2.
Measurement Process Characterization
2.5. Uncertainty analysis 2.5.8. Treatment of uncorrected bias


Definition of the bias and corrected measurement  If the bias is \(\delta\) and the corrected measurement is defined by $$ Y_{cor} = Y  \delta ,$$ the corrected value of \(Y\) has the usual expanded uncertainty interval which is symmetric around the unknown true value for the measurement process and is of the following type: $$ Y_{cor}  U \, \le \, \mbox{True} \, \mbox{Value} \, \le \, Y_{cor} + U $$  
Definition of asymmetric uncertainty interval to account for uncorrected measurement  If no correction is made for the bias, the uncertainty interval is contaminated by the effect of the bias term as follows: $$ Y  (U + \delta) \, \le \, \mbox{True} \, \mbox{Value} \, \le \, Y + (U  \delta) $$ and can be rewritten in terms of upper and lower endpoints that are asymmetric around the true value; namely, $$ Y  U_ \, \le \, \mbox{True} \, \mbox{Value} \, \le \, Y + U_+ $$  
Conditions on the relationship between the bias and U 
The definition above can lead to a negative uncertainty limit; e.g.,
if the bias is positive and greater than \(U\), the upper endpoint
becomes negative. The requirement that the uncertainty limits be
greater than or equal to zero for all values of the bias guarantees
nonnegative uncertainty limits and is accepted at the cost of
somewhat wider uncertainty intervals. This leads to the following
set of restrictions on the uncertainty limits:
$$ U_ = \left\{ \begin{array}{ll}
U + \delta & \mbox{if } \,\, U + \delta > 0 \\
0 & \mbox{if } \,\, U + \delta \le 0 \\
\end{array} \right. $$
$$ U_+ = \left\{ \begin{array}{ll} U  \delta & \mbox{if } \,\, U  \delta > 0 \\ 0 & \mbox{if } \,\, U  \delta \le 0 \end{array} \right. $$ 

Situation where bias is not known exactly but must be estimated  If the bias is not known exactly, its magnitude is estimated from repeated measurements, from sparse data or from theoretical considerations, and the standard deviation is estimated from repeated measurements or from an assumed distribution. The standard deviation of the bias becomes a component in the uncertainty analysis with the standard uncertainty restructured to be: $$ u_c = \sqrt{u^2 + u_{bias}^2} $$ and the expanded uncertainty limits become: $$ \mbox{Limits} = \left\{ \begin{array}{c} U_ = k \cdot u_c + \delta \\ U_+ = k \cdot u_c  \delta \end{array} \right. $$  
Interpretation  The uncertainty intervals described above have the desirable properties outlined on a previous page. For more information on theory and industrial examples, the reader should consult the paper by the authors of this technique (Phillips and Eberhardt). 