2.
Measurement Process Characterization
2.4.
Gauge R & R studies
2.4.4.
Analysis of variability
2.4.4.1.
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Analysis of repeatability
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Case study: Resistivity probes
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The repeatability quantifies the basic precision for the gauge. A
level-1 repeatability standard deviation is
computed for each group of J repetitions, and a graphical
analysis is recommended for deciding if repeatability is dependent on
the check standard, the operator, or the gauge. Two graphs are
recommended. These should show:
Typically, we expect the standard deviation to be gauge dependent -- in
which case there should be a separate standard deviation for each gauge.
If the gauges are all at the same level of precision, the values can be
combined over all gauges.
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Repeatability standard deviations can be pooled over operators, runs,
and check standards
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A repeatability standard deviation from J repetitions is not a
reliable estimate of the precision of the gauge. Fortunately, these
standard deviations can be pooled over days; runs; and check standards,
if appropriate, to produce a more reliable precision measure. The
table below shows a mechanism for pooling. The pooled repeatability
standard deviation,
 ,
has LK(J - 1) degrees of freedom for measurements taken over:
- J repetitions
- K days
- L runs
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Basic pooling rules
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The table below gives the mechanism for pooling repeatability standard
deviations over days and runs. The pooled value is an average of
weighted variances and is shown as the last entry in the right-hand
column of the table. The pooling can also cover check standards,
if appropriate.
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View of entire dataset from the
nested design
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To illustrate the calculations, a subset of data collected in a nested
design for one check standard (#140) and one probe (#2362) are shown
below. The measurements are resistivity (ohm.cm) readings with six
repetitions per day. The individual level-1 standard deviations from
the six repetitions and degrees of freedom are recorded in the last
two columns of the database.
Run Wafer Probe Month Day Op Temp Average Stddev df
1 140 2362 3 15 1 23.08 96.0771 0.1024 5
1 140 2362 3 17 1 23.00 95.9976 0.0943 5
1 140 2362 3 18 1 23.01 96.0148 0.0622 5
1 140 2362 3 22 1 23.27 96.0397 0.0702 5
1 140 2362 3 23 2 23.24 96.0407 0.0627 5
1 140 2362 3 24 2 23.13 96.0445 0.0622 5
2 140 2362 4 12 1 22.88 96.0793 0.0996 5
2 140 2362 4 18 2 22.76 96.1115 0.0533 5
2 140 2362 4 19 2 22.79 96.0803 0.0364 5
2 140 2362 4 19 1 22.71 96.0411 0.0768 5
2 140 2362 4 20 2 22.84 96.0988 0.1042 5
2 140 2362 4 21 1 22.94 96.0482 0.0868 5
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Pooled repeatability standard deviations over days, runs
| Source of Variability |
Degrees of Freedom |
Standard Deviations |
Sum of Squares (SS) |
| Probe 2362
|  |
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run 1 - day 1
run 1 - day 2
run 1 - day 3
run 1 - day 4
run 1 - day 5
run 1 - day 6
run 2 - day 1
run 2 - day 2
run 2 - day 3
run 2 - day 4
run 2 - day 5
run 2 - day 6
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5
5
5
5
5
5
5
5
5
5
5
5
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0.1024
0.0943
0.0622
0.0702
0.0627
0.0622
0.0996
0.0533
0.0364
0.0768
0.1042
0.0868
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0.05243
0.04446
0.01934
0.02464
0.01966
0.01934
0.04960
0.01420
0.00662
0.02949
0.05429
0.03767
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gives the total degrees of freedom
for s1
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60
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gives the total sum of squares
for s1
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0.37176
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The pooled value of s1 is given by
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0.07871
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Run software macro
for pooling standard deviations
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The Dataplot commands (corresponding to the calculations in the table above)
dimension 500 30
read mpc411.dat run wafer probe month day op temp avg s1i vi
let ssi=vi*s1i*s1i
let ss=sum ssi
let v = sum vi
let s1 = (ss/v)**0.5
print s1 v
return the following pooled values for the repeatability standard deviation
and degrees of freedom.
PARAMETERS AND CONSTANTS--
S1 -- 0.7871435E-01
V -- 0.6000000E+02
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