Page 107 - Six Sigma for electronics design and manufacturing

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sample averages, whereas specification limits are related to popula-
tion distributions of parts. It is desirable to have the specification lim-
its as large as possible compared to the process control limit.
The control limits represent the 3 s points, based on a sample of n
observations. To determine the standard deviation of the product pop-
ulation, the central limit theorem can be used:

s =
n
where Six Sigma for Electronics Design and Manufacturing (3.5)
s = standard deviation the distribution of sample averages
= population deviation
n = sample size
Multiplying 1/3 the distance from the centerline of the X chart to
one of the control limits by n will determine the total product popu-
lation deviation. A simpler approximation is the use of the formula
= R /d 2 from control chart factors in Table 3.1 to generate the total
product standard deviation directly from the control chart data. d 2
can be used as a good estimator for when using small numbers of
samples and their ranges.
3.2.4 X , R variable control chart calculations example
Example 3.1
In this example, a critical dimension for a part is measured as it is be-
ing inspected in a machining operation. To set up the control chart,
four measurements were taken every day for 25 successive days, to
approximate the daily production variability. These measurements
were then used to calculate the limits of the control charts. The meas-
urements are shown in Table 3.2.
It should be noted that the value n used in Equation 3.5 is equal to
4, which is the number of observations in each sample. This is not to
be confused with the 25 sets of subgroups or samples for the historical
record of the process. If the 25 samples are taken daily, they represent
approximately a one-month history of production.
During the first day, four samples were taken, measuring 9, 12, 11,
and 14 thousands of an inch. These were recorded in the top of the
four columns of sample #1. The average, or X , was calculated and en-
tered in column 5, and the R is entered in column 6.
X Sample 1 = (9 + 12 + 11 + 14)/4 = 11.50
The range, or R, is calculated by taking the highest reading (14 in
this case), minus the lowest reading (9 in this case).
R Sample 1 = 14 – 9 = 5