The Shewhart control charts have long been valuable for detecting when a production process has fallen out of Statistical control i e when assignable causes of erratic fluctuation have entered the process In particular the control charts for mean and range have been widely used together for controlling the average and variability of a process These charts sometimes are used to control the process with respect to pre specified standards Target values for the average and dispersion but they often are instead used with no standards given in order to detect lack of constancy of the cause system Unfortunately in the latter case sufficiently accurate control limits cannot be established by conventional methods until a large number of samples have been inspected For example Grant 1965 recommends that on statistical grounds it is desirable that control limits be based on at least 25 samples This has prevented the valid use of these charts during the crucial stage of initiating a new process during the start up of a process just brought in to statistical control again or for a process whose total output is not sufficiently large As is described in various books e g Grant 1965 Duncan 1965 Bowker and Liberman 1959 or Shewart 1939 the X charts are based upon the measurement of the single measurable quality characteristic such as dimension weight or tensile strength of sample items drawn from the production process The observations are taken periodically in small samples commonly of size five where each sample is as homogenous as possible
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The statistical measure calculated for each sample is the sample average x the average of the measurements within the sample The first stage of control chart procedure is to establish the appropriate control limits This is done by drawing a number of initial samples calculating the grand average X for these samples and then setting the control limits at X A2R for the X chart lower and upper control limits respectively If the X for any of the samples falls outside these control limits for the X chart it is concluded that the process probably was out of statistical control when this sample was drawn so its X would be thrown out This would be repeated as necessary for any other such sample until the only sample remaining seen in absence of contrary evidence to come from a constant cause system After recalculating X for these remaining samples the control limits for future use would be reset accordingly After establishing the control limits the mean for each new sample inspected would be plotted on the X chart As long as both points fall inside the control limits there is no statistical evidence of trouble When an X control chart having 3 σ limits is employed with a process that is normally distributed the Type I error associated with these control limits is 0 003 However this may not be the case for other underlying distributions In application of control charts in particular and in most practical applications in general Central Limit
Theorem is used Central limit theorem is the rate at which the distribution of sample means approaches the normal distribution Shewart 1931 has empirically shown that the standard control chart limits are approximately correct for the right triangular and rectangular distributions Burr 1967 had also presented a set of tables of 3 σ control limit factors for non normal distributions He studied the effect of non normality on these factors and concluded that we can use the ordinary normal curve control chart constants unless the population is markedly non normal when it is the tables provide guidance on what constants to use Burr 1967 Various members of the Burr family of distribution derived Burr s values from the expected values of the range for those distributions Based on the associated coefficients of Skewness and Kurtosis they provide an approximation to the limits for other non normal distributions The exact probability of exceeding these however remains unknown when the process is in control Shewart 1931 highlighted that most distributions showing control have been found to be in a close neighborhood of normality to be fitted by the first two terms of the Gram Charlier series But sometimes it seems logical and also necessary to consider a better form with terms including up to that in β _1 of the Edgeworth series The control limits for X chart and σ chart calculated on the basis of normal population may be seriously affected particularly in cases of variations showing significant departures of β _1 and β _2 from their respective normal theory values Dalporte 1951 If the β coefficients are unknown a priori they should be estimated by Fisher s g_1 and g_2 statistic in all cases by pooling a number of rational subgroups Dalporte 1951 recommends utilization of such estimates in the formulae for the β coefficients of the sample characteristics concerned in order to choose a suitable Pearson curve to represent its frequency distribution
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