2.
Measurement Process Characterization
2.3. Calibration


Important concept  Restraint  The designs are constructed for measuring differences among reference standards and test items, singly or in combinations. Values for individual standards and test items can be computed from the design only if the value (called the restraint = R*) of one or more reference standards is known. The methodology for constructing and solving calibration designs is described briefly in matrix solutions and in more detail in a NIST publication. (Cameron et al.).  
Designs listed in this catalog  Designs are listed by traditional subject area although many of the designs are appropriate generally for intercomparisons of artifact standards.  
Properties of designs in this catalog 
Basic requirements are:
Other desirable properties are:


Information:
Design Solution Factors for computing standard deviations 
Given
the following information is shown for each design:


Convention for showing the measurement sequence 
Nominal sizes of standards and test items are shown at the top of the
design. Pluses (+) indicate items that are measured together; and
minuses () indicate items are not measured together. The difference
measurements are constructed from the design of pluses and minuses.
For example, a 1,1,1 design for one reference standard and two test
items of the same nominal size with three measurements is shown below:
1 1 1 Y(1) = +  Y(2) = +  Y(3) = +  

Solution matrix Example and interpretation 
The crossproduct of the column of difference measurements and
R* with a column from the solution matrix, divided by the
named divisor, gives the value for an individual item. For example,
Solution matrix Divisor = 3 1 1 1 Y(1) 0 2 1 Y(2) 0 1 2 Y(3) 0 +1 1 R* +3 +3 +3 implies that estimates for the restraint and the two test items are: \begin{array} \( \widehat{R^*} = \frac{1}{3} \left\{ 0 Y_1 + 0 Y_2 + 0 Y_3 + 3 R^* \right\} = R^* \\ \widehat{Test}_1 = \frac{1}{3} \left\{ 2 Y_1  Y_2 + Y_3 + 3 R^* \right\} \\ \widehat{Test}_2 = \frac{1}{3} \left\{ Y_1  2 Y_2  Y_3 + 3 R^* \right\} \end{array} 

Interpretation of table of factors 
The factors in this table provide information on precision.
The repeatability standard deviation,
\( {\large s}_1 \),
is multiplied by the appropriate factor to obtain the standard
deviation for an individual item or combination of items. For
example,
Sum Factor 1 1 1 1 0.0000 + 1 0.8166 + 1 0.8166 + 2 1.4142 + + implies that the standard deviations for the estimates are: \begin{array} \( {\large s}_{R^*} = 0 \\ {\large s}_{{test}_{ \, 1}} = 0.8661 \cdot {\large s}_1 \\ {\large s}_{{test}_{ \, 2}} = 0.8661 \cdot {\large s}_1 \\ {\large s}_{{test}_{\, 1+2}} = 1.4142 \cdot {\large s}_1 \end{array} 