# 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

represents the jth repetition on the kth day, the mean for the kth day is

and the short-term (level-1) standard deviation with v = J - 1 degrees of freedom is

.

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

.

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:

where

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.