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

Sensitivity coefficients for measurements on the test item

From data on the test item itself If the temporal component is estimated from N short-term readings on the test item itself $$ Y_1, \,\, Y_2, \,\, \ldots , \,\, Y_N $$ and $$ s_1 = \frac{1}{\sqrt{N-1}} \sqrt{\sum_{i=1}^N (Y_i - \bar{Y})^2} $$ and the reported value is the average, the standard deviation of the reported value is $$ s_{reported \, value} = \frac{1}{\sqrt{N}} s_1 $$ with degrees of freedom \( \nu_1 = N-1 \).
Sensitivity coefficients The sensitivity coefficient is \( a_1 = \sqrt{\frac{1}{N}} \). The risk in using this method is that it may seriously underestimate the uncertainty.
To improve the reliability of the uncertainty calculation If possible, the measurements on the test item should be repeated over \(M\) days and averaged to estimate the reported value. The standard deviation for the reported value is computed from the daily averages, and the standard deviation for the temporal component is: $$ s_{reported value} = \frac{1}{\sqrt{M}} \sqrt{ \frac{1}{M-1} \sum_{m=1}^M \left( \bar{Y}_{m \small{\bullet}} - \bar{Y}_{\small{\bullet \bullet}} \right)^2 } $$ with degrees of freedom \( \nu_2 = M-1 \) where \( \bar{Y}_{m \small{\bullet}} \) are the daily averages and \( \bar{Y}_{\small{\bullet \bullet}} \) is the grand average.

The sensitivity coefficients are: \( a_1 = 0 \); \( a_2 = \sqrt{\frac{1}{M}} \).

Note on long-term errors Even if no day-to-day nor run-to-run measurements were made in determining the reported value, the sensitivity coefficient is non-zero if that standard deviation proved to be significant in the analysis of data.
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