Estimation of Short-Term Capability (Y1)
Step 1: Organize Data
The short-term 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
B-2 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
mid-term nature of mean performance was given as static even though the
long-term expectation would be dynamic.
TABLE B-2. RAW DATA AND SUMMARY STATISTICS FOR PARAMETER Y1
Notice that the range (R) was computed for each subgroup
Rj = Xmax -
Xmin Eq. (B-3)
For example, the range for subgroup (G1) was given as
R1 = Xmax - Xmin = 103 - 75 = 28
. Eq. (B-4)
Step 2: Compute Average Range
Next, the average range was computed using
|| Eq. (B-5)|
The result was given as
|| Rj/N = 145/6
= 24.17 .
|| Eq. (B-6)|
Step 3: Establish Constants
Following this, the appropriate d2* value was located in Table
B-3. Since in this instance, n = 5 and N = 6, it was determined that
d2* = 2.353.
Step 4: Compute Standard Deviation
The sample standard deviation was computed by dividing
the average range by the given d2
* value using the relation
S = /d2* Eq. (B-7)
In this case, the standard deviation was given
S = /d2* = 24.17/2.353 = 10.27 . Eq.
Step 5: Compute Z
Since the sampling objective associated with
Y1 was to estimate "instantaneous reproducibility," it was
determined that the next activity should be the calculation of the short-term
limiting standard normal deviate (ZST) using
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 ZST. The
result of that calculation was given as
| = 2.92 .
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 short-term
parts per million (PPM (ST)) was established with the aid of Table B-4. By entering the table in the
left-most 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)106
revealed that ppm = 1750; however, this was only for one side of the given
symmetrical bilateral specification. In the instance of Y1, the
total ppmST was determined to be 1750 x 2 = 3500. Thus, the
equivalent Eq (B-7) short- term, first-time yield was calculated as
YFT;ST = 1 - (3500/106) = .9965.
Step 7: Compute Capability Ratio
Continuing with the sequence of activity, the
short-term capability ratio for parameter Y1
was calculated using
CP = ZST/3 (Eq.
The result was given as
CP = ZST/3 = 2.92/3 = .973
. (Eq. (B-13)
As seen from the estimate of short-term parameter capability, Y1
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, Y1 proved to be a likely candidate
for short-term variance optimization.