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

2.4.4.3.

Analysis of stability

Case study: Resistivity probes

Run-to-run variability can be assessed graphically by a plot of check standard values (averaged over J repetitions) versus time with a separate graph for each check standard. Data on all check standards should be plotted on one page to obtain an overall view of the measurement situation.

Advantage of pooling

A level-3 standard deviation with (L - 1) degrees of freedom is computed from the run averages. Because there will rarely be more than two runs per check standard, resulting in one degree of freedom per check standard, it is prudent to have three or more check standards in the design to take advantage of pooling. The mechanism for pooling over check standards is shown in the table below. The pooled standard deviation has
Q(L - 1) degrees and is shown as the last entry in the right-hand column of the table.

Example of pooling

The following table shows how the level-3 standard deviations for a single gauge (probe #2362) are pooled over check standards. The table can be reproduced using
R code.

Level-3 standard deviations for a single gauge pooled over check standards

Source of variability	Standard deviation	Degrees of freedom	Sum of squares
Level-3	\( {\large s}_{3i} \)	\( \nu_i\)	\( SS_{i} = v_{i} \cdot {s_{3i}}^{2} \)
Chk std 138 Chk std 139 Chk std 140 Chk std 141 Chk std 142 Sum Pooled value \( s_{3} = \sqrt{ \sum{SS_{i}} / \sum{v_{i}} } \)	0.0223 0.0027 0.0289 0.0133 0.0205	1 1 1 1 1 ----- 5	0.0004973 0.0000073 0.0008352 0.0001769 0.0004203 ----------- 0.0019370 0.0197

Level-3 standard deviations

A subset of data collected in a nested design on one check standard (#140) with probe (#2362) for six days and two runs is analyzed for between-run effects. The level-3 standard deviation, computed from the averages of two runs, is 0.02885 with one degree of freedom. Dataplot code and R code can be used to perform the calculations for this data. The reader can download the data as a text file.

Relationship to long-term changes, days and gauge precision

The size of the between-run effect can be calculated by subtraction using the standard deviations for days and gauge precision as $$ {\large s}_{runs} = \sqrt{{\large s}_3^2 - \frac{1}{K} {\large s}_2^2} = \sqrt{{\large s}_3^2 - \frac{1}{k} {\large s}_{days}^2 - \frac{1}{KJ} {\large s}_1^2} $$