Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation
The Shewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-char...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/4/549 |
| _version_ | 1797539918561411072 |
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| author | Wei-Heng Huang |
| author_facet | Wei-Heng Huang |
| author_sort | Wei-Heng Huang |
| collection | DOAJ |
| description | The Shewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts are most commonly used for monitoring the process mean and variability based on the assumption of normality. However, many process distributions may follow a positively skewed distribution, such as the lognormal distribution. In this study, we discuss the construction of three combined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts for jointly monitoring the lognormal mean and the standard deviation. The simulation results show that the combined lognormal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts are more effective when the lognormal distribution is more skewed. A real example is used to demonstrate how the combined lognormal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts can be applied in practice. |
| first_indexed | 2024-03-10T12:52:38Z |
| format | Article |
| id | doaj.art-d6ad318de9f14dd0bf313fcd9d715fb8 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T12:52:38Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d6ad318de9f14dd0bf313fcd9d715fb82023-11-21T12:11:57ZengMDPI AGSymmetry2073-89942021-03-0113454910.3390/sym13040549Control Charts for Joint Monitoring of the Lognormal Mean and Standard DeviationWei-Heng Huang0Department of Statistics, Feng Chia University, Taichung 40724, TaiwanThe Shewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts are most commonly used for monitoring the process mean and variability based on the assumption of normality. However, many process distributions may follow a positively skewed distribution, such as the lognormal distribution. In this study, we discuss the construction of three combined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts for jointly monitoring the lognormal mean and the standard deviation. The simulation results show that the combined lognormal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts are more effective when the lognormal distribution is more skewed. A real example is used to demonstrate how the combined lognormal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>X</mi><mo>¯</mo></mover></semantics></math></inline-formula>- and <i>S</i>-charts can be applied in practice.https://www.mdpi.com/2073-8994/13/4/549average run lengthlognormal distributionphase II monitoringS-chart<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm900" display="block"> <mml:semantics> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:semantics> </mml:math> </inline-formula></named-content>-chart |
| spellingShingle | Wei-Heng Huang Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation Symmetry average run length lognormal distribution phase II monitoring S-chart <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm900" display="block"> <mml:semantics> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:semantics> </mml:math> </inline-formula></named-content>-chart |
| title | Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation |
| title_full | Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation |
| title_fullStr | Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation |
| title_full_unstemmed | Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation |
| title_short | Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation |
| title_sort | control charts for joint monitoring of the lognormal mean and standard deviation |
| topic | average run length lognormal distribution phase II monitoring S-chart <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm900" display="block"> <mml:semantics> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:semantics> </mml:math> </inline-formula></named-content>-chart |
| url | https://www.mdpi.com/2073-8994/13/4/549 |
| work_keys_str_mv | AT weihenghuang controlchartsforjointmonitoringofthelognormalmeanandstandarddeviation |