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

## Analysis and interpretation

Estimates of the repeatability standard deviation and level-2 standard deviation The level-1 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

s1 = 0.06139 ohm.cm

with K(J - 1) = 125 degrees of freedom. The level-2 standard deviation is computed from the daily averages to be

s2 = 0.02680 ohm.cm

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 level-2 designs.
Illustrative table showing computations of repeatability and level-2 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 level-2 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,

$$F_{0.05, 5, 125} = 2.29$$

$$UCL = s_1 \sqrt{F_{0.05, 5, 125}} = 0.09238 \,\, \mbox{ohm.cm}$$

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 in-control. 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 long-term variability of resistivity with a Shewhart control chart are given by

$$UCL = Average + 2 \cdot s_2 = 97.1234 \,\, \mbox{ohm.cm}$$

$$Centerline = Average = 97.0698 \,\, \mbox{ohm.cm}$$

$$LCL = Average - 2 \cdot s_2 = 97.0162 \,\, \mbox{ohm.cm}$$

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:
• No evidence of bias, change or drift in the measurement process.
• No evidence of long-term lack of control.
Future measurements that exceed the control limits must be evaluated for long-term changes in bias and/or variability.