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

Expanded uncertainties

Standard uncertainty The standard uncertainty for the test item is $$ u = \sqrt{ {\large s}_{test}^2 + \left( \frac{Nominal \,\, Test}{Nominal \,\, Restraint} \right)^2 \left( {\large s}_{R^*} \right)^2 } \,\, .$$
Expanded uncertainty The expanded uncertainty is computed as $$ U = k u $$ where k is either the critical value from the t table for degrees of freedom v or k is set equal to 2.
Problem of the degrees of freedom The calculation of degrees of freedom, v, can be a problem. Sometimes it can be computed using the Welch-Satterthwaite approximation and the structure of the uncertainty of the test item. Degrees of freedom for the standard deviation of the restraint is assumed to be infinite. The coefficients in the Welch-Satterthwaite formula must all be positive for the approximation to be reliable.
Standard deviation for test item from the 1,1,1,1 design For the 1,1,1,1 design, the standard deviation of the test items can be rewritten by substituting in the equation $$ {\large s}_{X_1} = {\large s}_{X_2} = \sqrt{\frac{3}{8}{\large s}_1^2 + \frac{3}{2} {\large s}_{days}^2 } = \sqrt{\frac{3}{8}{\large s}_1^2 + \frac{3}{2} \left\{ \frac{1}{2} {\large s}_2^2 - \frac{1}{4} {\large s}_1^2 \right\} } = \frac{\sqrt{3}}{2} {\large s}_2 $$ so that the degrees of freedom depends only on the degrees of freedom in the standard deviation of the check standard. This device may not work satisfactorily for all designs.
Standard uncertainty from the 1,1,1,1 design To complete the calculation shown in the equation at the top of the page, the nominal value of the test item (which is equal to 1) is divided by the nominal value of the restraint (which is also equal to 1), and the result is squared. Thus, the standard uncertainty is $$ u = \sqrt{\frac{3}{4} {\large s}_2^2 + {\large s}_{R^*}^2} \,\, .$$
Degrees of freedom using the Welch-Satterthwaite approximation Therefore, the degrees of freedom is approximated as $$ \nu = \frac{u^4}{\frac{1}{n-1} \left( \frac{9}{16}{\large s}_2^4 \right)} $$ where n - 1 is the degrees of freedom associated with the check standard uncertainty. Notice that the standard deviation of the restraint drops out of the calculation because of an infinite degrees of freedom.
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