The chart, it is concluded that the process

The Shewart 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

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 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. The
statistical measure calculated for each sample is the sample average 

 (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

 for these
samples 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 these
control limits for the

 chart, it
is concluded that the process probably was “out of statistical control” when
this sample was drawn, so its

 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 

  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

 chart. As
long as both points fall inside the control limits, there is no statistical
evidence of trouble.  When an

– control chart having 3? 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?
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

 of the
Edgeworth series. The control limits for

 chart and
?-chart calculated on the basis of normal population may be seriously affected
particularly in cases of variations showing significant departures of

 and

 from
their respective normal theory values (Dalporte (1951)). If the ? -coefficients
are unknown a priori, they should be estimated by Fisher’s

 and

 statistic
in all cases by pooling a number of ‘rational subgroups’. Dalporte (1951)
recommends utilization of such estimates in the formulae for the ?-coefficients of the sample
characteristics concerned in order to choose a suitable Pearson curve to
represent its frequency distribution. The results of extensive sampling
experiments on theoretical and empirical populations were used by Pearson and
Please (1975) to prepare a series of charts in which the robustness of the one
and two-sample ” Student’s t” statistic, the sample variance, and the variance
ratio are displayed as function of skewness and 
kurtosis  of the populations. The
robustness of both the sample correlation coefficient r and of Fisher’s Z
transformation of departures from bivariate normality were considered by Gayen
(1951).

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