# 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 *j*th repetition on the *k*th day, the mean for the *k*th 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.