# Calculating the Process Capability Ratio for Weibull Data

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## Abstract

The process capability ratio is one of many statistical process control widely used in manufacturing and service engineering based on normality assumption. If the data do not correspond with the assumption, they can lead to erroneous conclusions. So the wrong conclusions would cost manufacturing and service organization big financial losses and lost customers to competitors. One approach to dealing with this situation is to transform the data so that in the new, the transformed data have a normal distribution appearance. Many authors investigated the standard transformation such as square root, logarithmic, and so on including the well known Box-Cox transformation to transform data are not normally distributed to normality. However, transformations may behave sufficiently normal for statistical process control but they do not yield accurate process performance index. In this paper, the use of Manly transformation, Yeo-Johnson transformation, and Nelson transformation to transform Weibull data are investigated in sense of calculating the process capability ratio and the coefficient of variation. It is found that

**all of three transformations can be used for transforming Weibull data to data that are normally distributed and the average of the process capability ratio of transformed data via all of them is not different at significant level 0.05 although the average of coefficient of variation of data transformed by Nelson transformation is the lowest. However, Nelson transformation is easy to work because it does not need the transformation parameter.**## References

Aichouni, M., Al-Ghonamy, A. I., & Bachioua, L. (2014). Control charts for non-normal data: Illustrative Example from the construction industry business. In Proceedings of the 16th International Conference on Mathematical and Computational Methods in Science and Engineering, held in Malaysia, 23-25 April, 2006 (pp. 71-76). Malaysia: Kualalumpur.

Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society–Series B, 26, 211-252.

Doksum, K. A., & Wong, C. (1983). Statistical tests based on transformed data. Journal of the American Statistical Association, 78, 411-417.

John, J. A., & Draper, N. R. (1980). An alternative

family of transformations. Applied Statistics, 29, 190-197.

Levinson, W. A. (2010). Statistical process control for nonnormal distributions. Retrieved from http://www.ct-yankee.com/spc/nonnormal.html

Mach, P., Thuring, J., & Samal, D. (2006). Transformation of data for statistical processing. In Proceedings of the 29th International Spring Seminar on Electronics Technology, held in Germany, 10-14 May 2006 (pp. 278-282). Germany: St. Marienthal.

Manly, B. F. J. (1976). Exponential data transformations. Statistician, 25, 37-42.

Montgomery, D. C. (2001). Introduction to Statistical Quality Control (4th ed.). New York, USA: John Wiley & Sons.

Montgomery, D. C., Runger, G. C., & Hubele, N. F. (2004). Engineering Staistics (3rd ed.). New York, USA: John Wiley & Sons.

UCI Machine Learning Repository. (2007). Concrete Compressive Strength Data Set. Retrieved from https://archive.ics.uci.edu/ml/

datasets/Concrete+Compressive+Strength.

Weisberg, S. (2011). Yeo-Johnson Power Transformations. Retrieved from https://www.stat.umn.edu/arc/yjpower.pdf

WSDOT FOP for AASHTO T 22. (2013). Compressive strength of cylindrical concrete specimens. Retrieved from http://www.wsdot.

wa.gov/publications/manuals/fulltext/M46-01/t22.pdf

Yeo I. K., & Johnson R. A. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society–Series B, 26, 211-252.

Doksum, K. A., & Wong, C. (1983). Statistical tests based on transformed data. Journal of the American Statistical Association, 78, 411-417.

John, J. A., & Draper, N. R. (1980). An alternative

family of transformations. Applied Statistics, 29, 190-197.

Levinson, W. A. (2010). Statistical process control for nonnormal distributions. Retrieved from http://www.ct-yankee.com/spc/nonnormal.html

Mach, P., Thuring, J., & Samal, D. (2006). Transformation of data for statistical processing. In Proceedings of the 29th International Spring Seminar on Electronics Technology, held in Germany, 10-14 May 2006 (pp. 278-282). Germany: St. Marienthal.

Manly, B. F. J. (1976). Exponential data transformations. Statistician, 25, 37-42.

Montgomery, D. C. (2001). Introduction to Statistical Quality Control (4th ed.). New York, USA: John Wiley & Sons.

Montgomery, D. C., Runger, G. C., & Hubele, N. F. (2004). Engineering Staistics (3rd ed.). New York, USA: John Wiley & Sons.

UCI Machine Learning Repository. (2007). Concrete Compressive Strength Data Set. Retrieved from https://archive.ics.uci.edu/ml/

datasets/Concrete+Compressive+Strength.

Weisberg, S. (2011). Yeo-Johnson Power Transformations. Retrieved from https://www.stat.umn.edu/arc/yjpower.pdf

WSDOT FOP for AASHTO T 22. (2013). Compressive strength of cylindrical concrete specimens. Retrieved from http://www.wsdot.

wa.gov/publications/manuals/fulltext/M46-01/t22.pdf

Yeo I. K., & Johnson R. A. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

Keywords

Process capability ratio, Manly transformation, Yeo-Johnson transformation, Nelson transformation

Section

Research Articles

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How to Cite

WATTHANACHEEWAKUL, Lakhana.
Calculating the Process Capability Ratio for Weibull Data.

**Naresuan University Journal: Science and Technology (NUJST)**, [S.l.], v. 25, n. 1, p. 44-56, feb. 2017. ISSN 2539-553X. Available at: <http://www.journal.nu.ac.th/NUJST/article/view/1675>. Date accessed: 17 oct. 2019.