# A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes

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## Abstract

**:**

## 1. Introduction

## 2. Design of the Enhanced HEWMA-p Chart

- Step 1: Select a random sample of size $n$(${X}_{1},\dots ,{X}_{n}$) from the process at time t. Compute ${V}_{t}$ using (3) and $HEWM{A}_{{p}_{t}}$ using (5).
- Step 2: The process is declared to be as out-of-control if $HEWM{A}_{{p}_{t}}\ge UCL$ or $HEWM{A}_{{p}_{t}}\le LCL$ and to be in-control if $LCL<HEWM{A}_{{p}_{t}}<UCL$, here LCL and UCL show the lower control limit and upper control limit.

## 3. The Average Run Length of Enhanced HEWMA-p Control Chart

_{v0}) to be a specified value, usually 370. The ARL represents the expected number of samples until a control chart signals. The proposed control chart comprises of two control coefficients, ${k}_{1}$ and ${k}_{2}$, which are obtained by considering the desired in-control ARL. Once the coefficients ${k}_{1}$ and ${k}_{2}$ are determined, the control limits of the enhanced HEWMA-p control chart are obtained and the out-of-control ARLs (ARL

_{v1}) can be obtained according to various values of shift in proportion, p

_{v1}= c p

_{v0}, c ≠ 1, and 0 < p

_{v1}≤ 1. We use the following Monte Carlo simulation procedure to compute control coefficients ${k}_{1}$ and ${k}_{2}$, and to calculate the out-of-control ARL (ARL

_{v1}) under a specified n, p

_{v0}, ${\lambda}_{1},{\lambda}_{2}$ and ARL

_{v0}values.

- Step 1. Setting specified values of n, p
_{v0}, ${\lambda}_{1},{\lambda}_{2},$ and ARL_{v0}. - Step 2. Evaluation of proposed control chart coefficients ${k}_{1}$ and ${k}_{2}$ for in-control process
- 2.1. Generate 10,000 possible values of control chart coefficients ${k}_{1}$ and ${k}_{2}$.
- 2.2. When the process is in-control, from a binomial distribution with the in-control parameters $0.5n\text{}\mathrm{and}\text{}{p}_{v0}.$ a random sample of size 2000 is generated, i.e., ${V}_{t}~binomial\left(0.5n,{p}_{v0}\right)$ at time t.
- 2.3. The enhanced HEWMA-p statistic HEWMA-p is computed for each subgroup of size 2000.
- 2.4. The proposed statistic HEWMA-p is plotted and in-control if $LCL\le HEWM{A}_{p}\le UCL;$ go to step 2.5 and the run length for out of control process is noted.
- 2.5. Repeat 10,000 times steps 2.2 through 2.3, to compute run lengths. If the average of these run lengths (ARLs) is equal to the specified ARL
_{v0}note the corresponding values of ${k}_{1}$ and ${k}_{2}$, and move to step 3, otherwise select other possible values of ${k}_{1}$ and ${k}_{2}$, and repeat the procedure from steps 2.2. - Step 3 Evaluation of ARL
_{v1}for proposed control chart when the process is shifted - 3.1. Let the out-of-control proportion, p
_{v1}, be a proportion of the in-control proportion, p_{v0}. That is, p_{v1}= c p_{v0}, c ≠ 1, and 0 < p_{v1}≤ 1, where c is the amount of shift in the process proportion, p_{v0}. - 3.2. From binomial distribution, with the in-control parameters, $0.5n\text{}\mathrm{and}\text{}{p}_{v1}$, a random sample of size 2000 is generated, i.e., ${V}_{t}~binomial\left(0.5n,{p}_{v1}\right)$ at time t.
- 3.3. The Enhanced HEWMA-p Statistic HEWMA-p is Computed for Each Subgroup of Size 2000.
- 3.4. Using the Values of ${k}_{1}$ and ${k}_{2}$, the proposed statistic HEWMA-p is plotted and in-control if $LCL<HEWM{A}_{{p}_{t}}<UCL;$ go to step 3.5 and the run length for out of control process is noted.
- 3.5. Repeat 10,000 times steps 3.2 through 3.3, to compute run lengths. The average of run length (ARL
_{v1}) and standard error of run length (SERL_{v1}) for each specified amount of shift is computed.

