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Runs charts and associated hypothesis tests are available in many software
packages for examining the stability of a process.
Many students of statistics have heard of the magic sample size number
of N = 30 and apply it stability and validation cases. One of the reasons for
the recommendation of N = 30 is related to obtaining a precise estimate of
the standard deviation for process capability. Nelson [94] created a nomo-
graph that indicates the standard deviation reaches an error percentage of
<10% at N = 30 for standard normal distributions with 95% confidence. In
contrast, a sample of N = 10 could have error as high as 26% in the standard
deviation. Biased estimates of standard deviation are particularly problem-
atic in process capability calculations, thus the more data for estimating
the standard deviation, the better. Of course, this is assuming all impactful
sources of variation have been observed.
An example in the context of C pk is explored to further characterize the
sample size required to achieve a particular process capability. Recall
−
X − LSL XUSL (5.1)
C pk = min ,
s s
where X is the sample mean, s is the sample standard deviation, and LSL
and USL are the lower and upper specification limits, respectively.
While N = 30 provides a reasonable error percentage for standard de-
viation, this sample size may not be large enough to indicate that process
capability is above a particular threshold. An observed C pk of 1.4 may be
above 1.33, however, a C pk = 1.40 is a point estimate with no reflection of
the uncertainty in the estimate. There are two important factors that help
a manufacturer determine if C pk is above a prespecified threshold with a
particular level of confidence. They are the observed C pk from the sample
and the sample size.
A 95% lower confidence interval approach is used to conclude if the pro-
cess capability is above a threshold, often 1.33. The formula for the confidence
interval, denoted by LCL, is given in Eq. (5.2) (Montgomery, 2009) [95].
(5.2)
=
LCLC pk 1 − Z α 1 2 + 1
9 nC pk ( 2 n − ) 1
This bound should be greater than a specified minimum threshold, such
as 1.33. The value of Z 0.95 is 1.64 and n is the sample size. C pk is the estimate
