# A Revised New Plan of quality control Based on
## Probability Theory

Chung-How Yang

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Chung-How Yang. A Revised New Plan of quality control Based on Probability Theory. Annales de l'ISUP, 1970, XIX (1), pp.107-123. hal-04083346

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<figure><img src="image_2.png" /></figure>
A REVISED NEW PLAN OF QUALITY CONTROL
BASED ON PROBABILITY THEORY [7]

(31<sup>st</sup> MAR., 1966)

BY CHUNG - HOW YANG

Nanyang University, Singapore

# I. INTRODUCTION

The Shewhart Control Charts are the most important tools of statistical quality Control. They may be used for indicating when the production process has fallen out of statistical control. As Professor Hillier's words, «Unfortunately, sufficiently accurate control limits can not be established by present methods (It means the traditional methods) ... without first inspecting a large number of subgroups.» [3] Professor Hillier had published two articles [3,4] pertaining to $\bar{X}$ and R chart control limits based on small number of subgroups. His works are based on P.B. Patnaik and the others. It is noted that Professor Hillier's works are very useful in practical purpose and are more accurate than traditional ones.

In this paper the author has proposed some new quality control charts for use with small and large samples. The proposed charts permit most accurate probability statements in theoretical viewpoint. We know that Professor Patnaik's result [5] is an excellent approximation ($\bar{R}/\sigma$ is approximately distributed as $c\chi/\sqrt{V}$ where $\chi$ is a chi-variate with V degrees of freedom and c is a scale factor) but not the absolute true one. Therefore the $\bar{X}$ chart control limits presented in this article comparing with Professor Hillier's result [3] should be slightly more efficient.

## Notation and Assumption

Let us first introduce the symbols to be used in this paper.

* $n$ denotes the size of any sample

* $k$ denotes the number of samples

* $N=nk$

* $X_i$ denotes the mean of $i^{th}$ sample, $i = 1,2,..., k$
X denotes the mean of k sample-means

$\mu$ denotes the expected value of $X_i$, $i = 1, 2, ..., k$

$S_i^2$ denotes the variance of the $i^{th}$ sample, $i = 1, 2, ..., k$

$S^2$ denotes the sample variance on the N observations

$\sigma^2$ denotes the variance of a normal population from which the samples are drawn

$$ \hat{S}^2 = \frac{\sum_{i}^{k} \left( \frac{n}{n-1} S_i^2 \right)}{k} \quad \text{denotes an unbiased estimate of } \sigma^2 $$

$$ E(S_i) = \frac{\sigma \sqrt{2\pi}}{\sqrt{n} \cdot B\left(\frac{1}{2}, \frac{n-1}{2}\right)} \quad \bar{\bar{C}}_2 \sigma, \text{ the expected value of } S_i, i = 1, 2, \dots, k $$

$$ \bar{S} = \sum_{i}^{k} S_i / k \approx E(S_i), \text{ where } \bar{\bar{C}} \text{ denotes «approximately equal to»} $$

$d_i$ denotes the number of defectives in the $i^{th}$ samples, $i = 1, 2, ..., k$

$$ p_i = d_i / n $$

$E(d_i) = d$, the expected value of $d_i$

$E(p_i) = P = d/n$, the expected value of $p_i$

$$ \bar{d} = \sum_{i}^{k} d_i / k $$

$$ \bar{p} = \frac{\bar{d}}{n} \approx P $$

$\chi^2_{\alpha(n-1)}$ denotes n-1 degrees of freedom for $\chi^2$ when the significance level is $\alpha$

$$ E_1 = \frac{\chi_{0,995}^2 (n-1)}{n} $$

$$ E_2 = \frac{\chi_{0,005}^2 (n-1)}{n} $$

$$ F_1 = \sqrt{E_1} / C_2 $$

$$ F_2 = \sqrt{E_2} / C_2 $$

$F \alpha (n-1), (k-1) (n-1)$ denotes the F value with n-1 and (k-1) (n-1) degrees of freedom when significance level is $\alpha$
t α (N-1) denotes Student's t value with N-1 degrees of freedom when significance level is α

