# Measurement Uncertainty - Stages of Uncertainty Evaluation

Stages of Uncertainty Evaluation

The main stages of uncertainty evaluation constitute formulation and calculation, the latter consisting of propagation and summarizing. The formulation stage constitutes

1. defining the output quantity (the measurand),
2. identifying the input quantities on which depends,
3. developing a measurement model relating to the input quantities, and
4. on the basis of available knowledge, assigning probability distributions — Gaussian, rectangular, etc. — to the input quantities (or a joint probability distribution to those input quantities that are not independent).

The calculation stage consists of propagating the probability distributions for the input quantities through the measurement model to obtain the probability distribution for the output quantity, and summarizing by using this distribution to obtain

1. the expectation of, taken as an estimate of ,
2. the standard deviation of, taken as the standard uncertainty associated with, and
3. a coverage interval containing with a specified coverage probability.

The propagation stage of uncertainty evaluation is known as the propagation of distributions, various approaches for which are available, including

1. the GUM uncertainty framework, constituting the application of the law of propagation of uncertainty, and the characterization of the output quantity by a Gaussian or a -distribution,
2. analytic methods, in which mathematical analysis is used to derive an algebraic form for the probability distribution for, and
3. a Monte Carlo method, in which an approximation to the distribution function for is established numerically by making random draws from the probability distributions for the input quantities, and evaluating the model at the resulting values.

For any particular uncertainty evaluation problem, approach 1), 2) or 3) (or some other approach) is used, 1) being generally approximate, 2) exact, and 3) providing a solution with a numerical accuracy that can be controlled.

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