 2. Measurement Process Characterization
2.1. Characterization
2.1.2. What is a check standard?

## Analysis

Short-term or level-1 standard deviations from J repetitions An analysis of the check standard data is the basis for quantifying random errors in the measurement process -- particularly time-dependent errors.

Given that we have a database of check standard measurements as described in data collection where

$$Y_{kj}(k=1, \,\ldots, \, K, \,\, j=1, \,\ldots, \, J)$$ represents the jth repetition on the kth day, the mean for the kth day is
$$\overline{Y}_{k \, \small{\bullet}} = \frac{1}{J}\sum_{j=1}^{J} Y_{kj}$$ and the short-term (level-1) standard deviation with v = J - 1 degrees of freedom is
$${\large s}_k = \sqrt{\frac{1}{J-1} \sum_{j=1}^{J} ( Y_{kj} - \overline{Y}_{k \, \small{\bullet}} ) ^2} \,\,\,\, .$$

Drawback of short-term standard deviations An individual short-term standard deviation will not be a reliable estimate of precision if the degrees of freedom is less than ten, but the individual estimates can be pooled over the K days to obtain a more reliable estimate. The pooled level-1 standard deviation estimate with v = K(J - 1) degrees of freedom is $${\large s}_1 = \sqrt{\frac{1}{K} \sum_{k=1}^{K} {\large s}_k^2} \,\,\,\, .$$ This standard deviation can be interpreted as quantifying the basic precision of the instrumentation used in the measurement process.
Process (level-2) standard deviation The level-2 standard deviation of the check standard is appropriate for representing the process variability. It is computed with v = K - 1 degrees of freedom as:
$${\large s}_{chkstd} = {\large s}_2 = \sqrt{\frac{1}{K-1} \sum_{k=1}^{K} \left( \overline{Y}_{k \, \small{\bullet}} - \overline{Y}_{\small{\bullet} \small{\bullet}} \right) ^2}$$ where $$\overline{Y}_{\small{\bullet} \small{\bullet}} = \frac{1}{K} \sum_{k=1}^{K} \overline{Y}_{k \, \small{\bullet}}$$ is the grand mean of the KJ check standard measurements.
Use in quality control The check standard data and standard deviations that are described in this section are used for controlling two aspects of a measurement process:
Case study: Resistivity check standard For an example, see the case study for resistivity where several check standards were measured J = 6 times per day over several days. 