2. Measurement Process Characterization
2.4. Gauge R & R studies

## Analysis of variability

Analysis of variability from a nested design The purpose of this section is to show the effect of various levels of time-dependent effects on the variability of the measurement process with standard deviations for each level of a 3-level nested design. The graph below depicts possible scenarios for a 2-level design (short-term repetitions and days) to illustrate the concepts.
Depiction of 2 measurement processes with the same short-term variability over 6 days where process 1 has large between-day variability and process 2 has negligible between-day variability
            Process 1                Process 2
Large between-day variability   Small between-day variability
Distributions of short-term measurements over 6 days where distances from centerlines illustrate between-day variability
Hint on using tabular method of analysis An easy way to begin is with a 2-level table with J columns and K rows for the repeatability/reproducibility measurements and proceed as follows:
1. Compute an average for each row and put it in the J + 1 column.
2. Compute the level-1 (repeatability) standard deviation for each row and put it in the J + 2 column.
3. Compute the grand average and the level-2 standard deviation from data in the J + 1 column.
4. Repeat the table for each of the L runs.
5. Compute the level-3 standard deviation from the L grand averages.
Level-1: LK repeatability standard deviations can be computed from the data The measurements from the nested design are denoted by $$Y_{lkj}(l = 1, \, \ldots, \, L, \,\, k=1, \,\ldots, \, K, \,\, j=1, \,\ldots, \, J) \,\, .$$ Equations corresponding to the tabular analysis are shown below. Level-1 repeatability standard deviations, $${\large s}_{1lk}$$ , are pooled over the K days and L runs. Individual standard deviations with (J - 1) degrees of freedom each are computed from J repetitions as $$s_{1lk} = \sqrt{\frac{1}{J-1} \sum_{j=1}^{J}{(Y_{lkj} - \overline{Y}_{lk{\small \, \bullet}})^2}}$$ where $$\overline{Y}_{lk{\small \, \bullet}} = \frac{1}{J}\sum_{j=1}^{J}{\overline{Y}_{lkj}} \,\, .$$
Level-2: L reproducibility standard deviations can be computed from the data The level-2 standard deviation, $${\large s}_{2l}$$, is pooled over the L runs. Individual standard deviations with (K - 1) degrees of freedom each are computed from K daily averages as $${\large s}_{2l} = \sqrt{\frac{1}{K-1} \sum_{k=1}^{K}{\left( Y_{lk{\small \, \bullet}} - \overline{Y}_{l{\small \, \bullet \bullet}} \right)^2}}$$ where $$\overline{Y}_{l{\small \, \bullet \bullet}} = \frac{1}{K}\sum_{k=1}^{K}{\overline{Y}_{lk {\small \, \bullet}}} \,\,\, .$$
Level-3: A single global standard deviation can be computed from the L-run averages A level-3 standard deviation with (L - 1) degrees of freedom is computed from the L-run averages as $${\large s}_{3} = \sqrt{\frac{1}{L-1} \sum_{l=1}^{L}{\left( Y_{l{\small \, \bullet \bullet}} - \overline{Y}_{{\small \bullet \bullet \bullet}} \right)^2}}$$ where $$\overline{Y}_{{\small \bullet \bullet \bullet}} = \frac{1}{L}\sum_{l=1}^{L}{\overline{Y}_{l {\small \, \bullet \bullet}}} \,\,\, .$$
Relationship to uncertainty for a test item The standard deviation that defines the uncertainty for a single measurement on a test item is given by $${\large s}_R = \sqrt{{\large s}_{runs}^2 + {\large s}_{days}^2 + {\large s}_1^2} = \sqrt{{\large s}_3^2 + \frac{K-1}{K} {\large s}_2^2 + \frac{J-1}{J} {\large s}_1^2} \,\, ,$$ where the pooled values, $${\large s}_1$$ and $${\large s}_2$$, are the usual $${\large s}_1 = \sqrt{\frac{\sum_{l=1}^{L}{\sum_{k=1}^{K}{{\large s}_{1lk}^{2}}}}{LK}}$$ and $${\large s}_2 = \sqrt{\frac{1}{L} \sum_{l=1}^{L}{{\large s}_{2l}^{2}}} \,\,\, .$$ The time-dependent components can be computed individually as: $${\large s}_{runs} = \sqrt{{\large s}_3^2 - \frac{1}{K} {\large s_2^2}}$$ $${\large s}_{days} = \sqrt{{\large s}_2^2 - \frac{1}{J} {\large s_1^2}} \,\,\, .$$ There may be other sources of uncertainty in the measurement process that must be accounted for in a formal analysis of uncertainty.