# Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Brief Review of the Distribution of the Ratio Z

**W**is a bivariate normally distributed random vector with a mean vector and variance–covariance matrix, respectively, as below:

_{X}, γ

_{Y}, ω, and ρ. In addition, the p.d.f. (probability density function) ${f}_{Z}\left(z\mid {\gamma}_{X},{\gamma}_{Y},\omega ,\rho \right)$ of Z can be given as follows:

_{X}, γ

_{Y}, $p$, ω, and ρ. Moreover, Φ

^{−1}($\xb7$) is the i.d.f. of the standard normal distribution.

## 3. Construction of the TEWMA-RZ Control Charts

#### 3.1. A Brief Review of the VSI-EWMA-RZ Control Chart

^{+}) chart is defined as:

^{+}chart. In addition, an upper warning limit $UW{L}^{+}={W}^{+}\times {z}_{0}$ between [${z}_{0},UC{L}^{+}$] is added to the chart, where ${W}^{+}<{K}^{+}$ is the upper warning limit coefficient. A process is deemed to be out-of-control if the statistic ${Y}_{i}^{+}>UC{L}^{+}.$ Otherwise, the process is thought to be in-control if the statistic ${Y}_{i}^{+}$ falls within the warning region $\left(UW{L}^{+},UC{L}^{+}\right]$, and a shorter sampling interval ${h}_{s}$ is used to collect the next sampling point. The process is deemed to be in-control if the plotted statistic ${Y}_{i}^{+}$ falls within the safe region $[{z}_{0},UW{L}^{+}]$, and a longer sampling interval ${h}_{L}$ is used.

^{−}) chart is defined as:

^{−}chart, respectively. In addition, a lower warning limit $LW{L}^{-}={W}^{-}\times {z}_{0}$ between [$LC{L}^{-},{z}_{0}$] is added to the chart, where ${W}^{-}>{K}^{-}$ is the lower warning limit coefficient. A process is claimed to be out-of-control if the plotted statistic ${Y}_{i}^{-}<LC{L}^{-}.$ Otherwise, the process is deemed to be in-control if the plotted statistic ${Y}_{i}^{-}$ falls within the warning region $\left[LC{L}^{-}LW{L}^{-}\right)$ and a shorter sampling interval ${h}_{s}$ is used. The process is deemed to be in-control if the plotted statistic ${Y}_{i}^{-}$ falls within the safe region $[LW{L}^{-},{z}_{0}]$ and a longer sampling interval ${h}_{L}$ is used.

#### 3.2. A Brief Review of the VSI-DEWMA-RZ Chart

^{+}) chart is used to detect an increase in the process and the monitoring statistic ${U}_{i}^{-}$ is:

^{+}chart, respectively. The single control limit of the chart is $UC{L}^{+}={K}^{+}\times {z}_{0}$. Also, an upper warning limit $UW{L}^{+}={W}^{+}\times {z}_{0}$ between $\left[{z}_{0},UC{L}^{+}\right]$ is added, where ${W}^{+}<{K}^{+}$ is the upper warning limit coefficient. If the plotted statistic ${U}_{i}^{+}>UC{L}^{+},$ the process is considered to be out-of-control. Otherwise, the process is claimed to be in-control if $UW{L}^{+}<{U}_{i}^{+}\le UC{L}^{+}$ and a shorter sampling interval ${h}_{s}$ is used to collect the next sampling point. The process is considered to be in-control if ${z}_{0}\le {U}_{i}^{+}\le UW{L}^{+}$ and a longer sampling interval ${h}_{L}$. is used. The sampling interval ${h}_{i}$ can be expressed as follows:

^{−}) chart is used to detect a decrease in the process and the statistic ${U}_{i}^{-}$ can be similarly defined as:

#### 3.3. The Proposed TEWMA-RZ Charts

#### 3.3.1. The FSI-TEWMA-RZ Chart

^{+}) chart is used to detect an increase in the process, and the monitoring statistic ${V}_{i}^{+}$ is:

^{+}chart, respectively. The single control limit of the chart is $UC{L}^{+}={K}^{+}\times {z}_{0}$. A process is deemed to be out-of-control if the statistic ${V}_{i}^{+}$ falls above the $UC{L}^{+}$. Otherwise, the process is declared to be in-control.

