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

Solutions to 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
  • n = 3 difference measurements
  • m = 3 artifacts
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 one-kilogram 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 least-squares 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 cross-product 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.


           K1      1   1   1
      1  0.0000    +
      1  0.8165        +
      1  0.8165            +
      2  1.4142        +   +
      1  0.8165        +



           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 day-to-day changes in the measurement process. This standard deviation is in turn derived from the level-2 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:

  • Compute the repeatability standard deviation from the design or historical data.

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

  • Locate the factors, \( K_1 \) and \( K_2 \) for the check standard; for the 1,1,1 design the factors are 0.8165 and 1.4142, respectively, where the check standard entries are last in the tables.

  • Apply the unifying equation to the check standard to estimate the standard deviation for days. Notice that the standard deviation of the check standard is the same as the level-2 standard deviation, \( {\large s}_2 \), that is referred to on some pages. The equation for the between-days standard deviation from the unifying equation is $$ {\large s}_{days} = \frac{1}{K_2} \sqrt{{\large s}_2^2 - K_1^2 {\large s}_1^2} $$ Thus, for the example above $$ {\large s}_{days} = \frac{1}{\sqrt{2}} \sqrt{{\large s}_2^2 - \frac{2}{3} {\large s}_1^2} $$
  • This is the number that is entered into the NIST mass calibration software as the between-time standard deviation. If you are using this software, this is the only computation that you need to make because the standard deviations for the test items are computed automatically by the software.

  • If the computation under the radical sign gives a negative number, set \( {\large s}_{days} =0 \). (This is possible and indicates that there is no contribution to uncertainty from day-to-day effects.)

  • For completeness, the computations of the standard deviations for the test item and for the sum of the test and the check standard using the appropriate factors are shown below. $$ {\large s}_{test} = \sqrt{\frac{2}{3} {\large s}_1^2 + 2 {\large s}_{days}^2} = \sqrt{\frac{2}{3} {\large s}_1^2 + \frac{5}{2} \left( {\large s}_2^2 - \frac{2}{3} {\large s}_1^2 \right) } = {\large s}_2 $$ $$ {\large s}_{test+chk} = \sqrt{2 {\large s}_1^2 + 6 {\large s}_{days}^2} = \sqrt{2 {\large s}_1^2 + \frac{13}{2} \left( {\large s}_2^2 - \frac{2}{3} {\large s}_1^2 \right)} = \sqrt{3} \, {\large s}_2 $$
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