_{v0}= 0.1, ${\lambda}_{1}=0.2,\text{}\mathrm{and}\text{}{\lambda}_{2}$ = 0.2 with ARL

_{v0}≈ 370. Table 2 presents ARL

_{v1}and SERL

_{v1}values (in second row corresponding to each n value) for p

_{v1}= 0.025 (0.025) 0.200 at n = 8 (1) 30, p

_{v0}= 0.1, ${\lambda}_{1}=0.2,\text{}\mathrm{and}{\lambda}_{2}$ = 0.2 with ARL

_{v0}≈ 370. In Table 3, we present control chart coefficients ${k}_{1}$ and ${k}_{2}$, and corresponding upper and lower control limits of the enhanced HEWMA-p control chart for n = 8 (1) 30, p

_{v0}= 0.3, ${\lambda}_{1}=0.2,\text{}\mathrm{and}\text{}{\lambda}_{2}$ = 0.2 with ARL

_{v0}≈ 370. Table 4 presents ARL

_{v1}and SERL

_{v1}values (in second row corresponding to each n value) for p

_{v1}= 0.200 (0.025) 0.400 at n = 8 (1) 30, p

_{v0}= 0.3, ${\lambda}_{1}=0.2,\text{}\mathrm{and}\text{}{\lambda}_{2}$ = 0.2 with ARL

_{v0}≈ 370.

- 1. If n is increased, there is a decrease in ARL
_{v1}and SERL_{v1}values, as we expected. For example, for 0.5n = 4 and p_{v1}= 0.05 from Table 2 we have ARL_{v1}= 107.03 and SERL_{v1}= 0.8830, whereas if 0.5n = 15, we have ARL_{v1}= 18.12 and SERL_{v1}= 0.0921. We also observed a similar trend from Table 4. - 2. The ARL
_{v1}and SERL_{v1}values decrease when p_{v1}is far away from p_{v0}. - 3. The ARL
_{v1}and SERL_{v1}values decrease more rapidly as c increases rather than it decreases. For example, for 0.5n = 4 and p_{v1}= 0.05 (c = 0.5) from Table 2 we have ARL_{v1}= 107.03 and SERL_{v1}= 0.8830, whereas if p_{v1}= 0.15 (c = 1.5), we have ARL_{v1}= 47.74 and SERL_{v1}= 0.4055. We also observed a similar trend from Table 4.

## 4. Comparative Study

_{v1}for the proposed control chart as well as control charts given by [4,14] in Table 5 when in-control ARL

_{v0}≈ 370.

_{v1}as compared to the existing two control charts. For example, when 0.5n = 6, ${\lambda}_{1}={\lambda}_{2}=\lambda $ = 0.2, p

_{v0}= 0.3, p

_{v1}= 0.4 the proposed control chart gives ARL

_{v1}is 27.30, the ARL

_{v1}from the two existing control charts are 31.36 and 34.20, respectively. Thus, the proposed control chart performs better than the existing control charts.

_{v1}s comparison between the chart proposed by [4,14] for $\lambda $ = 0.2 and enhanced HEWMA-p control chart for ${\lambda}_{1}$ = 0.2, ${\lambda}_{2}$ = 0.2. Figure 1 depicts the ARL

_{v1}profile comparison at p

_{v0}= 0.1 and p

_{v1}= 0.2 for different values of n under HEWMA-p chart and two existing charts. From Figure 1, we noticed that ARL

_{v1}values of enhanced HEWMA-p control chart are smaller than in the two existing control charts. Hence, our proposed enhanced HEWMA-p control chart performed well as compared with existing charts.

## 5. Example

_{v0}≈ 370 are UCL = 0.4454 and LCL = 0.1963.

_{t}and the monitoring statistic $HEWM{A}_{{p}_{t}}={\lambda}_{1}EWM{A}_{{p}_{t}}+\left(1-{\lambda}_{1}\right)HEWM{A}_{{p}_{t-1}}$ where ${EWMA}_{{p}_{t}}={\lambda}_{2}{V}_{t}/0.5n+\left(1-{\lambda}_{2}\right)EWM{A}_{{p}_{t-1}}$ at time t, t = 1, 2, …, 10, were computed. The corresponding enhanced HEWMA-p control chart detected out-of-control variance signals from the third sample onward (samples 3–10 on the enhanced HEWMA-p control chart) (Figure 2). By comparing Figure 2 with the chart in [14], it can be seen that the existing chart indicated a shift at the 4th sample. Therefore, the proposed chart was more efficient in detecting a shift in the process as compared to existing chart of Yang and Arnold [14]. The same performance was also shown by the results in Table 2 and Table 4. For this study, we can conclude that the proposed chart shows better performance than the existing two charts.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ARL | Average run length |

