 2. Measurement Process Characterization
2.4. Gauge R & R studies
2.4.5. Analysis of bias

## Differences among gauges

Purpose A gauge study should address whether gauges agree with one another and whether the agreement (or disagreement) is consistent over artifacts and time.
Data collection For each gauge in the study, the analysis requires measurements on
• Q (Q > 2) check standards
• K (K > 2) days
The measurements should be made by a single operator.
Data reduction The steps in the analysis are:
1. Measurements are averaged over days by artifact/gauge configuration.
2. For each artifact, an average is computed over gauges.
3. Differences from this average are then computed for each gauge.
4. If the design is run as a 3-level design, the statistics are computed separately for each run.
Data from a gauge study The data in the table below come from resistivity (ohm.cm) measurements on Q = 5 artifacts on K = 6 days. Two runs were made which were separated by about a month's time. The artifacts are silicon wafers and the gauges are four-point probes specifically designed for measuring resistivity of silicon wafers. Differences from the wafer means are shown in the table.
Biases for 5 probes from a gauge study with 5 artifacts on 6 days
``` Table of biases for probes and silicon wafers (ohm.cm)
Wafers

Probe       138      139       140       141      142
---------------------------------------------------------
1    0.02476  -0.00356   0.04002   0.03938   0.00620

181    0.01076   0.03944   0.01871  -0.01072   0.03761

182    0.01926   0.00574  -0.02008   0.02458  -0.00439

2062   -0.01754  -0.03226  -0.01258  -0.02802  -0.00110

2362   -0.03725  -0.00936  -0.02608  -0.02522  -0.03830
```
Plot of differences among probes A graphical analysis can be more effective for detecting differences among gauges than a table of differences. The differences are plotted versus artifact identification with each gauge identified by a separate plotting symbol. For ease of interpretation, the symbols for any one gauge can be connected by dotted lines.
Interpretation Because the plots show differences from the average by artifact, the center line is the zero-line, and the differences are estimates of bias. Gauges that are consistently above or below the other gauges are biased high or low, respectively, relative to the average. The best estimate of bias for a particular gauge is its average bias over the Q artifacts. For this data set, notice that probe #2362 is consistently biased low relative to the other probes.
Strategies for dealing with differences among gauges Given that the gauges are a random sample of like-kind gauges, the best estimate in any situation is an average over all gauges. In the usual production or metrology setting, however, it may only be feasible to make the measurements on a particular piece with one gauge. Then, there are two methods of dealing with the differences among gauges.
1. Correct each measurement made with a particular gauge for the bias of that gauge and report the standard deviation of the correction as a type A uncertainty.
2. Report each measurement as it occurs and assess a type A uncertainty for the differences among the gauges. 