χ² (n-1) denotes χ² variate with n-1 degrees of freedom

F(n-1, (k-1) (n-1)) denotes the F variate with n-1 and (k-1) (n-1) degrees of freedom

$$G_1 = t_{0,005 (N-1)} \sqrt{\frac{k-1}{N-1}}$$

$$G_2 = \frac{G_1}{d_2} \sqrt{1 - \frac{1}{N}}$$

d₂ denotes the correction factor for the mean range

$$H_1 = \frac{(n-1) k F_{0,995} (n-1, (k-1) (n-1))}{n [(k-1) + F_{0,005} (n-1, (k-1) (n-1))]}$$

$$H_2 = \frac{(n-1) k F_{0,005} (n-1, (k-1) (n-1))}{n [(k-1) + F_{0,005} (n-1, (k-1) (n-1))]}$$

$$I_1 = \frac{1}{P + (1-P) F_{0,005} (2(n+1) (1-P), 2(n+1) P)}$$

$$I_2 = \frac{1}{P + (1-P) F_{0,995} (2(n+1) (1-P), 2(n+1) P)}$$

$$I_3 = \frac{1}{\bar{p} + (1-\bar{p}) F_{0,005} (2(n+1) (1-\bar{p}), 2(n+1) \bar{p})}$$

$$I_4 = \frac{1}{\bar{p} + (1-\bar{p}) F_{0,995} (2(n+1) (1-\bar{p}), 2(n+1) \bar{p})}$$

Cᵢ denotes the number of defects in iᵗʰ sample, i = 1,2,..., k

E(Cᵢ) = CᵢE, the expected value of Cᵢ

$$\bar{C} = \sum_{i}^{k} C_{i} / k \approx E(C_{i})$$

Rᵢ denotes the range of iᵗʰ sample, i = 1,2,..., k

$$\bar{R} = \sum_{i}^{k} R_{i} / k$$
$C_p$ denotes the scale factor (this is Patnaik's $C$)

$\bar{R}/\sigma = \frac{C_p \chi}{\sqrt{V}}$, where $\chi$ is a chi-variate with $V$ degrees of freedom, and

$$C_p = 2,3259 \left(1 + \frac{1}{4V} + \frac{1}{32V^2} - \frac{5}{128V^3}\right)$$

$A_2^{**} = \frac{\sqrt{\frac{1}{n}(1-\frac{1}{k})}}{C_p} \cdot t_{0,005(V)}$, where $A_2^{**}$ is different from $A_2^*$ as cited in [3]

$t_{0,005(V)}$ denotes the Student's t value with $V$ degrees of freedom when significance level is $\alpha = 0.01$

$D_3^*$ denotes control factor for lower control limit of R chart when sample size is five

$D_4^*$ denotes control factor for upper control limit of R chart when sample size is five

Let us assume that all the samples are randomly drawn from the normal population except the binomial variate.

## II. CONTROL CHARTS FOR MEAN AND STANDARD DEVIATION

We assume that the production process is in statistical control. In practice, $\bar{X}_i + A_1\bar{S}_i$, the common control limits of $\bar{X}$ chart, provides an unreliable basis for indicating when the process has fallen out statistical control. A new method for setting reliable control limits is now proposed. This method amounts to replace $A_1\bar{S}_i$ by a more appropriate multiple $G_1S$ as stated below. $G_1S$ would be chosen in such a way that the control limits, $\bar{X}_i + G_1S$, would give the desired probability statements. Let

$$= \frac{\sum_{i=1}^{k} \bar{X}_i - \bar{X}}{k-1},$$

then we have two independent normal variates as follows
$$t_1 = \frac{\bar{X}_i - \mu}{\frac{\sigma}{\sqrt{n}}} \quad \text{and} \quad t_2 = \frac{\bar{X}' - \mu}{\frac{\sigma}{\sqrt{n(k-1)}}}$$

The meanings of symbols are as mentioned above. Therefore, according to the properties of the normal distribution, we obtain

$$t = \frac{(\bar{X}_i - \bar{X}') - (\mu - \mu)}{\frac{\sigma}{\sqrt{\frac{1}{n} + \frac{1}{n(k-1)}}}} = \frac{\sqrt{n}(\bar{X}_i - \bar{X})}{\sigma\sqrt{1 - \frac{1}{k}}} \quad (1)$$

where $\bar{X} \neq \bar{X}'$. The right side of (1) is distributed like normal variate $N(0; 1)$. It follows that

$$t = \frac{\bar{X}_i - \bar{X}}{\sigma \sqrt{\frac{k-1}{N}}} \quad (1a)$$

and

$$t_{(N-1)} = \frac{\bar{X}_i - \bar{X}}{S \sqrt{\frac{k-1}{N-1}}} \quad (1b)$$

The right side of (1b) is distributed as Student's t with $N-1$ degrees of freedom. Defining that

