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

2.4.4.2.

Analysis of reproducibility

Case study: Resistivity gauges

Day-to-day variability can be assessed by a graph of check standard values (averaged over J repetitions) versus day with a separate graph for each check standard. Graphs for all check standards should be plotted on the same page to obtain an overall view of the measurement situation.

Pooling results in more reliable estimates

The level-2 standard deviations with (K - 1) degrees of freedom are computed from the check standard values for days and pooled over runs as shown in the table below. The pooled level-2 standard deviation has degrees of freedom
L(K - 1) for measurements made over:

• K days
• L runs

Mechanism for pooling

The table below gives the mechanism for pooling level-2 standard deviations over runs. The pooled value is an average of weighted variances and is the last entry in the right-hand column of the table. The pooling can be extended in the same manner to cover check standards, if appropriate.

The table was generated using a subset of data (shown on previous page) collected in a nested design on one check standard (#140) with probe (#2362) over six days. The data are analyzed for between-day effects. The level-2 standard deviations and pooled level-2 standard deviations over runs 1 and 2 are:

Level-2 standard deviations for a single gauge pooled over runs

Source of variability	Standard deviations	Degrees of freedom	Sum of squares
Days	\( {\large s}_{2i} \)	\( \nu_i \)	\( SS_i = \nu_i \cdot {\large s}_{2i}^2\)
Run 1 Run 2 Sum Pooled value \( {\large s}_2 = \sqrt{\sum SS_i / \sum \nu_i} \)	0.027280 0.027560	5 5 ------- 10	0.003721 0.003798 ----------- 0.007519 0.02742

Relationship to day effect

The level-2 standard deviation is related to the standard deviation for between-day precision and gauge precision by $$ {\large s}_{days} = \sqrt{{\large s}_2^2 - \frac{1}{J} {\large s}_1^2} $$ The size of the day effect can be calculated by subtraction using the formula above once the other two standard deviations have been estimated reliably.

Computation of variance component for days

For our example, the variance component for between days is -0.00028072. The negative number for the variance is interpreted as meaning that the variance component for days is zero. However, with only 10 degrees of freedom for the level-2 standard deviation, this estimate is not necessarily reliable. The standard deviation for days over the entire database shows a significant component for days.

Sample code

The calculations included in this section can be implemented using both Dataplot code and R code. The reader can download the data as a text file.