^{−}) chart is used to detect downward process sifts and the statistic ${V}_{i}^{-}$ can be similarly defined as:

^{−}chart, respectively. The single control limit of the chart is $LC{L}^{-}={K}^{-}\times {z}_{0}$. A process is deemed to be out-of-control if the charting statistic ${V}_{i}^{-}$ falls below the $LC{L}^{-}$. Otherwise, the process is declared to be in-control.

#### 3.3.2. The VSI-TEWMA-RZ Chart

^{+}) control chart, an upper warning limit $UW{L}^{+}={W}^{+}\times {z}_{0}$ between [${z}_{0},UC{L}^{+}$] is added, where ${W}^{+}<{K}^{+}$ is the upper warning limit coefficient. If the plotted statistic ${V}_{i}^{+}>UC{L}^{+},$ the process is considered to be out-of-control. Otherwise, the process is claimed to be in-control if $UW{L}^{+}<{V}_{i}^{+}\le UC{L}^{+}$ and a shorter sampling interval ${h}_{s}$ is used to collect the next sampling point. The process is considered to be in-control if ${z}_{0}\le {V}_{i}^{+}\le UW{L}^{+}$ and a longer sampling interval ${h}_{L}$ is used. The sampling interval ${h}_{i}$ can be expressed as follows:

^{−}) control chart, a lower warning limit $LW{L}^{-}={W}^{-}\times {z}_{0}$ between [$LC{L}^{-},{z}_{0}$] is added, where ${W}^{-}>{K}^{-}$ is the lower warning limit coefficient. If the plotted statistic ${V}_{i}^{-}<LC{L}^{-}$, the process is considered to be out-of-control. Otherwise, the process is claimed to be in-control if $LC{L}^{-}\le {V}_{i}^{-}<LW{L}^{-}$ and a shorter sampling interval ${h}_{s}$ is used. The process is considered to be in-control if $LW{L}^{-}\le {V}_{i}^{-}\le {z}_{0}$ and a longer sampling interval ${h}_{L}$ is used. The sampling interval ${h}_{i}$ can be expressed in a form similar to Equation (24).

## 4. Design of the Proposed TEWMA-RZ Charts

#### 4.1. Design of the Proposed FSI-TEWMA-RZ Chart

^{+}chart is summarized as follows:

_{0}= 200 and further studies the $AR{L}_{1}$ performance of the proposed chart under different shifts. The performance of the FSI-TEWMA-RZ chart can be expressed as:

#### 4.2. Design of the Proposed VSI-TEWMA Charts

^{+}chart is summarized as below:

^{+}and W

^{+}of the VSI-TEWMA-RZ control chart. Considering the space limitation, this article only gives the values of K

^{+}and W

^{+}under the condition that ${\gamma}_{X}={\gamma}_{Y}$. It is noted that the value of K

^{+}of the VSI-TEWMA-RZ chart presented in Table 1, which is the same as the one from the corresponding FSI-TEWMA-RZ chart.

## 5. Numerical Results and Analysis

#### 5.1. Comparisons between the VSI-TEWMA-RZ and the FSI-TEWMA-RZ Charts

^{+}and W

^{+}presented in Table 1, Figure 1, Figure 2, Figure 3 and Figure 4 compare the performance of the upper-sided FSI-TEWMA-RZ and VSI-TEWMA-RZ charts when monitoring the upward shifts. The $AR{L}_{1}$ and $AT{S}_{1}$ represent the performances of the corresponding FSI and VSI control charts, respectively. It is pointed out that since the sampling interval of the FSI chart is $h=1$, the FSI chart’s ARL is equal to its ATS value. Then, the $AR{L}_{1}$ performance of the FSI chart can be directly compared with the $AT{S}_{1}$ performance of the VSI chart.

