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OTE/SPH
OTE/SPH
Char Count= 0
2:55
JWBK119-06
August 31, 2006
Nested Design 79
1.0000
0.9500

0.9000
c 4
0.8500

0.8000 Exact
Approximation
0.7500
0 5 10 15 20 25 30 35
n

Figure 6.3 Values of c 4 (exact and approximate).

Figure 6.3 shows the c 4 curves for the values obtained from exact calculation as
well as the approximation given. From the curves, it is clear that the approximation
is closed enough to the values calculated using the exact formula.

6.2.2 A simulation example
When carrying out process characterization and process capability studies, in order
to avoid underestimating the process variation and overestimating the process capa-
bility, the unbiased estimator s/c 4 , instead of the biased estimator s, should be used
to estimate the process variation. Otherwise, an incorrect judgment might be made in
qualifying a product that is not designed for manufacturability, or in determining the
process capability during volume build.
Here, a Monte Carlo simulation is used to show the unbiasedness of the suggested
estimator. A normally distributed process output with μ = 50 and σ = 5 is simulated.
Five measurements are taken each day for 30 days. The process is replicated 12 times
to simulate the collection of data across a year. The average sample standard deviation
(both s Bar and s Overall ) over the month is shown in Table 6.2.
It may be observed that correction of s with c 4 provides an unbiased estimate of the
known population standard deviation (σ = 5).

6.3 NESTED DESIGN

A nested design (sometimes referred to as a hierarchical design) is used for experiments
involving a set of treatments where the experimental units are subsampled, that is,
a nested design is one in which random sampling is done from a number of groups