-up stage and control chart in process control for future observations are different.

2.1 Start-up Stage

In this stage, the realization of \(X_1\), \(X_2\), ..., \(X_m\) are considered as historical data. From the above proposition, we know that

\[\frac{(X_i - \overline{X})^2}{s^2} \sim \frac{(m-1)^2}{m} \operatorname{Beta}\left(\frac{1}{2}, \frac{m-2}{2}\right)\]

This distribution determines the exact control limits in start-up stage in X Chart. In fact, those exact control limits are

LCL = \[\overline{X}\] - A.s and
UCL = \(\overline{X}\) + A.s where \[A^2 = \frac{(m-1)^2}{m} Beta \left( (1-\alpha), \frac{1}{2}, \frac{m-2}{2} \right)\] and

Beta \[(1 - \alpha), \frac{1}{2}, \frac{m-2}{2}\] is \((1-\alpha)\)-quantile of beta

distribution with parameters \[\frac{1}{2}\] and \(\frac{m-2}{2}\)

Multivariate version of Proposition 1 can be seen in Tracy et al (1992) and Nomikos et al (1995). Its proof can be traced in Gnanadesikan and Kettenring (1972).

2.2 Process Control for Future Observation

Let again \(X_1\), \(X_2\), ..., \(X_m\) be random sample from normal distribution \(N(\mu, \sigma^2)\) used in start-up stage and \(\overline{X}\) and \(s^2\) represent respectively its sample mean and sample variance. If \(X_f\) represents future observation, then \(X_f\) and \(X_1\), \(X_2\), ..., \(X_m\) are independent. It is so between \(X_f\), \(\overline{X}\) and \(s^2\). Consequently, we have the following proposition which can be used to determine control limits in process control for future observations.

Proposition 2. Let again \(X_1\), \(X_2\), ..., \(X_m\) be random sample from normal distribution \(N(\mu, \sigma^2)\) used in startup stage and \(\overline{X}\) and \(s^2\) represent respectively its sample mean and sample variance. If \(X_f\) represents future observation, then

\[\frac{m}{m+1} = \frac{(X_f - \overline{X})^2}{s^2} \sim F_{1,(m-1)}\]

Proof.

The fact that \(X_f\) and \(\overline{X}\) are independent implies that

\[(X_f - \overline{X}) \sim N(0, \frac{m+1}{m} \sigma^2)\] and hence,

\[\frac{m}{m+1} \frac{(X_f - \overline{X})^2}{\sigma^2} \sim \chi^2(1)\]

Now consider the following expression

\[\frac{m}{m+1} \frac{(X_f - \overline{X})^2}{\sigma^2} = \frac{\frac{m}{m+1} \frac{(X_f - \overline{X})^2}{\sigma^2}}{\frac{s^2}{\sigma^2}}\]

The numerator is distributed as \(\chi^2_{(1)}\) and the denominator is distributed as \(\chi^2_{(m+1)}\) divided by its degree of freedom. Furthermore \(X_f\), \(\overline{X}\) and \(s^2\) are independent which implies that numerator and denominator are independent. Hence we proved the proposition.

Corollary

\[\frac{(X_f - \overline{X})}{s} \sqrt{\frac{m}{m+1}} \sim \mathbf{t}_{(m-1)}^{-1}\]

Based on Proposition 2 and its corollary, control limits in process control for future observations are determined as follows.

\[LCL = \overline{X} - B.s \text{ and}\] \[UCL = \overline{X} + B.s\] where B is the \((1 - \frac{\alpha}{2})\)-quantile of student-t distribution with (m-1) degree of freedom.

3 X-bar chart when subgroup sizes are equal n > 1

X-Bar Chart is often used in conjunction with standard deviation. In order to determine its control limits, it is customary to use the limiting distribution (see Badavas (1993). Doty (1991), and Smith (1995)). Here we try to identify the exact distribution which will be applied to calculate those control limits. Now suppose that m represents the number of subgroups and its sizes are equal n > 1. If \(X_{ij}\) is the j-th item in i-th subgroup; \(i = 1, 2, \ldots, m\), \(j = 1, 2, \ldots, n\) and

\[\overline{X}_i = \frac{1}{n} \sum_{j=1}^n X_{ij}\]; sample mean in i-th subgroup

\[\overline{\overline{X}} = \frac{1}{m} \sum_{i=1}^{m} \overline{X}_{i}\]; grand mean and \[s^2 = \frac{1}{m-1} \sum_{i=1}^{m} (\overline{X}_i - \overline{X})^2\] then, according to Proposition 1, we have

\[\frac{m}{(m-1)^2} \frac{(\overline{X}_i - \overline{\overline{X}})^2}{s^2} \sim \mathbf{Beta}\left(\frac{1}{2}, \frac{m-2}{2}\right)\] for all i = 1, 2, ..., m.

Suppose that those m subgroups are used in start-up stage. In this stage \(\overline{X}_i\) and \(\overline{\overline{X}}\) are not independent. It is so between \(\overline{X}_i\) and \(s^2\). If \(\overline{X}_f\) represents the sample mean of future subgroup, then \(\overline{X}_f\), \(\overline{\overline{X}}\) and \(s^2\) are independent. Hence the start-up stage and the process control for future subgroups are as follows.

3.1 Start-up stage

In this stage, the exact control limits are

\[LCL = \overline{\overline{X}} + A.s \text{ and}\] \[UCL = \overline{\overline{X}} + A.s\] where \(A^2\) is given in section 2.1.

3.2 Process control for future observation

Let again \(\overline{X}_i\) be the sample mean of i-th subgroup; \(i = 1, 2, \ldots\), m used in start-up stage, \(\overline{\overline{X}}\) be the grand mean and \(s^2\) be the variance of subgroup means. Then, according to Proposition 2, we have

\[\frac{m}{m+1} \frac{(\overline{X}_i - \overline{\overline{X}})^2}{s^2} \sim \mathrm{F}_{1,(m-1)}\]

Corollary

\[\frac{(\overline{X}_f - \overline{\overline{X}})}{s} \sqrt{\frac{m}{m+1}} \sim \mathfrak{t}_{(m-1)}\]

This corollary gives us the control limits in process control for future subgroups

\[LCL = \overline{\overline{X}} - B.s \text{ and}\] \[UCL = \overline{\overline{X}} + B.s\] where B is the \((1 - \frac{\alpha}{2})\)-quantile of student-t distribution with (m-1) degree of freedom.

4 Concluding remarks

X Chart and X-Bar Chart can be seen as having the same concept based on Proposition 1 for start-up stage and Proposition 2 for process control for future

observations. According to these propositions, the control limits in those two charts can be determined through exact distributions and not by an approximation method anymore. This is an advantage of those propositions.

5 Acknowledgment

I am very grateful to the anonymous referees for their valuable comments.