$$G_1 = t_{0,005(N-1)} \sqrt{\frac{k-1}{N-1}},$$

then we can easily write down the control limits of $\bar{X}$ and $S$ control charts as follows

$$ \left. \begin{array}{l} \text{Central line} \\ \quad \bar{X} \\ \\ \text{UC} \mathcal{L} \\ \\ \mathcal{L} \mathcal{C} \mathcal{L} \end{array} \right\} \begin{array}{l} \bar{X} + G_1 S \\ \\ \bar{X} - G_1 S \end{array} \quad (2) $$

Values of $G_1$ for selected values of $n$ and $k$ with $\alpha = 0.01$ are given in table 1.
<caption>Table 1 - G₁ table</caption>

Acceptance region : 99%

<table><thead><tr><th rowspan="2">n</th><th colspan="5">k</th></tr><tr><th>10</th><th>15</th><th>20</th><th>25</th><th>30</th></tr></thead><tbody><tr><td>2</td><td>2,133</td><td>1,915</td><td>1,897</td><td>1,876</td><td>1,863</td></tr><tr><td>3</td><td>2,535</td><td>1,518</td><td>1,509</td><td>1,506</td><td>1,504</td></tr><tr><td>4</td><td>1,303</td><td>1,299</td><td>1,295</td><td>1,292</td><td>1,291</td></tr><tr><td>5</td><td>1,150</td><td>1,150</td><td>1,150</td><td>1,151</td><td>1,151</td></tr><tr><td>6</td><td>1,037</td><td>1,045</td><td>1,047</td><td>1,048</td><td>1,049</td></tr><tr><td>8</td><td>0,892</td><td>0,897</td><td>0,900</td><td>0,903</td><td>0,905</td></tr><tr><td>10</td><td>0,793</td><td>0,798</td><td>0,802</td><td>0,805</td><td>0,808</td></tr></tbody></table>

### III. CONTROL CHARTS FOR MEAN AND RANGE

In actual practice, the grand average, $\bar{\bar{X}}$, and a certain multiple of the average range $A_2\bar{R}$, commonly are used to estimate $\mu$ and $3\sigma/\sqrt{n}$, respectively, and the control limits are set at $\bar{X} + A_2^*\bar{R}$ for taking the place of $\bar{X} + A_2^*\bar{R}$ as page 27 of [3] cited. Where $A_2^*$ is equivalent to

$$ \frac{\sqrt{\frac{k+1}{N}}}{C_p} \cdot t_{0,005(V)} \quad (3) $$

According to the author's notation mentioned above, we introduce a revised control limits of $\bar{X}$ and $\bar{R}$ charts to substitute for $\bar{X} + A_2^*\bar{R}$, the revised $\bar{X}$ and $\bar{R}$ chart control limits should be

$$ \left. \begin{array}{l} \text{Central line} \\ \text{UC}\mathcal{L} \\ \mathcal{L}\text{C}\mathcal{L} \end{array} \right\} \begin{array}{l} \bar{\bar{X}} \\ \bar{\bar{X}} + A_2^{**}\bar{R} \\ \bar{\bar{X}} - A_2^{**}\bar{R} \end{array} \quad (4) $$

where

$$ A_2^{**} = \frac{\sqrt{\frac{1}{n}(1-\frac{1}{k})}}{C_p} \cdot t_{0,005(V)} \quad (3a) $$
The relation between $A_2^*$ and $A_2^{**}$ is

$$A_2^{**} = \sqrt{\frac{k-1}{k+1}} \cdot A_2^* \quad (5)$$

Notes: From (1) and Professor Patnaik's result, $VR^2/C_p^2 \sigma^2$, we obtain

$$t_{(V)} = \frac{(\bar{X}_1 - \bar{\bar{X}}) / \sigma \sqrt{\frac{k-1}{N}}}{\sqrt{\frac{VR^2 / C_p^2 \sigma^2}{V}}} \quad i = 1, 2, \dots, k \quad (6)$$

$A_2^{**}$ is directly derived from (6). Values of $A_2^{**}$ and $A_2^*$ for selected values of k and n=5 with $\alpha = 0.01$ are given in table 2.

