2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.7. Standard and expanded uncertainties

2.5.7.1.

Degrees of freedom

Degrees of freedom for individual components of uncertainty

Degrees of freedom for type A uncertainties are the degrees of freedom for the respective standard deviations. Degrees of freedom for Type B evaluations may be available from published reports or calibration certificates. Special cases where the standard deviation must be estimated from fragmentary data or scientific judgment are assumed to have infinite degrees of freedom; for example,

• Worst-case estimate based on a robustness study or other evidence
• Estimate based on an assumed distribution of possible errors
• Type B uncertainty component for which degrees of freedom are not documented

Degrees of freedom for the standard uncertainty

Degrees of freedom for the standard uncertainty, \(u\), which may be a combination of many standard deviations, is not generally known. This is particularly troublesome if there are large components of uncertainty with small degrees of freedom. In this case, the degrees of freedom is approximated by the Welch-Satterthwaite formula (Brownlee). $$ {\large \nu = \frac{u^4}{\sum_{i=1}^R \frac{a_i^4 s_i^4}{\nu_i}} }$$

Case study: Uncertainty and degrees of freedom

A case study of type A uncertainty analysis shows the computations of temporal components of uncertainty; instrument bias; geometrical bias; standard uncertainty; degrees of freedom; and expanded uncertainty.