**Figure 3.**ARL values of the FSI-TEWMA-RZ (-○-) chart, ATS values of the VSI-DEWMA-RZ (-☆-), VSI-TEWMA-RZ (-□-), and VSI-EWMA-RZ (-✳-) charts for ${\gamma}_{X}\in \left\{0.01,0.2\right\},{\gamma}_{Y}\in \left\{0.01,0.2\right\},{\gamma}_{X}={\gamma}_{Y},{\rho}_{0}\in \left\{-0.8,-0.4,0,0.4,0.8\right\},{\rho}_{0}\ne {\rho}_{1},\tau \in \left\{1.001,1.005,1.01,1.02,1.05\right\}\mathrm{and}n\in \left\{1,5\right\}$.

**Figure 4.**ARL values of the FSI-TEWMA-RZ (-○-) chart, ATS values of the VSI-DEWMA-RZ (-☆-), VSI-TEWMA-RZ (-□-) and VSI-EWMA-RZ (-✳-) charts for ${\gamma}_{X}\in \left\{0.01,0.2\right\},{\gamma}_{Y}\in \left\{0.01,0.2\right\},{\gamma}_{X}\ne {\gamma}_{Y},{\rho}_{0}\in \left\{-0.8,-0.4,0,0.4,0.8\right\},{\rho}_{0}\ne {\rho}_{1},\tau \in \left\{1.001,1.005,1.01,1.02,1.05\right\}\mathrm{and}n\in \left\{1,5\right\}$.

_{1}= 2 × ρ

_{0}= −0.8, $n=1$. , $\lambda =0.2$, and $\tau =1.005$, we have $AT{S}_{1}=104$ for the FSI-TEWMA-RZ chart and $AT{S}_{1}=98.3$ for the VSI-TEWMA-RZ chart. While if ${\rho}_{1}={\rho}_{0}=-0.4$, we obtain $AT{S}_{1}=181.6$ for the FSI-TEWMA-RZ chart and $AT{S}_{1}=176.3$. for the VSI-TEWMA-RZ chart. On the contrary, when the level of the negative correlation coefficient decreases, that is $\left|{\rho}_{1}\right||{\rho}_{0}|$ the performances of the proposed FSI- and VSI-TEWMA-RZ charts deteriorate. For example, when ${\rho}_{1}=0.5\times {\rho}_{0}=-0.2$, we obtain $AT{S}_{1}=273.6$ for the FSI-TEWMA-RZ chart and $AT{S}_{1}=272.2$ for the FSI-TEWMA-RZ chart. These $AT{S}_{1}$ values are all smaller than the ones of the ${\rho}_{1}={\rho}_{0}=-0.4$ case, respectively.

#### 5.2. Comparisons between the VSI-TEWMA-RZ Chart and the VSI-EWMA-RZ Chart

^{+}and W

^{+}presented in Table 1, Figure 1, Figure 2, Figure 3 and Figure 4 also compare the performances of the VSI-TEWMA-RZ and VSI-EWMA-RZ control charts when monitoring the upward shifts. Figure 1 and Figure 2 present the out-of-control $AT{S}_{1}$ values of the VSI-EWMA-RZ chart for the condition ${\rho}_{0}={\rho}_{1}$. While for the condition ${\rho}_{0}\ne {\rho}_{1}$, the $AT{S}_{1}$ values of the VSI-EWMA-RZ chart are shown in Figure 3 and Figure 4. Some conclusions can be drawn from Figure 1, Figure 2, Figure 3 and Figure 4.

#### 5.3. Comparisons between the VSI-TEWMA-RZ Chart and the VSI-DEWMA-RZ Chart

## 6. An Illustrative Example

^{+}and VSI-TEWMA-RZ

^{+}${K}^{+}=1.009089$ and ${W}^{+}=1.000779$ of the VSI-EWMA-RZ

^{+}chart and ${K}^{+}=1.006163$ and ${W}^{+}=0.999942$ of the VSI-EWMA-RZ

^{+}chart and ${z}_{0}$ is set to be 1, then $UC{L}^{+}={K}^{+}$ and $UW{L}^{+}={W}^{+}$.

