2.
Measurement Process Characterization
2.6. Case studies 2.6.2. Check standard for resistivity measurements


Estimates of the repeatability standard deviation and level2 standard deviation 
The level1 standard deviations (with J  1 = 5 degrees of
freedom each) from the database are pooled over the K = 25 days
to obtain a reliable estimate of repeatability. This pooled value is
with K(J  1) = 125 degrees of freedom. The level2 standard deviation is computed from the daily averages to be
with K  1 = 24 degrees of freedom. 

Relationship to uncertainty calculations  These standard deviations are appropriate for estimating the uncertainty of the average of six measurements on a wafer that is of the same material and construction as the check standard. The computations are explained in the section on sensitivity coefficients for check standard measurements. For other numbers of measurements on the test wafer, the computations are explained in the section on sensitivity coefficients for level2 designs.  
Illustrative table showing computations of repeatability and level2 standard deviations  A tabular presentation of a subset of check standard data (J = 6 repetitions and K = 6 days) illustrates the computations. The pooled repeatability standard deviation with K(J  1) = 30 degrees of freedom from this limited database is shown in the next to last row of the table. A level2 standard deviation with K  1= 5 degrees of freedom is computed from the center averages and is shown in the last row of the table.  
Control chart for probe #2362 
The control chart for monitoring the precision of
probe #2362 is constructed as discussed in the section on
control charts for standard
deviations. The upper control limit (UCL) for testing for
degradation of the probe is computed using the critical value from the
F table with numerator
degrees of freedom J  1 = 5 and denominator degrees of freedom
K(J  1) = 125. For a 0.05 significance level,
\( UCL = s_1 \sqrt{F_{0.05, 5, 125}} = 0.09238 \,\, \mbox{ohm.cm} \) 

Interpretation of control chart for probe #2362  The control chart shows two points exceeding the upper control limit. We expect 5 % of the standard deviations to exceed the UCL for a measurement process that is incontrol. Two outliers are not indicative of significant problems with the repeatability for the probe, but the probe should be monitored closely in the future.  
Control chart for bias and variability 
The control limits for monitoring the bias and
longterm variability of resistivity with a
Shewhart control chart are given by
\( Centerline = Average = 97.0698 \,\, \mbox{ohm.cm} \) \( LCL = Average  2 \cdot s_2 = 97.0162 \,\, \mbox{ohm.cm} \) 

Interpretation of control chart for bias 
The control chart shows that the points scatter randomly about the
center line with no serious problems, although one point exceeds the
upper control limit and one point exceeds the lower control limit by a
small amount. The conclusion is that there is:
