2.4.4. Analysis of variability

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 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:

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.

Level-1: LK repeatability standard deviations can be computed from the data

The measurements from the nested design are denoted by

Y(ljk)(l=1, ..., L; k=1, ..., K; j=1, ..., J)

Equations corresponding to the tabular analysis are shown below. Level-1 repeatability standard deviations, 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{(1/(J-1))*SUM[j=1 to J](Y(lkj) - Ybar(lk..))**2}

where

Level-2: L reproducibility standard deviations can be computed from the data

The level-2 standard deviation, 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

s2l = SQRT{(1/(K-1))*SUM[k=1 to K](Ybar(lk.) - Ybar(l..))**2}

where

Ybar(l..) = (1/K)*SUM[k=1 to K]Ybar(lk.)

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

s3 = SQRT{(1/(L-1))*SUM[l=1 to L](Ybar(l..) - Ybar(...))**2}

where

Ybar(...) = (1/L)*SUM[l=1 to L]Ybar(l..)

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

s(R) = SQRT(s(runs)**2 + s(days)**2 + s1**2) =
SQRT{s3**2 + ((K-1)K)*s2**2 + ((J-1)/J)*s1**2

where the pooled values, s₁ and s₂, are the usual

$s(1) = SQRT{SUM{l=1 to L}{SUM{k=1 to K}{s(1lk)^2}}/(L*K)}$

and

$s(2) = SQRT{(1/L)*SUM{l=1 to L}{s(2l)^2}} </CENTER> <P> The time-dependent components can be computed individually as: <P> <CENTER> <IMG SRC=$

There may be other sources of uncertainty in the measurement process that must be accounted for in a formal analysis of uncertainty.