2.
Measurement Process Characterization
2.3. Calibration 2.3.3. What are calibration designs? 2.3.3.1. Elimination of special types of bias


Leftright bias which is not eliminated by differencing  A situation can exist in which a bias, P, which is constant and independent of the direction of measurement, is introduced by the measurement instrument itself. This type of bias, which has been observed in measurements of standard voltage cells (Eicke & Cameron) and is not eliminated by reversing the direction of the current, is shown in the following equations. \begin{eqnarray} Y_1 = X  R + B + error_1 \\ Y_2 = R  X + P + error_2 \end{eqnarray}  
Elimination of leftright bias requires two measurements in reverse direction  The difference between the test and the reference can be estimated without bias only by taking the difference between the two measurements shown above where P cancels in the differencing so that $$ D = Y_1  Y_2 = 2X  2R \,.$$  
The value of the test item depends on the known value of the reference standard, R*  The test item, X, can then be estimated without bias by $$ \widehat{Test} = X^* = \frac{1}{2}(Y_1  Y_2) + R^*$$ and P can be estimated by $$ \widehat{P} = \frac{1}{2}(Y_1 + Y_2) \, .$$  
Calibration designs that are leftright balanced  This type of scheme is called leftright balanced and the principle is extended to create a catalog of leftright balanced designs for intercomparing reference standards among themselves. These designs are appropriate ONLY for comparing reference standards in the same environment, or enclosure, and are not appropriate for comparing, say, across standard voltage cells in two boxes. 