<caption>Table 2 - $A_2^*$ and $A_2^{**}$ Table</caption>

Acceptance region: 99%

<table><thead><tr><th rowspan="2">k</th><th colspan="2">n = 5</th></tr><tr><th>A<sub>2</sub><sup>*</sup></th><th>A<sub>2</sub><sup>**</sup></th></tr></thead><tbody><tr><td>2</td><td>0,781</td><td>0,451</td></tr><tr><td>3</td><td>0,673</td><td>0,476</td></tr><tr><td>4</td><td>0,624</td><td>0,481</td></tr><tr><td>5</td><td>0,596</td><td>0,486</td></tr><tr><td>6</td><td>0,579</td><td>0,488</td></tr><tr><td>7</td><td>0,564</td><td>0,489</td></tr><tr><td>8</td><td>0,556</td><td>0,490</td></tr><tr><td>9</td><td>0,551</td><td>0,491</td></tr><tr><td>10</td><td>0,545</td><td>0,492</td></tr><tr><td>15</td><td>0,527</td><td>0,493</td></tr><tr><td>20</td><td>0,518</td><td>0,493</td></tr><tr><td>25</td><td>0,513</td><td>0,493</td></tr><tr><td>50</td><td>0,505</td><td>0,494</td></tr><tr><td>100</td><td>0,500</td><td>0,495</td></tr><tr><td>&infin;</td><td>0,495</td><td>0,495</td></tr></tbody></table>
Entries of $A_2^*$ in this table are taken from table II of page 26 in [3].

Let us return to (2), for the practical purpose we may use the relation

$$S \approx \bar{R} / d_2 \sqrt{\frac{N}{N-1}}$$

to shift (2) into the control limits of X and R control charts as followings

$$ 
\left.
\begin{array}{l}
\text{Central line} \\
\text{UC} \mathcal{L} \\
\mathcal{L} \mathcal{C} \mathcal{L}
\end{array}
\right\} \left.
\begin{array}{l}
\bar{X} \\
\bar{X} + G_2 \bar{R} \\
\bar{X} - G_2 \bar{R}
\end{array}
\right\} \quad (7)
$$

where $G_2 = \frac{G_1}{d_2} \cdot \sqrt{1 - \frac{1}{N}}$. Values of $G_2$ for selected values of n

and k with $\alpha = 0,01$ are given in table 3.

<caption>Table 3 - $G_2$ Table</caption>

Acceptance region: 99%

<table><thead><tr><th rowspan="2">n</th><th colspan="6">k</th></tr><tr><th>10</th><th>15</th><th>20</th><th>25</th><th>30</th><th>&infin;</th></tr></thead><tbody><tr><td>2</td><td>1,842</td><td>1,669</td><td>1,657</td><td>1,646</td><td>1,637</td><td></td></tr><tr><td>3</td><td>0,892</td><td>0,887</td><td>0,884</td><td>0,884</td><td>0,884</td><td></td></tr><tr><td>4</td><td>0,625</td><td>0,625</td><td>0,625</td><td>0,625</td><td>0,625</td><td></td></tr><tr><td>5</td><td>0,490</td><td>0,491</td><td>0,492</td><td>0,493</td><td>0,493</td><td>0,495</td></tr><tr><td>6</td><td>0,406</td><td>0,410</td><td>0,411</td><td>0,413</td><td>0,413</td><td></td></tr><tr><td>8</td><td>0,311</td><td>0,313</td><td>0,315</td><td>0,316</td><td>0,318</td><td></td></tr><tr><td>10</td><td>0,257</td><td>0,258</td><td>0,260</td><td>0,262</td><td>0,263</td><td></td></tr></tbody></table>

The readers may compare the values of $A_2^{**}$ (in table 2) and $G_2$ (in table 3) when n = 5 with the same k value. From the comparison, we know that both of the results are excellent approximations.
IV. CONTROL CHARTS FOR VARIANCE AND STANDARD DEVIATION

We know that $\frac{n S_i^2}{\sigma^2}$ is a $\chi^2$ variate with $n-1$ degrees of freedom and $[k(n-1)\hat{S}^2 - n S_i^2]/\sigma^2$ is a $\chi^2$ variate with $(k-1)(n-1)$ degrees of freedom. From the theory of Analysis of variance, we may write

$$F(n-1, (k-1)(n-1)) = \frac{\frac{n S_i^2}{\sigma^2} / (n-1)}{\left[ \frac{k(n-1)\hat{S}^2 - n S_i^2}{\sigma^2} \right] / (k-1)(n-1)} \quad (8)$$

$$= \frac{n (k-1) S_i^2}{k (n-1) \hat{S}^2 - n S_i^2} \quad (9)$$

From (9), we obtain

$$S_i^2 = \frac{(n-1) k F(n-1, (k-1)(n-1))}{n [(k-1) + F(n-1, (k-1)(n-1))]} \hat{S}^2. \quad (10)$$

