傳統的管制圖經常都伴隨著使用上的限制，其中樣本資訊須為常態分配是一個常見的管制圖使用條件。若樣本並非近似或服從常態分配而使用者忽略此條件使用管制圖的話可能會造成嚴重的影響。以Kullback-Leibler 資訊理論(KLI)為基礎所建構之KLI 管制圖相較於傳統的管制圖而言有著不需要設定管制圖參數優勢，而KLI 管制圖對於非常態樣本的效能相較於累積和管制圖好。指數加權移動平均管制圖雖然對非常態樣本較穩健，不過使用者須設定管制圖參數。KLI 管制圖在樣本為Gamma 分配時績效影響較樣本為T 分配時更大，我們可以藉由數學轉換或增加樣本數將非常態分配的樣本近似常態分配以使用KLI 管制圖進行製程監控。其中數學轉換包含了最常見的Box-Cox 轉換以及Yeo-Johnson 轉換。本研究先將非常態樣本減掉其平均數再除標準差，類似常態分配標準化的方式處理，再加入管制圖進行績效衡量。而為了公平的進行比較，本研究提出了Normal Score Index 將不同管制圖之間的績效進行比較。
當使用者手中的分配為接近T 分配之對稱型分配時，則可以使用Yeo-Johnson轉換將非常態樣本近似常態。而當非常態分配為類似Gamma 分配之右偏分配，則可以計算其偏態及超值峰態係數，當偏態系數大於1.265以及超值峰態係數大於2.4時Box-Cox 轉換以及Yeo-Johnson 轉換均適用。而偏態系數在1.633到1.265以及超值峰態係數在4到2.4之間時只能用Box-Cox 轉換處理。而當偏態係數小於1.633以及超值峰態係數小於2.4時，便需要增加樣本數以改善非常態樣本的KLI 管制圖績效。

This study investigates the performance of control charts when sample data deviates from or approximates a normal distribution and proposes corresponding improvement methods. The performance of the KLI control chart is more affected by the sample distribution when it follows a gamma distribution compared to a T-distribution. We can approximate non-normal distributions to a normal distribution by applying mathematical transformations or increasing the sample size to utilize the KLI control chart for process monitoring. The mathematical transformations include the commonly used Box-Cox transformation and Yeo-Johnson transformation. In this study, non-normal samples are normalized by subtracting their mean and dividing by their standard deviation, like the normalization of normal distributions, before applying control charts for performance evaluation. To ensure fair comparison, this study proposes the Normal Score Index to compare the performance of different control charts.
When the distribution in the user's possession approximates a symmetric distribution close to a T-distribution, the Yeo-Johnson transformation can be used to approximate the non-normal samples to a normal distribution. When the non-normal distribution resembles a right-skewed distribution such as a gamma distribution, the skewness and kurtosis coefficients can be calculated. If the skewness coefficient is greater than 1.265 and the kurtosis coefficient is greater than 2.4, both the Box-Cox and Yeo-Johnson transformations are applicable. If the skewness coefficient falls between 1.633 and 1.265 and the kurtosis coefficient falls between 4 and 2.4, only the Box-Cox transformation can be used. When the skewness coefficient is less than 1.633 and the kurtosis coefficient is less than 2.4, increasing the sample size is necessary to improve the performance of the KLI control chart for non-normal samples.

Atkinson, A. C. (2020) The Box-Cox transformation: review and extensions. Statistic Science. 36(2), 239-255.
Borror, C.M.,Montgomery, D.C., & Runger, G.C. (1999) Robustness of the EWMA control chart to non-normality. Journal of Quality Technology, 31(3), 309-316.
Bickel, P. J. & Doksum, K. A. (1981). An analysis of transformations revisited.Journal of the American Statistical Association, 76, 296–311.
Box, G. E. P. & Cox, D. R. (1964) An analysis of transformations (with discussion). Journal of the Royal Statistical Society, Series B.26(2), 211-252.
Box, G. E. P. & Cox, D. R. (1964). An analysis of transformations (with discussion). Journal of the Royal Statistical Society, Series B, 26, 211–252.
Chang, Y. C ., Li, T. W., & Mastrangelo, C.(2021).Designing a parameter-free Kullback-Leibler information control chart for monitoring process mean shift. Quality and Reliability Engineering International,37(3),1017-1034.
Draper, N. R. & Cox, D. R. (1969) On distributions and their transformation to normality. Journal of the Royal Statistical Society, Series B. 31(3), 472-476.
Hernández, F. & Johnson, R. A. (1980) The large sample behaviors of transformation, Journal of The American Statistical Association, 75, 855-861.
Hinkley, D.V. (1975). On power transformation to symmetry, Biometrika, 62, 101-111.
Gan, F. F. (1993) An optimal design of CUSUM control charts for binomial counts. Journal of Applied Statistics 20, 445-460.
Gan, F. F. (1994) Design of optimal exponential CUSUM control charts. Journal of Quality Technology 26, 109-124.
Hernandez, F., & Johnson, R. A. (1980). The large-sample behavior of transformations to normality. Journal of the American Statistical Association, 75(372), 855-861.
Hinkley, D. V. and Runger, G. (1984). The analysis of transformed data. Journal of the American Statistical Association, 79, 302–309.
John, J. A. and Draper, N. R. (1980). An alternative family of transformations. Applied Statistics, 29, 190–197.
Kanagawa, A., Arizono, I. & Ohta, H.. Design of the (x ̅-s) control chart based on Kullback-Leibler information, Frontiers in Statistical Quality Control (5th ed.), Springer.
Kullback S. Information Theory and Statistics. New York: Wiley; 1959.
Kullback, S. & Leibler, R.A. (1951) On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79-86.
Kupperman, M. (1956). Further applications of information-theory to multivariate-analysis and statistical-inference. Annals of Mathematical Statistics, 27(4), 1184-1186.
Montgomery, D. C. (1996) Introduction to Statistical Quality Control, 8th edition. New York: Wiley.
Moore, P. G. (1957) Normality in quality control charts. Applied Statistics 6(3),171-179.
Osborne, J. (2002). Notes on the use of data transformations. Practical Assessment, Research, and Evaluation, 8(1), 6.
Osborne, J. (2010). Improving your data transformations: Applying the Box-Cox transformation. Practical Assessment, Research, and Evaluation, 15(1), 12.
Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(2), 100-115.
Roberts, S. W. (1958). Properties of control chart zone tests. Bell System Technical Journal, 37(1), 83–114.
Ryan, T. P. & Faddy, B.J. (2001) The effect of non-normality on the performance of CUSUM procedures. Frontiers In Statistical Quality Control (6th ed.), Physica-Verlag.
Sakia, R. M. (1992) The Box-Cox transformation technique: a review. The Statistician. 41, 169-178.
Yanagimoto, T. (1994). The Kullback-Leibler risk of the Stein estimator and the conditional MLE. Annals of the Institute of Statistical Mathematics, 46(1), 29–41.
Yeo, I. K. & Johnson, R. A. (2000) A new family of power transformations to improve normality or symmetry. Biometrika. 87(4), 954-959.