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43.4 53.0

4:1 48.3 44.8

Dilution 46.4 47.6 top
ratio
Input
Location
2:1 40.2 45.6 bottom
off Stirring on

4−1
FIGURE 28.1 A 2 fractional factorial design and the average of duplicated measurements at each of the eight design
settings.
TABLE 28.5
Estimated Effects and Their Standard Errors
Contributing
Factors and Estimated Estimated
Effect Interactions Effect Standard Error

Average + 1234 Average(I) + SDLF 46.2 0.41
1 + 234 S + DLF 3.2 0.82
2 + 134 D + SLF 2.4 0.82
3 + 124 L + SDF 2.9 0.82
4 + 123 F + SDL 4.3 0.82
12 + 34 SD + LF −0.1 0.82
13 + 24 SL + DF 2.2 0.82
23 + 14 DL + SF −1.2 0.82

estimate the variance of the average response for each run. For a single pair of duplicate observations
(y 1i and y 2i ), the sample variance is:

s i = 1 ---d i 2
2
2
where d i = y 1i − y 2i is the difference between the two observations. The average of the duplicate observ-
ations is:

y i = y 1i – y 2i
------------------
2
and the variance of the average of the duplicates is:

2 2
s y = ---- = d i
s i
2
-----
2 4
The individual estimates for n pairs of duplicate observations can be combined to get a pooled estimate
of the variance of the average:
n 2 n
1
s y = 1 ∑ ----- = ------ ∑ d i 2
2
d i
---
n 4 4n
i=1 i=1
© 2002 By CRC Press LLC