Using (9), we can easily write down the control limits of control charts for variance and standard deviation as follows

S² chart (for variance)

$$ \left. \begin{array}{l} \text{Central line} \\ \hat{S}^2 \\ \frac{n-1}{n} \\ \hat{H}_2 \hat{S}^2 \\ \hat{H}_1 \hat{S}^2 \end{array} \right\} \begin{array}{l} \text{UCL} \\ \text{LCL} \end{array} \quad (11) $$

S chart (for standard deviation)

$$ \left. \begin{array}{l} \text{Central line} \\ \hat{S} \\ C_2 \\ \sqrt{H_2} \hat{S} \\ \sqrt{H_1} \hat{S} \end{array} \right\} \begin{array}{l} \text{UCL} \\ \text{LCL} \end{array} \quad (12) $$
where

$$H_1 = \frac{(n-1) k F_{0,995} (n-1, (k-1) (n-1))}{n [(k-1) + F_{0,995} (n-1, (k-1) (n-1))]},$$

$$H_2 = \frac{(n-1) k F_{0,005} (n-1, (k-1) (n-1))}{n [(k-1) + F_{0,005} (n-1, (k-1) (n-1))]},$$

and

$$C_2 = \sqrt{2\pi} / \sqrt{n} \cdot \beta \left( \frac{1}{2}, \frac{n-1}{2} \right).$$

As k-1 approaches infinity

$$ \left. \begin{array}{l} H_2 \hat{S}^2 \rightarrow E_2 \hat{S}^2 \\ H_1 \hat{S}^2 \rightarrow E_1 \hat{S}^2 \end{array} \right\} , \qquad (13) $$

and

$$ \left. \begin{array}{l} \sqrt{H_2} \hat{S} \rightarrow F_2 \bar{S} \\ \sqrt{H_1} \hat{S} \rightarrow F_1 \bar{S} \end{array} \right\} . \qquad (14) $$

where $\hat{S} \approx \bar{S} / C_2$. So that when $k-1 \to \infty$ or is large enough,

$$ \left. \begin{array}{l} \text{Central line} \\ \mathcal{U} \mathcal{C} \mathcal{L} \\ \mathcal{L} \mathcal{C} \mathcal{L} \end{array} \right\} \begin{array}{c} \frac{n-1}{n} \hat{S}^2 \\ H_2 \hat{S}^2 \\ H_1 \hat{S}^2 \end{array} \rightarrow \left. \begin{array}{l} \text{Central line} \\ \mathcal{U} \mathcal{C} \mathcal{L} \\ \mathcal{L} \mathcal{C} \mathcal{L} \end{array} \right\} \begin{array}{c} \frac{n-1}{n} \hat{S}^2 \\ E_2 \hat{S}^2 \\ E_1 \hat{S}^2 \end{array} \quad (15) $$

where $\rightarrow$ denotes «approaches». Similarly,

$$ \left. \begin{array}{l} \text{Central line} \\ \mathcal{U} \mathcal{C} \mathcal{L} \\ \mathcal{L} \mathcal{C} \mathcal{L} \end{array} \right\} \left. \begin{array}{c} C_2 \hat{S} \\ \sqrt{H_2} \hat{S} \\ \sqrt{H_1} \hat{S} \end{array} \right\} \rightarrow \left. \begin{array}{l} \text{Central line} \\ \mathcal{U} \mathcal{C} \mathcal{L} \\ \mathcal{L} \mathcal{C} \mathcal{L} \end{array} \right\} \begin{array}{c} \bar{S} \\ F_2 \bar{S} \\ F_1 \bar{S} \end{array} \quad (16) $$

In practice, $B_3 \bar{S}$ and $B_4 \bar{S}$ are commonly used as estimates of S chart control limits, now we use (12) or (16) to take the place of the said limits.
In (15) and (16),

$$E_1 = \frac{\chi_{0,995}^2 (n-1)}{n},$$

$$F_1 = \sqrt{E_1}/C_2,$$

$$E_2 = \frac{\chi_{0,005}^2 (n-1)}{n},$$

$$F_2 = \sqrt{E_2}/C_2,$$

and

$$C_2 = \sqrt{2\pi} / \sqrt{n} \cdot B\left(\frac{1}{2}, \frac{n-1}{2}\right).$$

Values of $H_1$, $H_2$, $\sqrt{H_1}$ and $\sqrt{H_2}$ are given in table 4 for selected values of n and k, with AR (acceptance region) = 99%.

