# Evaluation of Uncertainty

### Description

### Overview

The reliability of measurement results is a crucial prerequisite for today’s world-wide economy, e.g. when checking the conformity of products with an agreed quality standard. A complete measurement result hence requires a quantitative statement describing its associated uncertainty. Measurement uncertainty is particularly relevant in metrology, for example when the result of a measurement is traced back to an SI unit.

The uncertainty about a measurement result often results from random variations in measured data. Systematic deviations, which remain unchanged in repeated measurements, are another source of uncertainty. In order to evaluate measurement uncertainties, statistical methods are employed. Classical statistics, for example, can be used to determine confidence intervals for the sought quantities that account for observed random variations in the measured data. Bayesian statistics provides an alternative that allows also further information to be taken into account. This approach treats random variability and systematic deviations in a consistent way, and it results in probability statements about the quantities of interest. In particular, the Bayesian approach enables to account for prior knowledge, e.g., when negative results can be ruled out for physical reasons.

### GUM: Guide to the Expression of Uncertainty in Measurement

In metrology the uncertainty of a measurement result is often dominated by systematic deviations. The “Guide to the Expression of Uncertainty in Measurement“ (GUM) enables to coherently account for systematic deviations and random variations. The GUM can be seen as the de facto standard for the evaluation of measurement uncertainty in metrology. An important concept of the GUM methodology is that of a model which relates the measurand to so-called input quantities. In using information about the input quantities, this model is utilized to determine an estimate and its associated uncertainty for the measurand according to the rule of “propagation of uncertainties”.

Supplement 1 to the GUM (GUM S1) proposes Monte Carlo Method (MCM) for the calculation of uncertainties. Similarly to the GUM, a model relating the measurand and input quantities is taken as the basis. In using probability density functions (PDFs) expressing the knowledge about the incput quantities, MCM is then used to determine the PDF associated with the measurand by “propagation of distributions”. There is a certain relationship between GUM S1 and a Bayesian uncertainty analysis. A further supplement to the GUM deals with uncertainty evaluation for multivariate (e.g., complex valued) quantities (GUM S2).

**Fig. 1:** Illustration of the GUM S1 Monte-Carlo method

### For references see

### Coming Workshops

### Research

### Research

While the available guidelines are appropriate in many metrological applications, the development of procedures for the evaluation of measurement uncertainty in more involved applications is a topic of ongoing research. Current research topics in metrology are

- Development and application of procedures based on Bayesian statistics
- Markov Chain Monte Carlo methods for the numerical calculation of posterior PDFs
- Regression
- Analysis of dynamic measurements
- Virtual experiments

The joint EMRP project NEW04 has produced the following related best-practice guides

- Uncertainty evaluation for regression problems
- Uncertainty evaluation for computationally expensive models
- Decision-making and conformity assessment in multivariate cases

The joint EMRP project IND09 has produced the following related best-practice guide

### Official guidelines in metrology

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML 2008 Evaluation of measurement data—guide to the expression of uncertainty in measurement, Joint Committee for Guides in Metrology, JCGM 100:2008

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2008 Evaluation of Measurement Data—Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’—Propagation of Distributions using a Monte Carlo Method JCGM 101:2008

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2011 Evaluation of Measurement Data—Supplement 2 to the ‘Guide to the Expression of Uncertainty in Measurement’—Extension to any Number of Output Quantities JCGM 102:2011

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2008 Evaluation of Measurement Data—Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’—Propagation of Distributions using a Monte Carlo Method JCGM 101:2008

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2011 Evaluation of Measurement Data—Supplement 2 to the ‘Guide to the Expression of Uncertainty in Measurement’—Extension to any Number of Output Quantities JCGM 102:2011