Range Control Charts Revisited: Simpler Tippett-like Formulae, Its Practical Implementation, And The Study Of False Alarm

This article presents simpler alternative formulae and procedures of implementation to deal with the relative range statistic used in the range control chart for process dispersion monitoring. The chart performance is assessed considering false alarm comparison between our exact limits charts versus normal approximated ones, which shows the serious drawbacks of such misplaced control limits. These much simple tools introduced here, we believe, will permit the presentation of R charts more transparently and without unrealistic normal approximations, avoiding the serious limitations of such "ad hoc" practice. Copyright © Taylor & Francis Group, LLC.422247262Alwan, L., (2000) Statistical Process Analysis, , Boston: McGrawHillAlwan, L.C., Roberts, H.V., The problem of misplace control limits (1995) Applied Statistics, 44, pp. 269-278Brent, R.P., (1973) Algorithms for Minimization Without Derivatives, , Englewood Cliffs, NJ: Prentice Hall, Ch. 4Brown, R.A., Robustness of the studentized range statistic (1974) Biometrika, 61 (1), pp. 171-175Burr, I.W., (1966) On the Distribution of Range for a General System of Distributions, , Mimeo No 88, Department of Statistics, Purdue University, West Lafayette, INBurr, I.W., The effect of non-Normality on constants for X̄ and R charts (1967) Industrial Quality Control, 23, pp. 563-568Burr, I.W., (1974) Applied Statistical Methods, p. 479. , New York: Academic Press IncCopenhaver, M.D., Holland, B., Computation of the distribution of the maximum studentized range statistic with application to multiple significance testing (1988) Journal of Statistical Computation Simulation, 30 (1), pp. 1-15Costa, A.F.B., Epprecht, E.K., Carpinetti, L.C.R., (2004) Controle Estatistico Da Qualidade, , Edit. Brasil: AtlasCrawley, M.J., (2007) The R Book, , London: WileyDavid, H.A., Nagaraja, H.N., (2003) Order Statistics, , 3rd ed., New York: Wiley-InterscienceDevor, R.E., Chang, T., Sutherland, J.W., (2007) Statistical Quality Design and Control, , 2nd ed., Saddle River, NJ: Pearson / Prentice HallFeller, W., (1966) An Introduction to Probability Theory and It's Applications, 2, pp. 148-149. , New York: WileyFerreira, D., Demetrio, C., Manly, B., Machado, A., Quantiles from the maximum studentized range distribution (2007) Biometric Brazilian Journal, 25 (1), pp. 117-135Fornberg, B., Sloan, D., Richardson method for numerical derivatives (1994) Acta Numerica, 3 (3), pp. 203-267Grant, E.L., Leavenworth, R.S., (1996) Statistical Quality Control, , 7th ed. New York: McGrawHillHarter, H.L., Tables of range and studentized range (1960) The Annals of Mathemathical Statistics, 31, pp. 1122-1147Harter, H.L., Balakrishnan, N., (1998) Tables for the use of Range and Studentized Range in Tests of Hypotheses, , Boca Raton, Fl: CRC Press IncHoeffding, W., Masstabinvariate korrelation theorie (1940) Schrifte Math. Inst, 5, pp. 181-233. , University of BerlinHsu, J.C., (1996) Multiple Comparisons: Theory and Methods, , London: Chapman and Hall, CRCJones, M.C., Balakrishnan, N., How are moments of spacing related to distribution functions (2002) J. of Stat. Plan. and Infer, 103, pp. 377-390Kenett, R., Zachs, S., (1998) Modern Industrial Statistics: The Design and Control of Quality and Reliability, , Duxbury: Duxbury PressMardia, K.V., Tippet's formulas and other results on sample range and extremes (1965) Annals of the Institute of Statistical and Mathematics, 17, pp. 85-91Mardia, K.V., Thompson, J.W., Unified treatment of moment-formulae (1972) Sankhya: The Indian Journal of Statistics, 34, pp. 121-132. , Series AMontgomery, D.C., (2008) Introduction to Statistical Quality Control, , 6th ed. New York: WileyPiessens, R., De Doncker-Kapenga, E., Uberhuber, C., Kahaner, D., (1983) Quadpack: A Subroutine Package for Automathic Integration, , Chapter II: Theoretical Background, Berlin, Springer(2009) R: A Language and Environment for Statistical Computing, , R Development Core Team Vienna: R Foundation for Statistical Computing, ViennaRyan, T.P., (1989) Statistical Methods of Quality Improvement, , New York: WileySen, P.K., The impact of hoeffding's research: Measures of association (1994) Collected Works of W. Hoeffding, , Fisher N. I., Sen, P. K., eds. New York: Springer VerlagShewhart, W.A., Quality control charts (1926) Bell System Technical Journal, 5, pp. 593-603Tippett, L.H.C., On the extreme individuals and the range of samples from a normal population (1925) Biometrika, 17, pp. 364-387Tukey, J.W., (1953) The Problem of Multiple Comparisons, , Umpublished memorandum, US