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76 August 31, 2006 2:55 Process Variations and Their Estimates
are defined as the cumulative effect of many small, essentially unavoidable causes,
whereas the assignable causes are generally large compared to the chance causes, and
usually represent an unacceptable level of process performance.
It is obvious that the variability of the measurement system used should be as small
as possible, so that the variability observed is mainly from the process and so that
action can be taken accordingly to reduce process variability.

6.1.4 Sampling and rational subgroups
Distinguishing between the types of variation is paramount because the appropriate
managerial actions are different for each. Without this distinction, management will
never be able to identify the key elements for improvement. Sampling consists of
selecting some part of a population to observe so that one may have some idea about
the whole population. Due to time and/or cost constraints, sampling inspection is
preferred. From the previous section, it is clear that there are various sources of vari-
ation. In order to measure (or control) certain quality characteristics more effectively,
the concept of rational subgroups can be employed.
The rational subgroup concept means that subgroups or samples should be selected
so that if there is a change in population parameter, the chance of differences between
subgroups will be maximized, while the chance of differences due to the change in
parameter within a subgroup is minimized. For example, if some of the observations
in the sample are taken at the end of one shift and the remaining observations are
taken at the start of the next shift, then any differences between shifts might not
be detected. The undetected differences between shifts might consist of variation
between operators, between machines, and also between set-ups. Thus, if one were
to monitor the performances of certain machines, the rational subgroups should be
constructed accordingly.
A basic requirement for the applicability of a subgroup approach is that the obser-
vations within the subgroup naturally represent some appropriately defined unit of
time. In addition, subgroups should be chosen in such a way that the observations
within the sample have been, in all likelihood, measured under the same process con-
ditions. By doing so, the subgroup elements will be as homogeneous as possible and
thus special causes are more likely reflected in greater variability of the subgroups
themselves.

6.2 PROCESS VARIABILITY

2
The process variability can be measured by the population variance, σ . When the
population is finite and consist of N values, the population variance is given by

N 2
2 i=1 (x i − μ)
σ = ,
N
2
where μ is the process mean. The positive square root of σ ,or σ, is the popula-
tion standard deviation. The standard deviation was first proposed by Pearson as a
measurement of variation or scatter within a data set. 3