# 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

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).