^{+}chart, the VSI-DEWMA-RZ

^{+}chart, and the FSI- and VSI-TEWMA-RZ

^{+}control charts for the dataset in Table 2, where the index t in the axis is the cumulative time of the process monitoring. It can be seen from Figure 5 that the FSI- and VSI-TEWMA-RZ

^{+}chart triggers an out-of-control signal at sample #15 (in bold in Table 2), while the VSI-DEWMA-RZ

^{+}and VSI-EWMA-RZ

^{+}charts signal an out-of-control condition at sample #16 and #18 (in bold in Table 2), respectively. This example shows that the TEWMA-RZ charts outperform the VSI-DEWMA-RZ and the VSI-EWMA-RZ charts from the perspective of the number of samples.

^{+}chart, it is noted that the first six samples are in the warning region and a shorter sampling interval ${h}_{s}=0.1$ is used to collect the next sampling point. The plotted sample point ${V}_{7}^{+}$ falls within the safe region $[{z}_{0},UW{L}^{+}]$ and a longer sampling interval ${h}_{L}$ is used. The VSI-TEWMA-RZ

^{+}chart needs 10.5-times the units to detect the assignable cause. As a comparison, the VSI-DEWMA-RZ

^{+}chart needs 10.6-times the units to trigger an out-of-control signal, while the VSI-EWMA-RZ

^{+}chart needs 12.6-times the units to trigger an out-of-control signal. This shows the advantage of the VSI-TEWMA-RZ

^{+}chart over the VSI-EWMA-RZ

^{+}chart and VSI-DEWMA-RZ

^{+}chart. Moreover, since the sampling interval of the FSI-TEWMA-RZ

^{+}control chart is 1, then it needs 15-times the units to trigger an out-of-control signal. If a control chart indicates an out-of-control signal, then the quality engineering should take corrective actions to search the potential assignable causes and make the process as controlled as possible.

## 7. Conclusions

_{0}, and $\lambda $. Moreover, the comparison results show that the proposed VSI-TEWMA-RZ chart reacts faster than the FSI-TEWMA-RZ chart for all shifts, and the VSI-TEWMA-RZ chart also performs reacts faster than the VSI-EWMA-RZ and VSI-DEWMA-RZ charts in the detection of relatively small shifts.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**ARL values of the FSI-TEWMA-RZ (-○-) chart, ATS values of the VSI-DEWMA-RZ (-☆-), VSI-TEWMA-RZ (-□-), and VSI-EWMA-RZ (-✳-) charts for ${\gamma}_{X}\in \left\{0.01,0.2\right\},{\gamma}_{Y}\in \left\{0.01,0.2\right\},{\gamma}_{X}={\gamma}_{Y},{\rho}_{0}\in \left\{-0.8,-0.4,0,0.4,0.8\right\},{\rho}_{0}={\rho}_{1},\tau \in \left\{1.001,1.005,1.01,1.02,1.05\right\}\mathrm{and}n\in \left\{1,5\right\}$.

**Figure 2.**ARL values of the FSI-TEWMA-RZ (-○-) chart, ATS values of the VSI-DEWMA-RZ (-☆-), VSI-TEWMA-RZ (-□-), and VSI-EWMA-RZ (-✳-) charts for ${\gamma}_{X}\in \left\{0.01,0.2\right\},{\gamma}_{Y}\in \left\{0.01,0.2\right\},{\gamma}_{X}\ne {\gamma}_{Y},{\rho}_{0}\in \left\{-0.8,-0.4,0,0.4,0.8\right\},{\rho}_{0}={\rho}_{1},\tau \in \left\{1.001,1.005,1.01,1.02,1.05\right\}\mathrm{and}n\in \left\{1,5\right\}$.

**Figure 5.**Different charts monitoring the food industry example. (

**a**) The VSI-EWMA-RZ

^{+}chart, (

**b**) the VSI-DEWMA-RZ

^{+}chart, (

**c**) the proposed FSI-TEWMA-RZ

^{+}chart, and (

**d**) the proposed VSI-TEWMA-RZ

^{+}charts.

