Probe Run 1 DF Run 2 DF Pooled DF 2362. 0.0658 150 0.0758 150 0.0710 300 
Graphs of reproducibility and stability for probe #2362 
Averages of the 6 center measurements on each wafer
are plotted on a single graph for each wafer. The points (connected
by lines) on the left side of each graph are averages at the wafer
center plotted over 5 days; the points on the right are the same
measurements repeated after one month as a check on the stability of
the measurement process. The plots show daytoday variability as well
as slight variability from runtorun.
Earlier work discounts longterm drift in the gauge as the cause of these changes. A reasonable conclusion is that daytoday and runtorun variations come from random fluctuations in the measurement process. 
Level2 (reproducibility) standard deviations computed from day averages and pooled over wafers and runs 
Level2 standard
deviations (with K  1 = 5 degrees of freedom each) are
computed from the daily averages that are recorded in the database.
Then the level2 standard deviations
are pooled over:

Run 1 Run 2 Wafer Probe Average Stddev DF Average Stddev DF 138. 2362. 95.0928 0.0359 5 95.1243 0.0453 5 139. 2362. 99.3060 0.0472 5 99.3098 0.0215 5 140. 2362. 96.0357 0.0273 5 96.0765 0.0276 5 141. 2362. 101.0602 0.0232 5 101.0790 0.0537 5 142. 2362. 94.2148 0.0274 5 94.2438 0.0370 5 2362. Pooled 0.0333 25 0.0388 25 (over 2 runs) 0.0362 50 
Level3 (stability) standard deviations computed from run averages and pooled over wafers  Level3 standard deviations are computed from the averages of the two runs. Then the level3 standard deviations are pooled over the five wafers to obtain a standard deviation with 5 degrees of freedom as shown in the table below. 
Run 1 Run 2 Wafer Probe Average Average Diff Stddev DF 138. 2362. 95.0928 95.1243 0.0315 0.0223 1 139. 2362. 99.3060 99.3098 0.0038 0.0027 1 140. 2362. 96.0357 96.0765 0.0408 0.0289 1 141. 2362. 101.0602 101.0790 0.0188 0.0133 1 142. 2362. 94.2148 94.2438 0.0290 0.0205 1 
Graphs of probe biases  A graphical analysis shows the relative biases among the 5 probes. For each wafer, differences from the wafer average by probe are plotted versus wafer number. The graphs verify that probe #2362 (coded as 5) is biased low relative to the other probes. The bias shows up more strongly after the probes have been in use (run 2). 
Formulas for computation of biases for probe #2362 
Biases by probe are shown in the following table.
The reader can download the following
data as a text file.
Differences from the mean for each wafer Wafer Probe Run 1 Run 2 138. 1. 0.0248 0.0119 138. 281. 0.0108 0.0323 138. 283. 0.0193 0.0258 138. 2062. 0.0175 0.0561 138. 2362. 0.0372 0.0507 139. 1. 0.0036 0.0007 139. 281. 0.0394 0.0050 139. 283. 0.0057 0.0239 139. 2062. 0.0323 0.0373 139. 2362. 0.0094 0.0657 140. 1. 0.0400 0.0109 140. 281. 0.0187 0.0106 140. 283. 0.0201 0.0003 140. 2062. 0.0126 0.0182 140. 2362. 0.0261 0.0398 141. 1. 0.0394 0.0324 141. 281. 0.0107 0.0037 141. 283. 0.0246 0.0191 141. 2062. 0.0280 0.0436 141. 2362. 0.0252 0.0534 142. 1. 0.0062 0.0093 142. 281. 0.0376 0.0174 142. 283. 0.0044 0.0192 142. 2062. 0.0011 0.0008 142. 2362. 0.0383 0.0469 
How to deal with bias due to the probe 
Probe #2362 was chosen for the certification process because of its
superior precision, but its bias relative to the other probes creates
a problem. There are two possibilities for handling this problem:
