2.
Measurement Process Characterization
2.4.
Gauge R & R studies
2.4.4.
Analysis of variability
2.4.4.3.
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Analysis of stability
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Case study: Resistivity probes
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Run-to-run variability can be assessed graphically by a
plot of check standard values
(averaged over J repetitions) versus time with a separate graph
for each check standard. Data on all check standards should be plotted
on one page to obtain an overall view of the measurement situation.
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Advantage of pooling
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A level-3 standard deviation with
(L - 1) degrees of freedom is computed from the run averages.
Because there will rarely be more than 2 runs per check standard,
resulting in 1 degree of freedom per check standard, it is prudent to
have three or more check standards in the design in order to take
advantage of pooling. The mechanism for pooling over check standards
is shown in the table below. The pooled standard deviation has
Q(L - 1) degrees and is shown as the last entry in the
right-hand column of the table.
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Example of pooling
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Level-3 standard deviations for a single gauge pooled over
check standards
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Source of variability
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Standard deviation
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Degrees of freedom (DF)
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Sum of squares (SS)
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Level-3
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![SS = SUM[nu(i)*s3(i)]](mpc443d.gif)
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Chk std 138
Chk std 139
Chk std 140
Chk std 141
Chk std 142
Sum
Pooled value
![s3 = SS/SQRT(SUM[nu(i)])](mpc443c.gif)
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0.0223
0.0027
0.0289
0.0133
0.0205
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1
1
1
1
1
--------------
5
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0.0004973
0.0000073
0.0008352
0.0001769
0.0004203
-----------
0.0019370
0.0197
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Run software macro for computing level-3
standard deviation
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A subset of data collected in a nested design
on one check standard (#140) with probe (#2362) for six days and two
runs is analyzed for between-run effects. Dataplot commands to compute
the level-3 standard deviation from the averages of 2 runs are:
dimension 30 columns
read mpc441.dat run wafer probe mo ...
day op temp y s df
let y1 = average y subset run 1
let y2 = average y subset run 2
let ybar = (y1 + y2)/2
let ss = (y1-ybar)**2 + (y2-ybar)**2
let v3 = 1
let s3 = (ss/v3)**.5
print s3 v3
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Dataplot output
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Dataplot returns the level-3 standard deviation and degrees of freedom:
PARAMETERS AND CONSTANTS--
S3 -- 0.2885137E-01
V3 -- 0.1000000E+01
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Relationship to long-term changes, days and gauge precision
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The size of the between-run effect can be calculated by subtraction
using the standard deviations for days and gauge precision as
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