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

2.3.3.3.4.

Calculation of standard deviations for 1,1,1,1 design

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, R₁ and R₂, and two test items, X₁ and X₂, 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 level-2 standard deviations to uncertainty

     FACTORS FOR REPEATABILITY STANDARD DEVIATIONS
     WT  FACTOR
            K1     1   1   1   1
      1  0.3536    +
      1  0.3536        +
      1  0.6124            +
      1  0.6124                +
      0  0.7071    +   -

         
     FACTORS FOR LEVEL-2 STANDARD DEVIATIONS
     WT  FACTOR
            K2     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 between-day 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:

• Compute the repeatability standard deviation from historical data.

• Compute the standard deviation of the check standard from historical data.

• Locate the factors, K₁ and K₂, for the check standard.

• Compute the between-day variance (using the unifying equation for the check standard). For this example, $$ \begin{array}{l} {\large s}_2^2 = \left\{ \left( \sqrt{0.5} \, {\large s}_1 \right)^2 + \left( \sqrt{2} \, {\large s}_{days} \right)^2 \right\} = \left\{ \frac{1}{2} {\large s}_1^2 + 2 {\large s}_{days}^2 \right\} \\ \begin{array}{l} {\small implies} \\ \,\,\, \Longrightarrow \end{array} \\ {\large s}_{days}^2 = \frac{1}{2} \left\{ {\large s}_2^2 - \frac{1}{2}{\large s}_1^2 \right\} \,\, . \end{array} $$
• If this variance estimate is negative, set \( {\large s}_{days} = 0 \). (This is possible and indicates that there is no contribution to uncertainty from day-to-day effects.)

• Locate the factors, K₁ and K₂, for the test items, and compute the standard deviations using the unifying equation. For this example, $$ {\large s}_{X_1} = \sqrt{\frac{3}{8} {\large s}_1^2 + \frac{3}{2} {\large s}_{days}^2} = \sqrt{\frac{3}{8} {\large s}_1^2 + \frac{3}{2} \cdot \frac{1}{2} \left\{ {\large s}_2^2 - \frac{1}{2}{\large s}_1^2 \right\} } = \sqrt{\frac{3}{4} {\large s}_2^2} $$ and $$ {\large s}_{X_2} = {\large s}_{X_1} \,\, .$$