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.
Level 1 - repeatability/short-term precision
Level 2 - reproducibility/day-to-day
Level 3 - stability/run-to-run
The graph below depicts possible scenarios for a 2-level design (short-term
repetitions and days) to illlustrate 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
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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
| 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
| The measurements from the nested design are denoted by
 .
Equations corresponding to the tabular analysis are shown below.
Level-1 repeatability standard deviations 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
| Level-2 standard deviations are
pooled over the L runs where individual standard deviations
with (K - 1) degrees of freedom each are computed from K daily
averages as
where
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| Level-3: One stability can be computed from the data
| 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
The time-dependent components can be computed individually as:
There may be other sources of uncertainty in the measurement process
which must be accounted for in a formal analysis
of uncertainty.
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