create statistics of the individual samples. The means of each sample are called Xbar and the range of
each sample is called R. The average of the k sample standard deviations is σ
σ
. σ
Xbar
and σ
R
are the
standard deviations of the distributions of the sample means and sample ranges, respectively. These
distributions tend to be normal, regardless of the shape of the parent population (Shewhart's Law of
Sample Statistics).
So,
a. σ is the standard deviation of the population while σ
Xbar
is the standard deviation of the distribution
of sample average values
b. σ is the standard deviation of the population while σ
bar
is the average standard deviation for the
k
samples of size
n
taken from the population
c. σ
Xbar
, σ
R
and σ
σ
are the standard deviations of three distribution;
 the standard deviation of the distribution of the
k
sample averages,
 the standard deviation of the distribution of the
k
sample ranges,
 and the standard deviation of the distribution of the
k
sample standard deviations, respectively.
27. No! USL is given by the designer. UCL is a statistical parameter. There is no relationship.
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View Full Document28. The USL is 0.70 + 0.20 = 0.90. The LSL is 0.70 – 0.20 = 0.50
These are specifications on individual items. The charts reflect sample statistics and the numbers are
based on “n” items.
Problems:
6.
For the Parent Population, the parent population mean is estimated by
X
and so is 0.72
the parent population standard deviation is σ’ = R
bar
/d
2
, and for n = 4,
d
2
= 2.06 (Figure 3613) and R
bar
= 0.16.
σ’ = 0.16 / 2.06 = 0.078
Process Capability
C
p
= (USL – LSL) / 6 σ’
C
p
= (0.9 – 0.5) / ( 6 (0.078)) = 0.855
NOTE: This value is much smaller than the 1.33 value suggested as the minimum value for good
process capability.
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 Spring '08
 STAFF
 Standard Deviation, Mean, parent population

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