HEWMA-p | Hybrid exponentially weighted moving average proportion |

EWMA | Exponentially weighted moving average |

HEWMA | Hybrid exponentially weighted moving average |

EWMA–CUSUM | Exponentially weighted moving average–Cumulative sum |

LCL | Lower control limit |

UCL | Upper control limit |

SERL | Standard error of run length |

## Appendix A

_{v0})

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**Table 1.**The control limits for enhanced HEWMA-p control chart with ARL

_{0}= 370 when ${\lambda}_{1}$ = 0.2, ${\lambda}_{2}$ = 0.2, and p

_{v0}= 0.1. HEWMA-p is hybrid exponentially weighted moving average proportion, ARL is average run length, UCL is upper control limit, LCL is lower control limit.

n | 0.5n | UCL | LCL | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ |
---|---|---|---|---|---|

8 | 4 | 0.1892 | 0.0126 | 5.3509 | 5.2421 |

10 | 5 | 0.1823 | 0.0252 | 5.5211 | 5.0203 |

12 | 6 | 0.1751 | 0.0314 | 5.5216 | 5.0416 |

14 | 7 | 0.1699 | 0.0364 | 5.5459 | 5.0498 |

16 | 8 | 0.1689 | 0.0430 | 5.8435 | 4.8326 |

18 | 9 | 0.1612 | 0.0431 | 5.5045 | 5.1243 |

20 | 10 | 0.1573 | 0.0448 | 5.4378 | 5.2352 |

22 | 11 | 0.1618 | 0.0525 | 6.1448 | 4.7231 |

24 | 12 | 0.1553 | 0.0524 | 5.7484 | 4.9475 |

26 | 13 | 0.1566 | 0.0561 | 6.1275 | 4.7436 |

28 | 14 | 0.1492 | 0.0541 | 5.5185 | 5.1495 |

30 | 15 | 0.1477 | 0.0558 | 5.5422 | 5.1406 |

**Table 2.**The ARLs of the enhanced HEWMA-p control chart for ${\lambda}_{1}$ = 0.2, ${\lambda}_{2}$ = 0.2, and p

_{v0}= 0.1.

n | 0.5n | p_{v1} = 0.025 | p_{v1} = 0.050 | p_{v1} = 0.075 | p_{v0} = 0.100 | p_{v1} = 0.125 | p_{v1} = 0.150 | p_{v1} = 0.175 | p_{v1} = 0.200 |
---|---|---|---|---|---|---|---|---|---|