<caption>Table 4 - $H_1$, $H_2$, $\sqrt{H_1}$ and $\sqrt{H_2}$ Table</caption>

Acceptance region: 99%

<table><thead><tr><th rowspan="2">n</th><th colspan="4">k = 11</th><th colspan="4">k = 21</th></tr><tr><th>H<sub>1</sub></th><th>H<sub>2</sub></th><th>√H<sub>1</sub></th><th>√H<sub>2</sub></th><th>H<sub>1</sub></th><th>H<sub>2</sub></th><th>√H<sub>1</sub></th><th>√H<sub>2</sub></th></tr></thead><tbody><tr><td>2</td><td>0,0000227</td><td>3,090</td><td>0,0048</td><td>1,76</td><td>0,0000212</td><td>3,487</td><td>0,0046</td><td>1,87</td></tr><tr><td>3</td><td>0,00367</td><td>3,016</td><td>0,061</td><td>1,74</td><td>0,00351</td><td>3,258</td><td>0,059</td><td>1,81</td></tr><tr><td>4</td><td>0,0194</td><td>2,836</td><td>0,139</td><td>1,68</td><td>0,0186</td><td>3,012</td><td>0,136</td><td>1,74</td></tr><tr><td>5</td><td>0,0443</td><td>2,68</td><td>0,210</td><td>1,64</td><td>0,0429</td><td>2,83</td><td>0,207</td><td>1,68</td></tr><tr><td>6</td><td>0,0718</td><td>2,55</td><td>0,268</td><td>1,60</td><td>0,0698</td><td>2,67</td><td>0,264</td><td>1,63</td></tr><tr><td>7</td><td>0,1022</td><td>2,44</td><td>0,320</td><td>1,56</td><td>0,0944</td><td>2,54</td><td>0,315</td><td>1,59</td></tr></tbody></table>

Values of $E_1$, $E_2$, $F_1$ and $F_2$ are given in table 5 for selected values of n, with AR = 99%. Where the number of samples k becomes large enough.
<caption>Table 5 - E₁, E₂, F₁ and F₂ Table</caption>

Acceptance region : 99 %

<table><thead><tr><th>n</th><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>F<sub>1</sub></th><th>F<sub>2</sub></th></tr></thead><tbody><tr><td>2</td><td>0,00001964</td><td>3,9397</td><td>0,0078</td><td>3,518</td></tr><tr><td>3</td><td>0,00334</td><td>3,6866</td><td>0,080</td><td>2,653</td></tr><tr><td>4</td><td>0,0179</td><td>3,2095</td><td>0,168</td><td>2,245</td></tr><tr><td>5</td><td>0,0414</td><td>2,972</td><td>0,241</td><td>2,051</td></tr><tr><td>6</td><td>0,0686</td><td>2,792</td><td>0,302</td><td>1,923</td></tr><tr><td>7</td><td>0,0965</td><td>2,650</td><td>0,350</td><td>1,833</td></tr><tr><td>8</td><td>0,1237</td><td>2,536</td><td>0,390</td><td>1,764</td></tr><tr><td>9</td><td>0,149</td><td>2,439</td><td>0,422</td><td>1,709</td></tr><tr><td>10</td><td>0,173</td><td>2,359</td><td>0,451</td><td>1,665</td></tr><tr><td>11</td><td>0,196</td><td>2,290</td><td>0,476</td><td>1,627</td></tr><tr><td>13</td><td>0,236</td><td>2,177</td><td>0,516</td><td>1,567</td></tr><tr><td>16</td><td>0,288</td><td>2,050</td><td>0,564</td><td>1,504</td></tr><tr><td>21</td><td>0,354</td><td>1,905</td><td>0,617</td><td>1,431</td></tr><tr><td>25</td><td>0,395</td><td>1,822</td><td>0,648</td><td>1,391</td></tr></tbody></table>

