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

## Simple design

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 2-level and 3-level nested designs The disadvantage of this design is that there is minimal data for estimating variability over time. A 2-level nested design and a 3-level 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 level-1 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 level-2 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 time-dependent 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.