$\lambda $ | $\mathbf{\left(}{\mathit{\gamma}}_{\mathit{x}}\mathbf{=}\mathit{0.01}\mathbf{,}{\mathit{\gamma}}_{\mathit{Y}}\mathbf{=}\mathbf{0.01}\mathbf{\right)}$ | $\mathbf{\left(}{\mathit{\gamma}}_{\mathit{x}}\mathbf{=}\mathbf{0.2}\mathbf{,}{\mathit{\gamma}}_{\mathit{Y}}\mathbf{=}\mathbf{0.2}\mathbf{\right)}$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{n}\mathbf{=}\mathbf{1}$ | $\mathit{n}\mathbf{=}\mathbf{5}$ | $\mathit{n}\mathbf{=}\mathbf{1}$ | $\mathit{n}\mathbf{=}\mathbf{5}$ | |||||

$\mathit{K}$ | $\mathit{W}$ | $\mathit{K}$ | $\mathit{W}$ | $\mathit{K}$ | $\mathit{W}$ | $\mathit{K}$ | $\mathit{W}$ | |

${\rho}_{0}={\rho}_{1}=-0.8$ | ||||||||

0.2 | 1.0067 | 0.9998 | 1.0030 | 0.9999 | 1.2480 | 1.0601 | 1.0762 | 1.0083 |

0.5 | 1.0161 | 1.0000 | 1.0071 | 0.9999 | 1.5315 | 1.0581 | 1.1699 | 1.0106 |

${\rho}_{0}={\rho}_{1}=-0.4$ | ||||||||

0.2 | 1.0059 | 0.9998 | 1.0026 | 0.9999 | 1.2061 | 1.0467 | 1.0653 | 1.0061 |

0.5 | 1.0141 | 0.9999 | 1.0063 | 1.0000 | 1.4454 | 1.0452 | 1.1466 | 1.0069 |

${\rho}_{0}={\rho}_{1}=0$ | ||||||||

0.2 | 1.0050 | 0.9998 | 1.0022 | 0.9999 | 1.1618 | 1.0316 | 1.0534 | 1.0042 |

0.5 | 1.0119 | 0.9999 | 1.0053 | 1.0000 | 1.3544 | 1.0305 | 1.1210 | 1.0047 |

${\rho}_{0}={\rho}_{1}=0.4$ | ||||||||

0.2 | 1.0038 | 0.9998 | 1.0017 | 0.9999 | 1.1146 | 1.0179 | 1.0397 | 1.0019 |

0.5 | 1.0092 | 0.9999 | 1.0041 | 1.0000 | 1.2543 | 1.0179 | 1.0912 | 1.0028 |

${\rho}_{0}={\rho}_{1}=0.8$ | ||||||||

0.2 | 1.0022 | 0.9999 | 1.0010 | 1.0000 | 1.0574 | 1.0045 | 1.0216 | 1.0002 |

0.5 | 1.0053 | 1.0000 | 1.0024 | 1.0000 | 1.1316 | 1.0051 | 1.0504 | 1.0008 |

Sample Number | Box Size | $\begin{array}{c}{\mathit{W}}_{\mathit{p}\mathbf{,}\mathit{i}\mathbf{,}\mathit{j}}\mathbf{\left[}\mathbf{g}\mathbf{r}\mathbf{\right]}\\ {\mathit{W}}_{\mathit{f}\mathbf{,}\mathit{i}\mathbf{,}\mathit{j}}\mathbf{\left[}\mathbf{g}\mathbf{r}\mathbf{\right]}\end{array}$ | $\begin{array}{c}{\overline{\mathit{W}}}_{\mathit{p}\mathbf{,}\mathit{i}}\mathbf{\left[}\mathbf{g}\mathbf{r}\mathbf{\right]}\\ {\overline{\mathit{W}}}_{\mathit{f}\mathbf{,}\mathit{i}}\mathbf{\left[}\mathbf{g}\mathbf{r}\mathbf{\right]}\end{array}$ | ${\widehat{\mathit{Z}}}_{\mathit{i}}=\frac{{\overline{\mathit{W}}}_{\mathit{p}\mathbf{,}\mathit{i}}}{{\overline{\mathit{W}}}_{\mathit{f}\mathbf{,}\mathit{i}}}$ | VSI-EWMA | VSI-DEWMA | VSI-TEWMA | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{Y}}_{\mathit{i}}^{\mathbf{+}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{U}}_{\mathit{i}}^{\mathbf{+}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{V}}_{\mathit{i}}^{\mathbf{+}}$ | ${\mathit{t}}_{\mathit{i}}$ | |||||||||