V. CONTROL CHART FOR RANGE

In accordance with Professor Hillier's work [4] if D₃* and D₄* are control factors for lower and upper control limits, respectively, of R chart when sample size is five, then we have

$$ \left. \begin{array}{l} \text{Central line} \\ \text{UC}\bar{L} \\ \text{L}\bar{C}\bar{L} \end{array} \right\} \left. \begin{array}{l} \bar{R} \\ D_{4}^{*} \bar{R} \\ D_{3}^{*} \bar{R} \end{array} \right\} \quad (17) $$

Values of D₃* and D₄* for selected values of k and n = 5 with α = 0,005 (one tailed as stated in [4] are given in table 6).
<caption>Table 6 - D₃* and D₄* Table #</caption>

Acceptance region : 99 %

<table><thead><tr><th rowspan="2">k</th><th colspan="2">n = 5</th></tr><tr><th>D<sub>3</sub>*</th><th>D<sub>4</sub>*</th></tr></thead><tbody><tr><td>2</td><td>0,2189</td><td>3,194</td></tr><tr><td>3</td><td>0,2248</td><td>2,758</td></tr><tr><td>4</td><td>0,2280</td><td>2,573</td></tr><tr><td>5</td><td>0,2300</td><td>2,468</td></tr><tr><td>6</td><td>0,2305</td><td>2,400</td></tr><tr><td>7</td><td>0,2323</td><td>2,354</td></tr><tr><td>8</td><td>0,2331</td><td>2,320</td></tr><tr><td>9</td><td>0,2337</td><td>2,294</td></tr><tr><td>10</td><td>0,2342</td><td>2,274</td></tr></tbody></table>

\# Entries in this table are taken from table 3 of page in [4].

Commonly D₃, R̄ and D₄, R̄ are used as estimates of R chart control limits, now we replace D₃ and D₄ by more accurate constants D₃* and D₄*.

## VI. CONTROL CHART FOR FRACTION DEFECTIVES

In practice, $\bar{p} + 3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$ are commonly used as estimates of p

chart control limits, now we know that this common control limits of p chart also provides an unreliable basis for indicating when the process has fallen out of statistical control.

Suppose that the beta function is

$$ \beta p = \frac{n!}{(m-1)! (n-m)!} p^{m-1} (1-p)^{n-m}, \quad 1 \le m \le n \quad (18) $$

the density function of p is
$$dp' = \frac{n!}{(m-1)! (n-m)!} p^{m-1} (1-p)^{n-m} dp. \quad (19)$$

Now let

$$p = \frac{2m}{2m + 2(n-m+1)} F \quad \text{or} \quad F = \frac{2m}{2(n-m+1)} \cdot \frac{1-p}{p}$$

From (19) we obtain

$$dp' = \frac{K \cdot F^{\frac{1}{2}(V_1-2)}}{(V_1 F + V_2)^{\frac{1}{2}(V_1 + V_2)}} \quad (20)$$

Where K is a constant, $V_1 = 2(n-m+1)$ and $V_2 = 2m$.

Thus we have [2]

$$p' = \frac{n!}{(m-1)! (n-m)!} \int_{0}^{p} p^{m-1} (1-p)^{n-m} dp \quad (21)$$

$$= \sum_{\gamma=m}^{n} \binom{n}{\gamma} p^{\gamma} q^{n-\gamma} \quad (21a)$$

n and m are positive integers $p+q=1$

From (18) to (21a), we deduce [1]

$$F_{(2(n-m+1), 2m)} = \frac{2m}{2(n-m+1)} \cdot \frac{1-p}{p} \quad (22)$$

From (21), the expected value of p is

$$P = \frac{1}{\beta(m, n-m+1)} \int_{0}^{1} p^{m} (1-p)^{n-m} dp = \frac{m}{n+1}. \quad (23)$$

Substituting (23) into (22), we obtain

$$F_{(2(n+1), p, 2(n+1), (1-p))} = \frac{1-p}{p} \cdot \frac{p}{1-p} \quad (24)$$
The right side of (24) is distributed like F with 2(n+1) P and 2(n+1) (1-P) degrees of freedom. Now we have a new control limits of p chart as follows

P known

$$ \left. \begin{array}{ll} \text{Central line} & \text{P} \\ U \mathcal{C} \mathcal{L} & I_2 \text{P} \\ \mathcal{L} \mathcal{C} \mathcal{L} & I_1 \text{P} \end{array} \right\} , \qquad (25) $$