8 | 4 | 35.29 | 107.03 | 227.89 | 370.30 | 108.55 | 47.74 | 27.62 | 18.52 |

0.2125 | 0.8830 | 4.1031 | 3.7450 | 1.0118 | 0.4055 | 0.2032 | 0.1169 | ||

10 | 5 | 22.51 | 55.24 | 204.00 | 370.36 | 107.43 | 45.55 | 25.30 | 16.80 |

0.1067 | 0.4342 | 1.9517 | 3.7060 | 0.9798 | 0.3844 | 0.1749 | 0.0992 | ||

12 | 6 | 19.22 | 44.42 | 168.19 | 370.36 | 101.22 | 40.28 | 22.00 | 14.99 |

0.0794 | 0.3273 | 1.5208 | 3.5468 | 0.9420 | 0.3305 | 0.1508 | 0.0837 | ||

14 | 7 | 16.81 | 37.03 | 141.92 | 370.09 | 98.87 | 36.21 | 19.87 | 13.64 |

0.0630 | 0.2572 | 1.3294 | 3.7582 | 0.9133 | 0.2828 | 0.1276 | 0.0716 | ||

16 | 8 | 14.15 | 28.21 | 100.20 | 370.12 | 106.17 | 37.22 | 18.71 | 12.36 |

0.0473 | 0.1753 | 0.8503 | 3.5983 | 0.9782 | 0.2861 | 0.1260 | 0.0680 | ||

18 | 9 | 14.17 | 29.27 | 116.90 | 370.35 | 82.10 | 29.98 | 16.52 | 11.66 |

0.0443 | 0.1870 | 1.0169 | 3.6899 | 0.7268 | 0.2192 | 0.0986 | 0.0549 | ||

20 | 10 | 13.41 | 27.90 | 114.63 | 370.35 | 74.67 | 26.99 | 15.20 | 9.83 |

0.0411 | 0.1772 | 1.0130 | 3.7271 | 0.6372 | 0.1921 | 0.0860 | 0.0478 | ||

22 | 11 | 11.07 | 20.20 | 65.97 | 370.20 | 101.72 | 31.72 | 15.01 | 9.60 |

0.0285 | 0.1097 | 0.5498 | 3.5711 | 0.9293 | 0.2351 | 0.0942 | 0.0491 | ||

24 | 12 | 10.12 | 19.51 | 71.04 | 370.29 | 78.48 | 24.22 | 12.61 | 8.29 |

0.0270 | 0.1122 | 0.5978 | 3.6796 | 0.7026 | 0.1885 | 0.0756 | 0.0420 | ||

26 | 13 | 9.21 | 17.79 | 57.10 | 370.28 | 90.92 | 24.92 | 11.07 | 7.55 |

0.0219 | 0.0912 | 0.4547 | 3.6574 | 0.7930 | 0.1952 | 0.0787 | 0.0420 | ||

28 | 14 | 9.69 | 19.33 | 70.98 | 370.34 | 62.21 | 21.78 | 11.49 | 7.28 |

0.0231 | 0.1001 | 0.5895 | 3.8081 | 0.5213 | 0.1400 | 0.0600 | 0.0339 | ||

30 | 15 | 9.28 | 18.12 | 65.59 | 370.11 | 60.42 | 20.57 | 11.07 | 7.07 |

0.0213 | 0.0921 | 0.5349 | 3.8897 | 0.5151 | 0.1291 | 0.0566 | 0.0326 |

**Table 3.**The control constants with ARL

_{0}= 370 for enhanced HEWMA-p control chart. when ${\lambda}_{1}$ = 0.2, ${\lambda}_{2}$ = 0.2, and p

_{v0}= 0.3.

n | 0.5n | UCL | LCL | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ |
---|---|---|---|---|---|

8 | 4 | 0.4404 | 0.1691 | 5.5158 | 5.1435 |

10 | 5 | 0.4342 | 0.1873 | 5.8915 | 4.9485 |

12 | 6 | 0.4137 | 0.1911 | 5.4695 | 5.2405 |

14 | 7 | 0.4049 | 0.1990 | 5.4499 | 5.2481 |

16 | 8 | 0.4040 | 0.2094 | 5.7765 | 5.0350 |

18 | 9 | 0.3931 | 0.2111 | 5.4839 | 5.2395 |

20 | 10 | 0.3924 | 0.2186 | 5.7395 | 5.0565 |

22 | 11 | 0.3883 | 0.2225 | 5.7495 | 5.0485 |

24 | 12 | 0.3803 | 0.2230 | 5.4635 | 5.2414 |

26 | 13 | 0.3815 | 0.2287 | 5.7725 | 5.0515 |

28 | 14 | 0.3925 | 0.2352 | 6.7985 | 4.7655 |

30 | 15 | 0.3887 | 0.2376 | 6.7485 | 4.7465 |

**Table 4.**The ARL

_{v1}of the enhanced HEWMA-p control chart for ${\lambda}_{1}$ = 0.2, ${\lambda}_{2}$ = 0.2, and p

_{v0}= 0.3.

n | 0.5n | p_{v1} = 0.20 | p_{v1} = 0.225 | p_{v1} = 0.250 | p_{v1} = 0.275 | p_{v0} = 0.300 | p_{v1} = 0.325 | p_{v1} = 0.350 | p_{v1} = 0.375 | p_{v1} = 0.400 |
---|---|---|---|---|---|---|---|---|---|---|

8 | 4 | 35.28 | 60.05 | 118.01 | 245.61 | 370.90 | 226.17 | 107.71 | 61.38 | 38.35 |

0.2524 | 0.4932 | 1.0515 | 2.3753 | 3.8363 | 2.2024 | 0.9818 | 0.5119 | 0.3029 | ||

10 | 5 | 26.68 | 43.27 | 84.94 | 188.98 | 370.31 | 280.15 | 124.92 | 63.86 | 38.04 |

0.1754 | 0.3237 | 0.7445 | 1.7778 | 3.5587 | 2.7424 | 1.1877 | 0.5410 | 0.2896 | ||

