2.
Measurement Process Characterization
2.4.
Gauge R & R studies
2.4.4.

Analysis of variability


Analysis of variability from a nested design

The purpose of this section is to show the effect of various levels of
timedependent effects on the variability of the measurement process
with standard deviations for each level of a 3level nested design.
The graph below depicts possible scenarios for a 2level design
(shortterm repetitions and days) to illustrate the concepts.

Depiction of 2 measurement processes with the same shortterm
variability over 6 days where process 1 has large betweenday
variability and process 2 has negligible betweenday variability

Process 1 Process 2
Large betweenday variability Small betweenday variability
Distributions of shortterm measurements over 6 days
where distances from centerlines illustrate betweenday variability

Hint on using tabular method of analysis

An easy way to begin is with a 2level
table with J columns and K rows for the
repeatability/reproducibility measurements and proceed as follows:
 Compute an average for each row and put it in the J + 1
column.
 Compute the level1 (repeatability) standard deviation for each
row and put it in the J + 2 column.
 Compute the grand average and the level2 standard deviation
from data in the J + 1 column.
 Repeat the table for each of the L runs.
 Compute the level3 standard deviation from the L grand
averages.

Level1: 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.
Level1 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
$$ s_{1lk} = \sqrt{\frac{1}{J1} \sum_{j=1}^{J}{(Y_{lkj}  \overline{Y}_{lk{\small \, \bullet}})^2}} $$
where
$$ \overline{Y}_{lk{\small \, \bullet}} = \frac{1}{J}\sum_{j=1}^{J}{\overline{Y}_{lkj}} \,\, . $$

Level2: L reproducibility standard deviations can be computed from
the data

The level2 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
$$ {\large s}_{2l} = \sqrt{\frac{1}{K1} \sum_{k=1}^{K}{\left( Y_{lk{\small \, \bullet}}  \overline{Y}_{l{\small \, \bullet \bullet}} \right)^2}} $$
where
$$ \overline{Y}_{l{\small \, \bullet \bullet}} = \frac{1}{K}\sum_{k=1}^{K}{\overline{Y}_{lk {\small \, \bullet}}} \,\,\, .$$

Level3: A single global standard deviation can be computed from
the Lrun averages

A level3 standard deviation with (L  1) degrees of freedom
is computed from the Lrun averages as
$$ {\large s}_{3} = \sqrt{\frac{1}{L1} \sum_{l=1}^{L}{\left( Y_{l{\small \, \bullet \bullet}}  \overline{Y}_{{\small \bullet \bullet \bullet}} \right)^2}} $$
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
$$ {\large s}_R = \sqrt{{\large s}_{runs}^2 + {\large s}_{days}^2 + {\large s}_1^2}
= \sqrt{{\large s}_3^2 + \frac{K1}{K} {\large s}_2^2 + \frac{J1}{J} {\large s}_1^2} \,\, , $$
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 timedependent 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.
