2.
Measurement Process Characterization
2.4. Gauge R & R studies 2.4.4. Analysis of variability


Case study: Resistivity probes  The repeatability quantifies the basic precision for the gauge. A level1 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.  
Repeatability standard deviations can be pooled over operators, runs, and check standards 
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, \( {\large s}_1 \),
has \(LK(J  1)\) degrees of freedom for measurements taken over:


Basic pooling rules  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 righthand column of the table. The pooling can also cover check standards, if appropriate.  
View of entire dataset from the nested design 
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 level1 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 

Dataplot code and R code. 