1 | 250 gr | 25.479 | 25.355 | 24.027 | 25.792 | 24.960 | 25.122 | 1.003 | 1.00150 | 0.1 | 1.00075 | 0.1 | 1.00038 | 0.1 |

25.218 | 25.171 | 24.684 | 25.052 | 25.107 | 25.046 | |||||||||

2 | 250 gr | 25.359 | 25.172 | 24.508 | 25.292 | 24.449 | 24.956 | 1.000 | 1.00075 | 0.2 | 1.00075 | 0.2 | 1.00056 | 0.2 |

25.211 | 25.115 | 24.679 | 24.933 | 24.831 | 24.954 | |||||||||

3 | 250 gr | 24.574 | 24.864 | 25.865 | 25.107 | 24.811 | 25.044 | 1.005 | 1.00288 | 2.1 | 1.00181 | 0.3 | 1.00119 | 0.3 |

24.784 | 24.868 | 25.377 | 24.879 | 24.734 | 24.929 | |||||||||

4 | 250 gr | 25.313 | 24.483 | 24.088 | 25.184 | 25.681 | 24.950 | 0.999 | 1.00094 | 2.2 | 1.00138 | 0.4 | 1.00128 | 0.4 |

25.338 | 24.859 | 24.305 | 25.115 | 25.251 | 24.974 | |||||||||

5 | 250 gr | 25.557 | 24.959 | 25.023 | 24.482 | 25.531 | 25.111 | 0.998 | 1.00000 | 2.3 | 1.00042 | 0.5 | 1.00085 | 0.5 |

25.277 | 25.402 | 25.012 | 24.937 | 25.148 | 25.163 | |||||||||

6 | 250 gr | 24.882 | 24.473 | 24.814 | 25.418 | 24.732 | 24.864 | 0.997 | 1.00000 | 4.2 | 0.99933 | 0.6 | 1.00009 | 0.6 |

24.962 | 24.644 | 24.817 | 25.419 | 24.818 | 24.932 | |||||||||

7 | 500 gr | 49.848 | 48.685 | 49.994 | 49.910 | 49.374 | 49.562 | 0.999 | 1.00000 | 6.1 | 0.99897 | 2.5 | 0.99953 | 0.7 |

49.993 | 49.128 | 49.830 | 49.566 | 49.422 | 49.588 | |||||||||

8 | 500 gr | 49.668 | 50.338 | 49.149 | 47.807 | 49.064 | 49.205 | 0.990 | 1.00000 | 8 | 0.99664 | 4.4 | 0.99809 | 2.6 |

49.695 | 50.681 | 49.640 | 48.969 | 49.612 | 49.720 | |||||||||

9 | 500 gr | 51.273 | 48.303 | 48.510 | 50.594 | 48.591 | 49.454 | 0.993 | 1.00000 | 9.9 | 0.99515 | 6.3 | 0.99662 | 4.5 |

50.366 | 49.210 | 49.844 | 49.890 | 49.595 | 49.781 | |||||||||

10 | 500 gr | 48.720 | 51.566 | 49.677 | 50.651 | 50.344 | 50.192 | 1.002 | 1.00100 | 11.8 | 0.99649 | 8.2 | 0.99655 | 6.4 |

