2. Measurement Process Characterization
2.4. Gauge R & R studies
2.4.4. Analysis of variability

2.4.4.2.

Analysis of reproducibility

Case study: Resistivity gauges

Day-to-day variability can be assessed by a graph of check standard values (averaged over J repetitions) versus day with a separate graph for each check standard. Graphs for all check standards should be plotted on the same page to obtain an overall view of the measurement situation.

Pooling results in more reliable estimates

The level-2 standard deviations with (K - 1) degrees of a freedom are computed from the check standard values for days and pooled over runs as shown in the table below. The pooled level-2 standard deviation has degrees of freedom L(K - 1) for measurements made over:

• K days
• L runs

Mechanism for pooling

The table below gives the mechanism for pooling level-2 standard deviations over runs. The pooled value is an average of weighted variances and is the last entry in the right-hand column of the table. The pooling can be extended in the same manner to cover check standards, if appropriate.

Level-2 standard deviations for a single gauge pooled over runs

Source of variability	Standard deviations	Degrees freedom	Sum of squares (SS)
Days
Run 1 Run 2 Pooled value	0.027280 0.027560	5 5 ------- 10	0.003721 0.003798 ------------- 0.007519 0.02742

Run software macro for computing level-2 standard deviations and pooling over runs

A subset of data (shown on previous page) collected in a nested design on one check standard (#140) with probe (#2362) on six days are analyzed for between-day effects. Dataplot commands to compute the level-2 standard deviations and pool over runs 1 and 2 are:

dimension 500 30
read mpc441.dat run wafer probe mo day op temp y s df
let n1 = count y subset run 1
let df1 = n1 - 1
let n2 = count y subset run 2
let df2 = n2 - 1
let v2 = df1 + df2
let s2run1 = standard deviation y subset run 1
let s2run2 = standard deviation y subset run 2
let s2 = df1*(s2run1)**2 + df2*(s2run2)**2
let s2 = (s2/v2)**.5
print s2run1 df1
print s2run2 df2
print s2 v2

Dataplot output

Dataplot returns the following level-2 standard deviations and degrees of freedom:

 PARAMETERS AND CONSTANTS--

     S2RUN1  --  0.2728125E-01
     DF1     --  0.5000000E+01

 PARAMETERS AND CONSTANTS--

     S2RUN2  --  0.2756367E-01
     DF2     --  0.5000000E+01

 PARAMETERS AND CONSTANTS--

     S2  --  0.2742282E-01
     v2  --  0.1000000E+02

Relationship to day effect

The level-2 standard deviation is related to the standard deviation for between-day precision and gauge precision by

The size of the day effect can be calculated by subtraction using the formula above once the other two standard deviations have been estimated reliably.

Computation of component for days

The Dataplot commands:

let J = 6
let varday = s2**2 - (s1**2)/J

returns the following value for the variance for days:

THE COMPUTED VALUE OF THE CONSTANT 
VARDAY = -0.2880149E-03 

The negative number for the variance is interpreted as meaning that the variance component for days is zero. However, with only 10 degrees of freedom for the level-2 standard deviation, this estimate is not necessarily reliable. The standard deviation for days over the entire database shows a significant component for days.