2.
Measurement Process Characterization
2.4.
Gauge R & R studies
2.4.4.
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Analysis of variability
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Analysis of variability from a nested design
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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.
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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
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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
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Hint on using tabular method of analysis
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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:
- Compute an average for each row and put it in the J+1
column.
- Compute the level-1 (repeatability) standard deviation for each
row and put it in the J+2 column.
- Compute the grand average and the level-2 standard deviation
from data in the J+1 column.
- Repeat the table for each of the L runs.
- Compute the level-3 standard deviation from the L grand
averages.
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Level-1: LK repeatability standard deviations can be
computed from the data
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The measurements from the nested design are denoted by
Equations corresponding to the tabular analysis are shown below.
Level-1 repeatability standard deviations,
s1lk, 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
where
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Level-2: L reproducibility standard deviations can be computed from
the data
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The level-2 standard deviation,
s2l, is pooled over the L runs.
Individual standard deviations with (K - 1) degrees of freedom
each are computed from K daily averages as
where
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Level-3: A single global standard deviation can be computed from
the L-run averages
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A level-3 standard deviation with (L - 1) degrees of freedom
is computed from the L-run averages as
where
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Relationship to uncertainty
for a test item
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The standard deviation that defines the uncertainty for a single
measurement on a test item is given by
where the pooled values, s1 and
s2, are the usual
and
There may be other sources of uncertainty in the measurement process
that must be accounted for in a formal
analysis of uncertainty.
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