An alternative expression for the constant c4[n] with desirable properties

Document Type : Article


Faculty of Economics and Business, Campus Cartuja s/n. 18071 Granada, Spain


The constant c4[n] is commonly used in the construction of control charts and the estimation of process capability indices, where n denotes the sample size. Assuming the Normal distribution the unbiased estimator of the population standard deviation is obtained by dividing the sample standard deviation by the constant c4[n]. An alternative expression for c4[n] is proposed, and the mathematical induction technique is used to prove its validity. Some desirable properties are described. First, the suggested expression provides the exact value of c4[n]. Second, it is not a recursive formula in the sense it does not depend on the previous sample size. Finally, the value of c4[n] can be directly computed for large sample sizes. Such properties suggest that the proposed expression may be a convenient solution in computer programming, and it has direct applications in statistical quality control.


1. Mitra, A., Fundamentals of Quality Control and Improvement, John Wiley & Sons (2016).
2. Montgomery, D.C., Statistical Quality Control, 7, New York: Wiley (2009).
3. Chen, G. "The mean and standard deviation of the run length distribution of X charts when control limits are estimated", Statistica Sinica, 7(3), pp. 789-798 (1997).
4. Chou, Y.M., Mason, R.L., and Young, J.C. "The SPRT control chart for standard deviation based on individual observations", Quality Technology & Quantitative Management, 3(3), pp. 335-345 (2006).
5. Huang, W.H., Yeh, A.B., and Wang, H. "A control chart for the lognormal standard deviation", Quality Technology & Quantitative Management, 15(1), pp. 1- 36 (2018).
6. Ajadi, J.O. and Riaz, M. "New memory-type control charts for monitoring process mean and dispersion", Scientia Iranica, Transaction E: Industrial Engineering, 24(6), pp. 3423-3438 (2017).
7. Diko, M.D., Chakraborti, S., and Does, R.J.M.M. "Guaranteed in-control performance of the EWMA chart for monitoring the mean", Quality Reliability and Engineering International, 35(4), pp. 1144-1160 (2019).
8. Chen, C.H. and Chou, C.Y. "Economic design of product and process parameters under the specified process capability value", Quality Technology & Quantitative Management, 15(6), pp. 686-701 (2017).
9. Liao, M.Y. "Process capability control chart for nonnormal data- evidence of on-going capability assessment", Quality Technology & Quantitative Management, 13(2), pp. 165-181 (2016).
10. Polansky, A.M. and Maple, A. "Using Bayesian models to assess the capability of a manufacturing process in the presence of unobserved assignable causes", Quality Technology & Quantitative Management, 13(2), pp. 139-164 (2016).
11. Keshteli, R.N., Kazemzadeh, R.B., Amiri, A., et al. "Developing functional process capability indices for simple linear profile", Scientia Iranica, Transaction E: Industrial Engineering, 21(3), pp. 1044-1050 (2014).
12. Kasprikova, N. and Klufa, J. "AOQL sampling plans for inspection by variables and attributes versus the plans for inspection by attributes", Quality Technology & Quantitative Management, 12(2), pp. 133-142 (2015).
13. Robertson, B.L., McDonald, T., Price, C.J., et al. "A modification of balanced acceptance sampling", Statistics & Probability Letters, 129, pp. 107-112 (2017).
14. Mahmoud, M.A., Henderson, G.R., Epprecht, E.K., et al. "Estimating the standard deviation in qualitycontrol applications", Journal of Quality Technology, 42(4), pp. 348-357 (2010).
15. Munoz-Rosas, J.F., Alvarez-Verdejo, E., Perez-Arostegui, M.N., et al. "Empirical comparisons of Xbar charts when control limits are estimated", Quality and Reliability Engineering International, 32(2), pp. 453-464 (2016).
16. Chen, S.M., Liaw, J.T., and Hsu, Y. S. "Sample size determination for Cp comparisons", Scientia Iranica, Transactions E: Industrial Engineering, 23(6), pp. 3072-3085 (2016).
17. Parry, G., Vendrell-Herrero, F., and Bustinza, O.F. "Using data in decision-making: Analysis from the music industry", Strategic Change, 23(3-4), pp. 265- 277 (2014).
18. Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality, Springer Science & Business Media (2009).
19. Bolch, B.W. "More on unbiased estimation of the standard deviation", The American Statistician, 22(3), p. 27 (1968).
20. Cryer, J.D. and Ryan, T.P. "The estimation of sigma for an X chart: MR=d2 or S=c4?", Journal of Quality Technology, 22(3), pp. 187-192 (1990).
21. Gurland, J. and Tripathi, R.C. "A simple approximation for unbiased estimation of the standard deviation", The American Statistician, 25(4), pp. 30-32 (1971).
22. Team, R.C. "R: A language and environment for statistical computing", Vienna, Austria: R Foundation for Statistical Computing (2016).
23. Huberts, L.C., Schoonhoven, M., Goedhart, R., et al. "The performance of control charts for large nonnormally distributed datasets", Quality and Reliability Engineering International, 34(6), pp. 979-996 (2018).
24. Franklin, J., Proof in Mathematics: An Introduction, Kew Books (1996).
25. Hermes, H., Introduction to Mathematical Logic, Springer Science & Business Media (2013).
26. Chen, C.P. and Qi, F. "The best bounds in Wallis' inequality", Proceedings of the American Mathematical Society, pp. 397-401 (2005).