2.
Measurement Process Characterization
2.5. Uncertainty analysis 2.5.6. Uncertainty budgets and sensitivity coefficients


Sensitivity coefficients from a 3level design
Case study showing sensitivity coefficients for 3level design 
If the temporal components are estimated from a
3level nested design and the
reported value is an average over
 
Problem with estimating degrees of freedom  If degrees of freedom are required for the uncertainty, the formula above cannot be used directly and must be rewritten in terms of the standard deviations \(s_1\), \(s_2\), and \(s_3\). $$ s_{reported \, value} = \sqrt{ \frac{1}{P} s_3^2 + \frac{KM}{PMK} s_2^2 + \frac{JN}{PMNJ} s_1^2 } $$  
Sensitivity coefficients 
The sensitivity coefficients are:
$$ \begin{eqnarray*}
a_1 & = & \sqrt{\frac{(JN)}{PMNJ}} \\
a_2 & = & \sqrt{\frac{(KM)}{PMK}} \\
a_3 & = & \sqrt{\frac{1}{P}} \\
\end{eqnarray*} $$
Specific sensitivity coefficients are shown in the table below for selections of \(N, \, M, \, P\). In addition, the following constraints must be observed:

Sensitivity coefficients for three components of uncertainty
\(N\) 
\(M\) 
\(P\) 



 















