2.
Measurement Process Characterization
2.4. Gauge R & R studies 2.4.3. Data collection for timerelated sources of variability


Constraints on time and resources  In planning a gauge study, particularly for the first time, it is advisable to start with a simple design and progress to more complicated and/or labor intensive designs after acquiring some experience with data collection and analysis. The design recommended here is appropriate as a preliminary study of variability in the measurement process that occurs over time. It requires about two days of measurements separated by about a month with two repetitions per day.  
Relationship to 2level and 3level nested designs  The disadvantage of this design is that there is minimal data for estimating variability over time. A 2level nested design and a 3level nested design, both of which require measurments over time, are discussed on other pages.  
Plan of action 
Choose at least \( Q = 10 \)
work pieces or check standards, which
are essentially identical insofar as their expected responses to the
measurement method. Measure each of the check standards twice with
the same gauge, being careful to randomize the order of the check
standards.
After about a month, repeat the measurement sequence, randomizing anew the order in which the check standards are measured. 

Notation  Measurements on the check standards are designated: \begin{eqnarray} Y_{11}, & Y_{12} \\ Y_{21}, & Y_{22} \end{eqnarray} with the first index identifying the month of measurement and the second index identifying the repetition number.  
Analysis of data 
The level1 standard deviation, which describes the basic precision of
the gauge, is
$$ s_1 = \sqrt{\frac{1}{4Q}\sum_{i=1}^{Q}{\{(Y_{11}  Y_{12})^2 + (Y_{21}  Y_{22})^2\}}} \,\, ,$$
with \( \nu_1 = 2Q \)
degrees of freedom.
The level2 standard deviation, which describes the variability of the measurement process over time, is $$ s_2 = \sqrt{\frac{1}{Q}\sum_{i=1}^{Q}{\left\{\frac{(Y_{11} + Y_{12})  (Y_{21} + Y_{22})}{2}\right\}^2}} \,\, , $$ with \( \nu_2 = Q \) degrees of freedom. 

Relationship to uncertainty for a test item  The standard deviation that defines the uncertainty for a single measurement on a test item, often referred to as the reproducibility standard deviation (ASTM), is given by $$ {\large s}_R = \sqrt{{\large s}_{days}^2 + {\large s}_1^2} = \frac{{\large s}_2}{\sqrt{2}} \,\, . $$ The timedependent component is $$ {\large s}_{days} = \sqrt{\frac{1}{2} {\large s}_2^2  {\large s}_1^2} \,\, . $$ There may be other sources of uncertainty in the measurement process that must be accounted for in a formal analysis of uncertainty. 