P Unknown

$$ \left. \begin{array}{ll} \text{Central line} & \bar{p} \\ U \mathcal{C} \mathcal{L} & I_4 \bar{p} \\ \mathcal{L} \mathcal{C} \mathcal{L} & I_3 \bar{p} \end{array} \right\} . \qquad (26) $$

Where

$$ I_1 = \frac{1}{\text{P} + (1-\text{P}) \text{F}_{0,005} (2(n+1)(1-\text{P}), 2(n+1)\text{P})}, $$

$$ I_2 = \frac{1}{\text{P} + (1-\text{P}) \text{F}_{0,995} (2(n+1)(1-\text{P}), 2(n+1)\text{P})}, $$

$$ I_3 = \frac{1}{\bar{p} + (1-\bar{p}) \text{F}_{0,005} (2(n+1)(1-\bar{p}), 2(n+1)\bar{p})}, $$

and

$$ I_4 = \frac{1}{\bar{p} + (1-\bar{p}) \text{F}_{0,995} (2(n+1)(1-\bar{p}), 2(n+1)\bar{p})}. $$

## VII. C CHART

We know that $C \pm 3\sqrt{C}$ are commonly used as estimates of C
chart control limits. This common control limits also provides an unreliable
basis.

As n approaches infinity, and p and P approach zero, then (24)
approaches

$$ \chi_{(2 \mathcal{C} \mathcal{E})}^2 = 2 \mathcal{C} \qquad (27) $$
Where C, the number of defects, has the Poisson distribution, and

$$C_E = nP \approx (n+1)P,$$

$$C = np \approx (n+1)p.$$

The right side of (27) is distributed like $\chi^2$ with $2C_E$ degrees of freedom, where $C_E$ is the expected value of C.

Therefore the control limits of C chart should be

$C_E$ known

$$
\left.
\begin{array}{ll}
\text{Central line} & C_E \\
\text{UC}\L & \frac{1}{2} \chi_{0,005 (2C_E)}^2 \\
\L C\L & \frac{1}{2} \chi_{0,995 (2C_E)}^2
\end{array}
\right\}
\qquad (28)
$$

$C_E$ unknown

$$
\left.
\begin{array}{ll}
\text{Central line} & \bar{C} \\
\text{UC}\L & \frac{1}{2} \chi_{0,005 (2\bar{C})}^2 \\
\L C\L & \frac{1}{2} \chi_{0,995 (2\bar{C})}^2
\end{array}
\right\}
\qquad (29)
$$

where

$$\bar{C} = \sum_{i}^{k} C_{i} / k \approx C_{E}.$$
REFERENCES

[1] ANDERSON (T.W.) (Chief Editor), *Wald: Selected Papers in Statistics and Probability*, Stanford University Press, (1957), p. 256.

[2] BAYES (T.) An Essay Toward Solving a Problem in the Doctrine of Chances, *Phil. Trans. Roy. Soc. Vol. 53* (1763), pp. 370-418. Re-produced by the Graduate School, United States Dept. of Agriculture, with an editorial preface and remarks by W. E. Deming and Comments by E. C. Molina, Hafner Publishing Co. (1963), pp. 11-12.

[3] HILLIER (F.S.) $\bar{X}$ Chart Control Limits Based on a small Number of subgroups, *Industrial Quality Control, Vol. XX, N° 8* (Feb., 1964), pp. 24-29.

[4] HILLIER (F.S.) R Chart Control Limits Based on Small Number of subgroups, *Technical Raport N° 83*, Contract Nonr - 225 (53) (NR - 042 - 002), Stanford University, (Oct. 4, 1965), pp. 1 - 18.

[5] PATNAIK (P.B.) The Use of Mean Range as an Estimator of Variance in Statistical Tests, *Biometrika*. Vol. 37 (1950), pp. 78 - 87.

[6] SHEWHART (W.A.) With the editorial assistance of W. Edwards Deming
*Statistical Method from the viewpoint of Quality Control*, Graduate
School, U. S. Dept. of Agriculture, Washington D.C., 1939.

[7] YANG (C.H.) *A New Plan of Quality Control Based on Probability Theory* (discussed with Professor Dudley J. Cowden on Feb., to March., 1965).

<figure><img src="image_3.png" /></figure>

Reçu en 1970
Nanyang University,
Jurong Road,
Singapore 22.