2.
Measurement Process Characterization
2.3. Calibration 2.3.3. What are calibration designs?


Solutions for designs listed in the catalog  Solutions for all designs that are cataloged in this Handbook are included with the designs. Solutions for other designs can be computed from the instructions on the following page given some familiarity with matrices.  
Measurements for the 1,1,1 design 
The use of the tables shown in the catalog are illustrated for three
artifacts; namely, a reference standard with known value R* and
a check standard and a test item with unknown values. All artifacts are
of the same nominal size. The design is referred to as a
1,1,1 design for


Convention for showing the measurement sequence and identifying the reference and check standards 
The convention for showing the measurement sequence is shown below.
Nominal values are underlined in the first line showing that this design
is appropriate for comparing three items of the same nominal size such
as three onekilogram weights. The reference standard is the first
artifact, the check standard is the second, and the test item is the
third.
1 1 1 Y(1) = +  Y(2) = +  Y(3) = +  Restraint + Check standard + 

Limitation of this design  This design has degrees of freedom $$ \nu = n  m + 1 = 1 $$  
Convention for showing leastsquares estimates for individual items 
The table shown below lists the coefficients for finding the estimates
for the individual items. The estimates are computed by taking the
crossproduct of the appropriate column for the item of interest with
the column of measurement data and dividing by the divisor shown at the
top of the table.
SOLUTION MATRIX DIVISOR = 3 OBSERVATIONS 1 1 1 Y(1) 0 2 1 Y(2) 0 1 2 Y(3) 0 1 1 R* 3 3 3 

Solutions for individual items from the table above  For example, the solution for the reference standard is shown under the first column; for the check standard under the second column; and for the test item under the third column. Notice that the estimate for the reference standard is guaranteed to be R*, regardless of the measurement results, because of the restraint that is imposed on the design. The estimates are as follows: $$ \widehat{R^*} = \frac{1}{3}(0 Y_1 + 0 Y_2 + 0 Y_3) + R^* $$ $$ \widehat{Chk} = \frac{1}{3}(2 Y_1  1 Y_2 + 1 Y_3) + R^* $$ $$ \widehat{Test} = \frac{1}{3}(1 Y_1  2 Y_2  1 Y_3) + R^* $$  
Convention for showing standard deviations for individual items and combinations of items 
The standard deviations are computed from two tables of factors as
shown below. The standard deviations for combinations of items include
appropriate covariance terms.
FACTORS FOR REPEATABILITY STANDARD DEVIATIONS WT FACTOR K1 1 1 1 1 0.0000 + 1 0.8165 + 1 0.8165 + 2 1.4142 + + 1 0.8165 + FACTORS FOR BETWEENDAY STANDARD DEVIATIONS WT FACTOR K2 1 1 1 1 0.0000 + 1 1.4142 + 1 1.4142 + 2 2.4495 + + 1 1.4142 + 

Unifying equation  The standard deviation for each item is computed using the unifying equation: $$ {\large s}_{test} = \sqrt{K_1^2 {\large s}_1^2 + K_2^2 {\large s}_{days}^2} $$  
Standard deviations for 1,1,1 design from the tables of factors  For the 1,1,1 design, the standard deviations are: $$ {\large s}_{R^*} = 0 $$ $$ {\large s}_{chk} = \sqrt{(0.8165 \cdot {\large s}_1)^2 + (1.4142 \cdot {\large s}_{days})^2)} = \sqrt{\frac{2}{3} {\large s}_1^2 + 2 {\large s}_{days}^2} $$ $$ {\large s}_{test} = \sqrt{(0.8165 \cdot {\large s}_1)^2 + (1.4142 \cdot {\large s}_{days})^2} = \sqrt{\frac{2}{3} {\large s}_1^2 + 2 {\large s}_{days}^2} $$ $$ {\large s}_{chk+test} = \sqrt{(1.4142 \cdot {\large s}_1)^2 + (2.4495 \cdot {\large s}_{days})^2)} = \sqrt{2 {\large s}_1^2 + 6 {\large s}_{days}^2} $$  
Process standard deviations must be known from historical data 
In order to apply these equations, we need an estimate of the standard
deviation, \( {\large s}_{days} \),
that describes daytoday changes in
the measurement process. This standard deviation is in turn
derived from the level2 standard deviation,
\( {\large s}_2 \), for the check standard.
This standard deviation
is estimated from historical data on the check standard; it can be
negligible, in which case the calculations are simplified.
The repeatability standard deviation, \( {\large s}_1 \), is estimated from historical data, usually from data of several designs. 

Steps in computing standard deviations 
The steps in computing the standard deviation for a test item are:
