2. Measurement Process Characterization
2.5. Uncertainty analysis

2.5.7.

Standard and expanded uncertainties

Definition of standard uncertainty

The sensitivity coefficients and standard deviations are combined by root sum of squares to obtain a 'standard uncertainty'. Given \(R\) components, the standard uncertainty is: $$ u = \sqrt{\sum_{i=1}^R a_i^2 s_i^2} $$

Expanded uncertainty assures a high level of confidence

If the purpose of the uncertainty statement is to provide coverage with a high level of confidence, an expanded uncertainty is computed as $$ U = k \cdot u $$ where \(k\) is chosen to be the \( t_{1-\alpha/2, \,\nu}\) critical value from the t-table with \(\nu\) degrees of freedom. For large degrees of freedom, \(k = 2\) approximates 95 % coverage.

Interpretation of uncertainty statement

The expanded uncertainty defined above is assumed to provide a high level of coverage for the unknown true value of the measurement of interest so that for any measurement result, \(Y\), $$ Y - U \le \mbox{True} \, \mbox{Value} \le Y + U $$