49.721 | 50.215 | 50.178 | 50.324 | 50.071 | 50.102 | |||||||||

11 | 500 gr | 53.173 | 51.079 | 51.636 | 49.187 | 49.779 | 50.971 | 1.015 | 1.00800 | 11.9 | 1.00145 | 10.1 | 0.99900 | 8.3 |

51.081 | 50.660 | 50.468 | 49.787 | 49.197 | 50.239 | |||||||||

12 | 500 gr | 51.255 | 48.578 | 49.657 | 49.971 | 50.675 | 50.027 | 1.004 | 1.00600 | 12 | 1.00333 | 10.2 | 1.00116 | 10.2 |

49.899 | 49.476 | 49.400 | 49.909 | 50.365 | 49.810 | |||||||||

13 | 500 gr | 48.760 | 50.206 | 51.216 | 51.997 | 49.818 | 50.399 | 1.010 | 1.00800 | 12.1 | 1.00547 | 10.3 | 1.00332 | 10.3 |

48.919 | 50.032 | 50.497 | 50.627 | 49.483 | 49.912 | |||||||||

14 | 500 gr | 51.599 | 49.257 | 52.077 | 49.874 | 48.791 | 50.319 | 1.004 | 1.00600 | 12.2 | 1.00563 | 10.4 | 1.00447 | 10.4 |

50.351 | 49.885 | 51.044 | 49.898 | 49.506 | 50.137 | |||||||||

15 | 500gr | 49.178 | 51.188 | 50.602 | 50.221 | 50.433 | 50.325 | 1.006 | 1.00600 | 12.3 | 1.00577 | 10.5 | 1.00512 | 10.5 |

49.104 | 50.348 | 50.621 | 50.018 | 50.085 | 50.035 | |||||||||

16 | 500gr | 50.667 | 50.600 | 50.601 | 49.517 | 50.578 | 50.393 | 1.010 | 1.00800 | 12.4 | 1.00686 | 10.6 | 1.00599 | 10.6 |

50.011 | 49.870 | 49.779 | 50.020 | 49.877 | 49.911 | |||||||||

17 | 500gr | 50.925 | 49.036 | 50.971 | 51.888 | 50.741 | 50.712 | 1.009 | 1.00850 | 12.5 | 1.00767 | 10.7 | 1.00683 | 10.7 |

50.579 | 49.735 | 50.196 | 50.740 | 49.959 | 50.242 | |||||||||

18 | 500gr | 50.673 | 50.653 | 50.346 | 50.749 | 51.338 | 50.752 | 1.010 | 1.00925 | 12.6 | 1.00845 | 10.8 | 1.00764 | 10.8 |

50.459 | 49.990 | 50.281 | 50.251 | 50.281 | 50.252 | |||||||||

19 | 250gr | 25.390 | 25.554 | 25.799 | 23.869 | 25.041 | 25.131 | 1.006 | 1.00763 | 12.7 | 1.00804 | 10.9 | 1.00784 | 10.9 |

25.158 | 25.278 | 25.073 | 24.349 | 25.085 | 24.989 | |||||||||

20 | 250gr | 24.343 | 26.087 | 25.431 | 24.799 | 26.440 | 25.420 | 1.002 | 1.00481 | 12.8 | 1.00642 | 11.0 | 1.00713 | 11.0 |

24.771 | 25.427 | 25.005 | 24.711 | 25.258 | 25.035 |

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## Share and Cite

**MDPI and ACS Style**

Hu, X.; Sun, G.; Xie, F.; Tang, A. Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals. *Symmetry* **2022**, *14*, 1236.
https://doi.org/10.3390/sym14061236

**AMA Style**

Hu X, Sun G, Xie F, Tang A. Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals. *Symmetry*. 2022; 14(6):1236.
https://doi.org/10.3390/sym14061236

**Chicago/Turabian Style**

Hu, Xuelong, Guan Sun, Fupeng Xie, and Anan Tang. 2022. "Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals" *Symmetry* 14, no. 6: 1236.
https://doi.org/10.3390/sym14061236