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Multivariate control charts for mean vector and covariance matrix with supplementary runs rules
Journal of the Korean Data & Information Science Society 2022;33:311-23
Published online March 31, 2022;
© 2022 Korean Data and Information Science Society.

Inho Oh1 · Gyo-Young Cho2

12Department of Statistics, Kyungpook National University
Correspondence to: 1 Graduate student, Department of Statistics, Kyungpook National University, Daegu, 41566, Korea.
2 Professor, Department of Statistics, Kyungpook National University, 80 Dae-hakro, Bukgu, Daegu, 41566, Korea. E-mail:
Received February 8, 2022; Revised February 22, 2022; Accepted February 25, 2022.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Shewhart control chart developed to statistically control the production process is not implemented in the actual process. It is simple and has the advantage of effectively detecting large shifts. But it is not effective in small shifts. To compensate for this, there are many several methods. In 1956, Western Electric Company presented a control chart with runs-rules added. In this paper, to compensate for the shortcomings of traditional Shewhart charts, we propose a chart for monitoring the mean vector and covariance matrix using runs rules proposed by Western Electric Company. In the run length distribution of the control chart to which the finite Markov chain embedding method and run rule were applied, it was confirmed that the mean of the run length rapidly decreased even with a small shift of the mean vector and covariance matrix. It is sensitive to small shifts. Therefore, it can be seen that the Shewhart control chart using runs rules in the production process is effective in detecting small uctuations.
Keywords : Covariance matrix, Markov chain imbedding, mean vector, quartiles, run length.