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

## 2.5.6.3. Sensitivity coefficients for measurements from a 2-level design

Sensitivity coefficients from a 2-level design If the temporal components are estimated from a 2-level nested design, and the reported value for a test item is an average over
• $$N$$ short-term repetitions
• $$M$$ ($$M = 1$$ is permissible) days
of measurements on the test item, the standard deviation for the reported value is: $$s_{reported \, value} = \sqrt{\frac{1}{M}s_{days}^2 + \frac{1}{MN} s_1^2}$$ See the relationships in the section on 2-level nested design for definitions of the standard deviations and their respective degrees of freedom.
Problem with estimating degrees of freedom If degrees of freedom are required for the uncertainty of the reported value, the formula above cannot be used directly and must be rewritten in terms of the standard deviations, $$s_1$$ and $$s_2$$. $$s_{reported \, value} = \sqrt{\frac{1}{M}s_2^2 + \frac{J-N}{MNJ} s_1^2}$$
Sensitivity coefficients The sensitivity coefficients are: $$a_1 = \sqrt{\frac{(J-N)}{MNJ}} ; \, a_2 = \sqrt{\frac{1}{M}}$$.

Specific sensitivity coefficients are shown in the table below for selections of $$N, \, M$$.

  Sensitivity coefficients for two components
of uncertainty
 Number short-term$$N$$ Number day-to-day$$M$$ Short-term sensitivity coefficient $$a_1$$ Day-to-day sensitivity coefficient $$a_2$$ $$1$$ $$1$$ $$\sqrt{(J-1)/J}$$ $$1$$ $$N$$ $$1$$ $$\sqrt{(J-N)/(NJ)}$$ $$1$$ $$N$$ $$M$$ $$\sqrt{(J-N)/(MNJ)}$$ $$\sqrt{1/M}$$