12 | 6 | 25.94 | 43.63 | 88.21 | 218.61 | 370.36 | 193.66 | 82.10 | 44.11 | 27.30 |

0.1651 | 0.3323 | 0.7546 | 2.1131 | 3.7020 | 1.8944 | 0.7278 | 0.3442 | 0.1879 | ||

14 | 7 | 22.86 | 38.41 | 77.95 | 203.34 | 370.12 | 180.32 | 72.49 | 38.13 | 23.90 |

0.1408 | 0.2824 | 0.6669 | 1.9692 | 3.6434 | 1.7636 | 0.6397 | 0.2950 | 0.1580 | ||

16 | 8 | 19.08 | 30.41 | 59.02 | 155.32 | 370.12 | 213.84 | 80.71 | 40.05 | 23.99 |

0.1092 | 0.2102 | 0.4795 | 1.4364 | 3.6315 | 2.0796 | 0.7144 | 0.3078 | 0.1549 | ||

18 | 9 | 18.64 | 29.35 | 61.47 | 169.83 | 370.12 | 161.54 | 62.54 | 31.54 | 19.87 |

0.0991 | 0.1991 | 0.5082 | 1.6226 | 3.5624 | 1.5162 | 0.5296 | 0.2273 | 0.1222 | ||

20 | 10 | 16.21 | 25.20 | 49.94 | 139.81 | 370.11 | 185.81 | 65.54 | 29.22 | 19.90 |

0.0827 | 0.1632 | 0.3935 | 1.3342 | 3.6760 | 1.8122 | 0.5570 | 0.2378 | 0.1183 | ||

22 | 11 | 15.02 | 23.56 | 45.65 | 128.87 | 370.29 | 176.40 | 61.05 | 27.67 | 16.29 |

0.0716 | 0.1513 | 0.3694 | 1.1736 | 3.5888 | 1.6707 | 0.5079 | 0.2084 | 0.1034 | ||

24 | 12 | 14.60 | 23.50 | 46.43 | 141.41 | 370.11 | 132.92 | 49.45 | 22.94 | 16.06 |

0.0683 | 0.1484 | 0.3717 | 1.2653 | 3.5991 | 1.3040 | 0.4106 | 0.1661 | 0.0836 | ||

26 | 13 | 13.54 | 20.49 | 39.45 | 115.26 | 370.11 | 161.59 | 52.72 | 26.01 | 15.38 |

0.0593 | 0.1236 | 0.3016 | 1.0572 | 3.6564 | 1.5568 | 0.4251 | 0.1730 | 0.0858 | ||

28 | 14 | 12.08 | 17.50 | 32.06 | 88.77 | 370.20 | 356.82 | 88.25 | 29.94 | 13.28 |

0.0489 | 0.0966 | 0.2331 | 0.7514 | 3.6537 | 3.4724 | 0.7457 | 0.2628 | 0.1168 | ||

30 | 15 | 11.49 | 16.59 | 30.72 | 83.96 | 370.08 | 326.41 | 79.90 | 27.51 | 13.22 |

0.0457 | 0.0902 | 0.2163 | 0.7208 | 3.6923 | 3.1849 | 0.6792 | 0.2246 | 0.1001 |

n | 0.5n | p_{v0} = 0.1 | p_{v0} = 0.3 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | ||

p_{v1} = 0.2 | p_{v1} = 0.2 | p_{v1} = 0.2 | p_{v1} = 0.2 | p_{v1} = 0.2 | p_{v1} = 0.2 | p_{v1} = 0.4 | p_{v1} = 0.4 | p_{v1} = 0.4 | ||

