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OTE/SPH
OTE/SPH
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JWBK119-06
82 August 31, 2006 2:55 Process Variations and Their Estimates
Lots (21) Lot 1 Lot 19 Lot 20 Lot 21

Wafers (4) Wafer 1 Wafer 2 Wafer 3 Wafer 4
Reading 1
Readings (2)
Reading 2
Reading 1 Reading 2
Figure 6.5 Nested design for oxide thickness example.

6.3.1 Variance components and their calculations
Here, an analysis of variance (ANOVA) for the data in Table 6.3 is performed to show
the differences between the variance components. From the table, the correlation
factor, c, is given by
2
¯2
c = N • Y = 168(963.04) = 155 809 548.21,
¯2
where N is the total number of readings (measurements) and Y is the grand mean.
The total sum of squares, SS tot is
SS tot = ss y − c

where SS Y is the sum of square for each measurement. The sum of squares for lots is
L
2
y
SS L = MW (¯ k − ¯y) ,
k=1
where L is the total number of lots, M is the total number of measurements in each
wafer, and W is the total number of wafers in each lot. Thus, the sum of squares for
wafers is
L W
2
y
y
SS W = N (¯ jk − ¯ k )
k=1 j=1
and the sum of squares for measurements is
L W M
2
y
y
SS M = (¯ ijk − ¯ jk ) ;
k=1 j=1 i=1
alternatively, SS M may be obtained by
SS M = SS tot − SS L − SS W .
The degrees of freedom are given in Table 6.4.

Table 6.4 Degree of freedom.

Degree of freedom Total Lot Wafer Measurement
df tot = N − 1 df L = L − 1 df w = LW − 1 df M = LW(M − 1)