The Shewart control charts have long been valuablefor detecting when a production process has fallen out of Statistical control, i.e.,when assignable causes of erratic fluctuation have entered the process. Inparticular, the control charts for mean and range have been widely usedtogether for controlling the average and variability of a process.
These chartssometimes are used to control the process with respect to pre specifiedstandards (Target values) for the average and dispersion, but they often areinstead used with no standards given in order to detect lack of constancy ofthe cause system. Unfortunately, in the latter case, sufficiently accuratecontrol limits cannot be established by conventional methods until a largenumber of samples have been inspected. For example, Grant (1965) recommendsthat on “statistical grounds it is desirable that control limits be based on atleast 25 samples.”This has prevented the valid use of these charts during thecrucial stage of initiating a new process, during the start-up of a processjust brought in to statistical control again, or for a process whose totaloutput is not sufficiently large. As is described in various books e.g.,Grant (1965), Duncan (1965), Bowker and Liberman (1959) or Shewart (1939), the chartsare based upon the measurement of the single measurable quality characteristic (suchas dimension, weight, or tensile strength) of sample items drawn from theproduction process. The observations are taken periodically in small samples,commonly of size five, where each sample is as homogenous as possible.
Thestatistical measure calculated for each sample is the sample average (theaverage of the measurements within the sample). The first stage of controlchart procedure is to establish the appropriate control limits. This is done bydrawing a number of initial samples calculating the grand average for thesesamples and then setting the control limits at for the chart(lower and upper control limits respectively). If the for any of the samples falls outside thesecontrol limits for the chart, itis concluded that the process probably was “out of statistical control” whenthis sample was drawn, so its would bethrown out. This would be repeated as necessary for any other such sample untilthe only sample remaining seen (in absence of contrary evidence) to come from aconstant cause system. After recalculating forthese remaining samples, the control limits for future use would be resetaccordingly. After establishing the control limits, the mean for each newsample inspected would be plotted on the chart.
Aslong as both points fall inside the control limits, there is no statisticalevidence of trouble. When an – control chart having 3? limits is employedwith a process that is normally distributed, the Type-I error associated withthese control limits is 0.003.
However, this may not be the case for otherunderlying distributions. In application of control charts in particular and inmost practical applications in general, Central Limit Theorem is used. Centrallimit theorem is the rate at which the distribution of sample means approachesthe normal distribution.
Shewart (1931) has empirically shown that the standardcontrol chart limits are approximately correct for the right triangular andrectangular distributions. Burr (1967) had also presented a set of tables of 3?control limit factors for non-normal distributions. He studied the effect ofnon-normality on these factors and concluded that “we can use the ordinarynormal curve control chart constants unless the population is markedly non-normalwhen it is, the tables provide guidance on what constants to use “(Burr(1967)). Various members of the Burr family of distribution derived Burr’svalues from the expected values of the range for those distributions. Based onthe associated coefficients of Skewness and Kurtosis they provide anapproximation to the limits for other non-normal distributions.
The exactprobability of exceeding these, however, remains unknown when the process is incontrol. Shewart (1931) highlighted that most distributions showing control havebeen found to be in a close neighborhood of normality to be fitted by the first two terms of the Gram-Charlierseries. But sometimes it seems logicaland also necessary to consider a better form with terms including up to that in of theEdgeworth series.
The control limits for chart and?-chart calculated on the basis of normal population may be seriously affectedparticularly in cases of variations showing significant departures of and fromtheir respective normal theory values (Dalporte (1951)). If the ? -coefficientsare unknown a priori, they should be estimated by Fisher’s and statisticin all cases by pooling a number of ‘rational subgroups’. Dalporte (1951)recommends utilization of such estimates in the formulae for the ?-coefficients of the samplecharacteristics concerned in order to choose a suitable Pearson curve torepresent its frequency distribution. The results of extensive samplingexperiments on theoretical and empirical populations were used by Pearson andPlease (1975) to prepare a series of charts in which the robustness of the oneand two-sample ” Student’s t” statistic, the sample variance, and the varianceratio are displayed as function of skewness and kurtosis of the populations.
Therobustness of both the sample correlation coefficient r and of Fisher’s Ztransformation of departures from bivariate normality were considered by Gayen(1951).