8 | 4 | 28.2 | 25.50 | 18.52 | 41.90 | 43.59 | 35.28 | 50.50 | 47.55 | 38.35 |

10 | 5 | 22.7 | 19.02 | 16.80 | 33.70 | 34.06 | 26.68 | 40.80 | 40.29 | 38.04 |

12 | 6 | 19 | 17.43 | 14.99 | 28.20 | 31.24 | 25.94 | 34.20 | 31.36 | 27.30 |

14 | 7 | 16.4 | 14.54 | 13.64 | 24.20 | 25.69 | 22.86 | 29.40 | 28.23 | 23.90 |

16 | 8 | 16.4 | 12.55 | 12.36 | 21.30 | 22.42 | 19.08 | 25.80 | 24.58 | 23.99 |

18 | 9 | 13 | 11.85 | 11.66 | 19.00 | 19.96 | 18.64 | 23.00 | 22.13 | 19.87 |

20 | 10 | 11.8 | 10.67 | 9.83 | 17.20 | 17.27 | 16.21 | 20.70 | 20.96 | 19.90 |

22 | 11 | 10.9 | 9.67 | 9.60 | 15.70 | 17.92 | 15.02 | 18.90 | 17.03 | 16.29 |

24 | 12 | 10.1 | 7.47 | 8.29 | 14.50 | 14.77 | 14.90 | 17.40 | 17.25 | 15.96 |

26 | 13 | 9.4 | 7.95 | 7.55 | 13.40 | 14.07 | 13.54 | 16.10 | 15.62 | 15.38 |

28 | 14 | 8.9 | 7.87 | 7.28 | 12.60 | 13.45 | 12.08 | 15.10 | 14.26 | 13.28 |

30 | 15 | 8.4 | 7.51 | 7.07 | 11.80 | 11.98 | 11.59 | 14.10 | 14.23 | 13.22 |

**Table 6.**The new service times from 10 counters in a bank branch. EWMA is exponentially weighted moving average.

t | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | X_{8} | X_{9} | X_{10} | V_{t} | EWMA_{pt} | HEWMA_{pt} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3.54 | 0.01 | 1.33 | 7.27 | 5.52 | 0.09 | 1.84 | 1.04 | 2.91 | 0.63 | 0 | 0.2480 | 0.2976 |

2 | 0.86 | 1.61 | 1.15 | 0.96 | 0.54 | 3.05 | 4.11 | 0.63 | 2.37 | 0.05 | 0 | 0.1984 | 0.2381 |

3 | 1.45 | 0.19 | 4.18 | 0.18 | 0.02 | 0.70 | 0.80 | 0.97 | 3.60 | 2.94 | 0 | 0.1587 | 0.1905 |

4 | 1.37 | 0.14 | 1.54 | 1.58 | 0.45 | 6.01 | 4.59 | 1.74 | 3.92 | 4.82 | 0 | 0.1270 | 0.1524 |

5 | 3.00 | 2.46 | 0.06 | 1.80 | 3.25 | 2.13 | 2.22 | 1.37 | 2.13 | 0.25 | 0 | 0.1016 | 0.1219 |

6 | 1.59 | 3.88 | 0.39 | 0.54 | 1.58 | 1.70 | 0.68 | 1.25 | 6.83 | 0.31 | 0 | 0.0813 | 0.0975 |

7 | 5.01 | 1.85 | 3.10 | 1.00 | 0.09 | 1.16 | 2.69 | 2.79 | 1.84 | 2.62 | 0 | 0.0650 | 0.0780 |

8 | 4.96 | 0.55 | 1.43 | 4.12 | 4.06 | 1.42 | 1.43 | 0.86 | 0.67 | 0.13 | 0 | 0.0520 | 0.0624 |

9 | 1.08 | 0.65 | 0.91 | 0.88 | 2.02 | 2.88 | 1.76 | 2.87 | 1.97 | 0.62 | 0 | 0.0416 | 0.0499 |

10 | 4.56 | 0.44 | 5.61 | 2.79 | 1.73 | 2.46 | 0.53 | 1.73 | 7.02 | 2.13 | 0 | 0.0333 | 0.0399 |

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## Share and Cite

**MDPI and ACS Style**

Aslam, M.; Rao, G.S.; AL-Marshadi, A.H.; Jun, C.-H.
A Nonparametric HEWMA-*p* Control Chart for Variance in Monitoring Processes. *Symmetry* **2019**, *11*, 356.
https://doi.org/10.3390/sym11030356

**AMA Style**

Aslam M, Rao GS, AL-Marshadi AH, Jun C-H.
A Nonparametric HEWMA-*p* Control Chart for Variance in Monitoring Processes. *Symmetry*. 2019; 11(3):356.
https://doi.org/10.3390/sym11030356

**Chicago/Turabian Style**

Aslam, Muhammad, G. Srinivasa Rao, Ali Hussein AL-Marshadi, and Chi-Hyuck Jun.
2019. "A Nonparametric HEWMA-*p* Control Chart for Variance in Monitoring Processes" *Symmetry* 11, no. 3: 356.
https://doi.org/10.3390/sym11030356