2. Measurement Process Characterization
2.5. Uncertainty analysis

2.5.6.

Uncertainty budgets and sensitivity coefficients

Case study showing uncertainty budget

Uncertainty components are listed in a table along with their corresponding sensitivity coefficients, standard deviations and degrees of freedom. A table of typical entries illustrates the concept.

Typical budget of type A and type B uncertainty components

Type A components	Sensitivity coefficient	Standard deviation	Degrees freedom
1. Time (repeatability)	\(a_1\)	\(s_1\)	\(\nu_1\)
2. Time (reproducibility)	\(a_2\)	\(s_2\)	\(\nu_2\)
3. Time (long-term)	\(a_3\)	\(s_3\)	\(\nu_3\)
Type B components
4. Reference standard	(nominal test / nominal ref)	\(s_4\)	\(\nu_4\)

Sensitivity coefficients show how components are related to result	The sensitivity coefficient shows the relationship of the individual uncertainty component to the standard deviation of the reported value for a test item. The sensitivity coefficient relates to the result that is being reported and not to the method of estimating uncertainty components where the uncertainty, \(u\), is $$ u = \sqrt{\sum_{i=1}^R a_i^2 s_i^2} $$
Sensitivity coefficients for type A components of uncertainty	This section defines sensitivity coefficients that are appropriate for type A components estimated from repeated measurements. The pages on type A evaluations, particularly the pages related to estimation of repeatability and reproducibility components, should be reviewed before continuing on this page. The convention for the notation for sensitivity coefficients for this section is that: \(a_1\) refers to the sensitivity coefficient for the repeatability standard deviation, \(s_1\) \(a_2\) refers to the sensitivity coefficient for the reproducibility standard deviation, \(s_2\) \(a_3\) refers to the sensitivity coefficient for the stability standard deviation, \(s_3\) with some of the coefficients possibly equal to zero.
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.
Sensitivity coefficients for other type A components of random error	Procedures for estimating differences among instruments, operators, etc., which are treated as random components of uncertainty in the laboratory, show how to estimate the standard deviations so that the sensitivity coefficients = 1.
Sensitivity coefficients for type A components for bias	This Handbook follows the ISO guidelines in that biases are corrected (correction may be zero), and the uncertainty component is the standard deviation of the correction. Procedures for dealing with biases show how to estimate the standard deviation of the correction so that the sensitivity coefficients are equal to one.
Sensitivity coefficients for specific applications	The following pages outline methods for computing sensitivity coefficients where the components of uncertainty are derived in the following manner: From measurements on the test item itself From measurements on a check standard From measurements in a 2-level design From measurements in a 3-level design and give an example of an uncertainty budget with sensitivity coefficients from a 3-level design.
Sensitivity coefficients for type B evaluations	The majority of sensitivity coefficients for type B evaluations will be one with a few exceptions. The sensitivity coefficient for the uncertainty of a reference standard is the nominal value of the test item divided by the nominal value of the reference standard.
Case study-sensitivity coefficients for propagation of error	If the uncertainty of the reported value is calculated from propagation of error, the sensitivity coefficients are the multipliers of the individual variance terms in the propagation of error formula. Formulas are given for selected functions of: functions of a single variable functions of two variables several variables