2.
Measurement Process Characterization
2.3. Calibration 2.3.3. What are calibration designs? 2.3.3.3. Uncertainties of calibrated values


Design with two reference standards and two test items  An example is shown below for a 1,1,1,1 design for two reference standards, R_{1} and R_{2}, and two test items, X_{1} and X_{2}, and six difference measurements. The restraint, R*, is the sum of values of the two reference standards, and the check standard, which is independent of the restraint, is the difference between the values of the reference standards. The design and its solution are reproduced below.  
Check standard is the difference between the two reference standards 
OBSERVATIONS 1 1 1 1 Y(1) +  Y(2) +  Y(3) +  Y(4) +  Y(5) +  Y(6) +  RESTRAINT + + CHECK STANDARD +  DEGREES OF FREEDOM = 3 SOLUTION MATRIX DIVISOR = 8 OBSERVATIONS 1 1 1 1 Y(1) 2 2 0 0 Y(2) 1 1 3 1 Y(3) 1 1 1 3 Y(4) 1 1 3 1 Y(5) 1 1 1 3 Y(6) 0 0 2 2 R* 4 4 4 4 

Explanation of solution matrix  The solution matrix gives values for the test items of \begin{eqnarray} X_1^* = \frac{1}{8} \left( 3Y_2  Y_3  3Y_4  Y_5 + 2Y_6 \right) + \frac{1}{2} R^* \\ X_2^* = \frac{1}{8} \left( Y_2  3Y_3  Y_4  3Y_5  2Y_6 \right) + \frac{1}{2} R^* \end{eqnarray}  
Factors for computing contributions of repeatability and level2 standard deviations to uncertainty 
FACTORS FOR REPEATABILITY STANDARD DEVIATIONS WT FACTOR K_{1} 1 1 1 1 1 0.3536 + 1 0.3536 + 1 0.6124 + 1 0.6124 + 0 0.7071 +  FACTORS FOR LEVEL2 STANDARD DEVIATIONS WT FACTOR K_{2} 1 1 1 1 1 0.7071 + 1 0.7071 + 1 1.2247 + 1 1.2247 + 0 1.4141 +  The first table shows factors for computing the contribution of the repeatability standard deviation to the total uncertainty. The second table shows factors for computing the contribution of the betweenday standard deviation to the uncertainty. Notice that the check standard is the last entry in each table. 

Unifying equation  The unifying equation is: $$ {\large s}_{test} = \sqrt{K_1^2 {\large s}_1^2 + K_2^2 {\large s}_{days}^2 } \,\, .$$  
Standard deviations are computed using the factors from the tables with the unifying equation 
The steps in computing the standard deviation for a test item are:
