But due to the sampling cost the sample size n is not sufficiently large Therefore for non normally distributed measurements the traditional way for designing the control chart may reduce the ability that a control chart detects the assignable causes Yourstone and Zimmer 1992 used the Burr distribution to represent various non normal distributions and consequently to statistically design the control limits of an X chart Chou and Cheng 1997 extended the model presented by Yourstone and Zimmer to design the control limits of the ranges control chart under non normality Schilling and Nelson 1976 studied the effect of non normality on the control limits of X charts They concluded that the sample size n should be at least 4 in order to assure P_0to be 0014 or less Padgett et al 1992 examine the effect of non normality on the design schemes when µ and σ are estimated by their usual estimators i e the mean of the sample means µ and the mean sample standard deviation σ They also observed that the in control probability of signaling of both charts highly increase under non normality Schoonhoven et al 2009 studied the design schemes for the X control charts under normality
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Dalporte 1951 revealed the marked effect of non normality on the limits of the control chart for mean of samples of size ten from the population of an industrial variable The behavior of X control chart schemes for skewed distributions by listing ARL values for various non normal parent populations were discussed by Burrows 1962 The behavior of upper and lower control limits of X chart in class of non normal variations were discussed by Gayen 1957 Bradley 1977 claimed that many areas of statistical application can easily lead to quite non normal distributions It is well known that parametric tests can be more powerful than the distribution free tests when the usual assumption of normality and independence are satisfied Wilks 1962 One might apply some tests of normality in order to recognize situations in which distribution free procedures would be likely to produce greater power than the parametric counterpart Nelson 1981 has observed that normality testing with both mean and variance unknown is one most common and important problems on statistical practice Two of the older tests of normality are the third and fourth moment tests also known as tests of Skewness and Kurtosis respectively Testing for normality to choose between parametric and non parametric tests may not work well with small sample size n 10 Best tests of normality may lack power for detecting non normality with small n testing of normality can provide useful information even if one is not concerned about the choice between parametric and nonparametric tests Olejnik and Olgina 1987 indicates that two different non parametric tests of variance should be applied depending upon the population being Platykurtic or Leptokurtic Bradley 1977 discussed a number of situations in which unexpected results in an experiment can be attributed by markedly non normal distributions Other than the normality assumption on designing control charts one usually assumes that the measurements within the samples are independently distributed But this assumption may not be workable in some specific processes For the assumption of independent measurements when the production process consists of multiple but similar units of single part Grant and Leaven 1988 the collected measurements within a sample may be correlated Neuhardt 1987 examined the effect of correlation within a sub group on control charts For normality assumption the Statistic X is normally distributed if the measurements are really normally distributed For asymmetrically distributed measurements the statistic X will be approximately normally distributed only when the sample size n is sufficiently large
They consider different estimators of the standard deviation and for each scheme the correction factor is derived by controlling P _0 Marit and Ronald 2009 examined the behavior of X chart under non normality Soleimani et al 2009 presented an example of situation for error terms within profiles are correlated Noorossana et al 2008b studied the effect of non normality of the error terms using gamma and t distribution to model the behavior of observations Noorossana et al 2008a investigated the effect of correlation on the linear profile monitoring models Noorossana et al 2010 studied the effect of simultaneous violation of normality and independence on the performance of three common methods of linear profile monitoring To study the effect of non normality and correlation they consider the use of both heavy tailed symmetrical and skewed non normal distributions with dependent error terms generated by first order autocorrelation model They consider t distribution and gamma distribution as a skewed distribution to study the effect of non normality on the performance of linear profile monitoring methods In this chapter the upper and lower control limits for mean under non normal and correlated data are constructed The non normal distribution has been represented by first four terms of an Edgeworth series The values of standardized cumulants λ _ 3 β _1 and λ _4 β _2 3 considered are with Barton and Dannis 1952 limits which means that for such values the population is positive definite and unimodal For non normal populations and different values of correlation coefficient ρ the value of upper and lower control limits are tabulated and compared with those of the normal population For various non normal populations with various non normality parameters λ _3 λ _4 and correlation coefficient ρ the values of upper and lower control limits are given in tables for different values of n
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