Estimation of ShortTerm Capability (Y_{1})
Step 1: Organize Data
The shortterm data was gathered using a sequential
sampling strategy under the constraint n = 5, N = 6. The individual
measurements (X) were recorded in an appropriate manner. Table
B2 displays the raw data and related summary statistics. From this
table, the mean of Y_{ 1} was declared as a "static" parameter, since the sampling conditions present in the given
situation tended to reveal a static mean offset versus dynamic mean behavior.
In other words, it was unlikely that X would approach the target after
sampling a highly limited number of subgroups. Because of this, the expected
midterm nature of mean performance was given as static even though the
longterm expectation would be dynamic.
TABLE B2. RAW DATA AND SUMMARY STATISTICS FOR PARAMETER Y_{1}
__________________________________ 
Subgroup 
Observation 
_______________________ 
(X_{i}) 
G_{1} 
G_{2} 
G_{3} 
G_{4} 
G_{5} 
G_{6} 
__________________________________ 
X_{1} X_{2} X_{3} X_{4} X_{5} 
103_{ } 90_{ } 95_{
} 81_{ } 75_{ } 
94_{ } 89_{ } 82_{
} 73_{ } 68_{ } 
86_{ } 78_{ } 71_{
} 65_{ } 61_{ } 
101_{ } 94_{ } 85_{
} 80_{ } 79_{ } 
89_{ } 77_{ } 72_{
} 69_{ } 68_{ } 
92_{ } 84_{ } 81_{
} 72_{ } 69_{ } 
__________________________________

R

28 444 88.8 
26 406 81.2 
25 361 72.2 
22 439 87.8 
21 375 75.0 
23 398 79.6 
__________________________________ 
Notice that the range (R) was computed for each subgroup
using
R_{j} = X_{max} 
X_{min} Eq. (B3)
For example, the range for subgroup (G_{1}) was given as
R_{1} = X_{max}  X_{min} = 103  75 = 28
. Eq. (B4)
Step 2: Compute Average Range
Next, the average range was computed using
= 

R_{j}/N 
Eq. (B5) 
The result was given as
= 

R_{j}/N = 145/6
= 24.17 . 
Eq. (B6) 
Step 3: Establish Constants
Following this, the appropriate d_{2}* value was located in Table
B3. Since in this instance, n = 5 and N = 6, it was determined that
d_{2}* = 2.353.
Step 4: Compute Standard Deviation
The sample standard deviation was computed by dividing
the average range by the given d_{2}
* value using the relation
S = /d_{2}* Eq. (B7)
In this case, the standard deviation was given
as
S = /d_{2}* = 24.17/2.353 = 10.27 . Eq.
(B8)
Step 5: Compute Z
Since the sampling objective associated with
Y_{1} was to estimate "instantaneous reproducibility," it was
determined that the next activity should be the calculation of the shortterm
limiting standard normal deviate (Z_{ST}) using
Z_{ST} 
= 
TSL /d_{2} 
Eq.
(B9) 
In this instance, the data was first plotted in the form of a histogram.
Subsequent analysis of the histogram gave no reason to dispute normality. The
analytical process was therefore continued by calculating Z_{ST}. The
result of that calculation was given as
Z_{ST} 
= 
SLT S 
= 
12090 10.27 
= 2.92 .
Eq.
(B9) 
It should be noted that the sample standard deviation (S) was substituted
for the population standard deviation (s).
Step 6: Establish the Yield
Once Z (Short Term) had been estimated, the shortterm
parts per million (PPM (ST)) was established with the aid of Table B4. By entering the table in the
leftmost column, the row Z value of 2.9 was located. Next, the .02 column was
located. The Z value of 2.92 proved to be the intersect of the given row and
column. Application of
ppm = (d/um)10^{6
} Eq. (B11)
revealed that ppm = 1750; however, this was only for one side of the given
symmetrical bilateral specification. In the instance of Y_{1}, the
total ppm_{ST} was determined to be 1750 x 2 = 3500. Thus, the
equivalent Eq (B7) short term, firsttime yield was calculated as
Y_{FT;ST} = 1  (3500/10^{6}) = .9965.
Step 7: Compute Capability Ratio
Continuing with the sequence of activity, the
shortterm capability ratio for parameter Y_{1}
was calculated using
C_{P} = Z_{ST}/3 (Eq.
(B12)
The result was given as
C_{P} = Z_{ST}/3 = 2.92/3 = .973
. (Eq. (B13)
As seen from the estimate of shortterm parameter capability, Y_{1}
exhibited a very marginal level of performance. If the mean of this parameter
was to vary over time, the estimates of capability would deteriorate
appreciably. As a consequence, Y_{1} proved to be a likely candidate
for shortterm variance optimization.