Uncertainty of the result of a measurement reflects the lack of exact knowledge of the value of the measurand. The result of a measurement after correction for recognized systematic effects is still only an *estimation *of the value of the measurand because of the uncertainty arising from random effects andfrom imperfect correction of the result for systematic effects.

This article aims to give a brief description with examples of the steps in determining uncertainties.

“La incertidumbre del resultado de una medida refleja la falta de conocimiento sobre el verdadero valor del mensurando. El resultado de medir tal mensurando tras aplicar las correcciones debidas a efectos sistemáticos es todavía una estimación debido a la incertidumbre proveniente de los efectos aleatorios y a la falta de conocimiento completo de las correcciones aplicadas por los efectos sistemáticos.”

“En este artículo se pretende dar una breve descripción, acompañada de ejemplos, de los pasos a seguir en la determinación de incertidumbres.”

**1.- Introduction**

One measurement without a quantitative indication of the quality of the result is useless, this indication is what we will call uncertainty. The word “uncertainty” means doubt, doubt about the validity of the result of a measure and reflects the impossibility of knowing exactly the measurand value.

Uncertainty estimation is not a simple work in which there is consensus. Work is continuing and guidelines are being developed.

Thanks to a working group, in 1993 the ISO presented the first edition of the Guide to Expressing Uncertainty of Measurement (GUM). This Guide sets out general rules for assessing and expressing uncertainty in measurement, not how this estimation can be used to make decisions. For this last one is the norm UNE-EN ISO 14253-1: 1999 Geometric specification of products (GPS). Inspection by measuring parts and measuring equipment. Part 1: Decision rules for testing compliance or non-compliance with specifications. Although the existence of this guide has made us much easier, we must not lose sight of the fact that the evaluation of uncertainty is not a purely mathematical process that must be performed after each measurement, but rather more complex, such as the guide itself Clarifies

“Although this Guide provides an action framework for the assessment of uncertainty, it can never substitute for critical reflection, intellectual honesty and professional competence. The assessment of uncertainty is neither a routine task nor something purely mathematical; Depends on the detailed knowledge of the nature of the measurand and measurement. The quality and usefulness of the uncertainty associated with the outcome of a measurement ultimately depends on the knowledge, critical analysis and integrity of those who contribute to its assessment.” (Guide for Expression of Measurement Uncertainty, Section 3.4.8)

“Aunque la presente Guía proporciona un marco de actuación para la evaluación de la incertidumbre, este no puede nunca sustituir a la reflexión crítica, la honradez intelectual y la competencia profesional. La evaluación de la incertidumbre no es ni una tarea rutinaria ni algo puramente matemático; depende del conocimiento detallado de la naturaleza del mensurando y de la medición. La calidad y utilidad de la incertidumbre asociada al resultado de una medición dependen en último término del conocimiento, análisis crítico e integridad de aquellos que contribuyen a su evaluación”. (Guía para la expresión de la incertidumbre de medida, Sección 3.4.8)

According to the GUM definition, uncertainty is the “parameter associated with the outcome of a measure, which characterizes the dispersion of values that can reasonably be attributed to the measurand.”

The base documents for the estimation of uncertainties are as follows, the first being the fundamental reference document:

⎯ Evaluation of measurement data – Guide to the expression of uncertainty in measurement

⎯ Evaluation of measurement data – Supplement 1 to the “Guide to the expression of uncertainty in measurement” – Propagation of distributions using a Monte Carlo method

⎯ Evaluation of measurement data – Supplement 2 to the “Guide to the expression of uncertainty in measurement” – Extension to any number of output quantities

**2.- Uncertainty sources.**

Inside measurement practices there are many potential sources of uncertainty, including:

a) incomplete measurand definition;

b) imperfect realization of the measurand definition;

c) non-representative measurand sample;

d) inadequate knowledge of the effects of environmental conditions over the measurement, or imperfect measurement of such environmental conditions;

e) biased reading of analogue instruments by the operator;

f) finite instrument resolution or discrimination threshold;

g) inaccurate values of measurement standards or reference materials;

h) inaccurate values of constants and other parameters obtained from external sources, used inside data processing algorithm;

i) approximations and assumptions established in the measurement procedure method;

j) variations in the repetition of measurand observations under apparently identical conditions.

These sources are not necessarily independent, and some of them, a) to i), can contribute to source j). Of course, an unrecognized systematic effect cannot be taken into account in the evaluation of the uncertainty of the result of a measurement but contributes to its error

Instrumental uncertainty is the measure uncertainty component that comes from the instrument or measurement system used and is obtained by its calibration. In the case of a primary pattern it is usually obtained from key inter-laboratory comparisons participations.

Uncertainty assessment associated with a measurement is fundamental to be able to subsequently check the product conformity or accordingly to the specs applicable to it, whether of design, legal or otherwise (UNE-EN ISO 14253 -1).

**3.- Uncertainty & Probability.**

In order to begin to evaluate the uncertainty of a measurement in an operational way, the values obtained inside the measurement from the point of view of the probability theory must be seen.

The set of all possible outcomes of a random experiment is called the sample space. The function that assigns a real number to each element of the sample space is the random variable. The range of this function is the set of all possible values of this variable.

In our case we are interested in knowing a magnitude X after the realization of an experiment. Denoting x to any of the elements of the sample space, we will define as function probability density pdf the function f(x) in the range of (0, ∞) such that the infinitesimal probability *dp* that the value of the variable is in between the Values x and x + *d*x is *f *(x) *d*x. Hence the probability that a random variable takes values i between the xa and xb limits is:

The distribution function of a measured quantity describes our knowledge of the reality of that magnitude.

For our purpose, the most interesting parameters of the distribution function are: mathematical expectation or expected value, variance and covariance in the case of two related magnitudes.

Given a magnitude *Xi* whose probability density function is *f(xi),* the expected value *E(Xi)* is defined as:

The variance of a random variable or a probability distribution is defined as:

As properties we can emphasize that the mathematical expected value is a linear operator and the variance is not.The best estimator of X*i* will be the one that minimizes the expression:

with respect to xi’. The minimum appears when **xi´= E(Xi)**

On the other hand the variance of a density function gives us idea of the dispersion of the values. Uncertainty is defined as the positive square root of the variance.

**4.- Evaluating standard uncertainty**

Following the GUM we can group the components of uncertainty into two categories according to the evaluation method, “type A” and “type B”. Classification in type A and type B does not imply any difference in nature between the components of these types, it consists only of two different ways of evaluating the components of uncertainty, and both are based on probability distributions.

**4.1.- Type A evaluation of standard uncertainty**

The uncertainty type A evaluation is used when n independent observations are made from one of the input quantities *Xi *under the same measurement conditions.

In most cases, the best available estimate of the mathematical expectation *q* of a random variable *q*, from which *n* independent observations *qk* have been obtained under the same measurement conditions, is the arithmetic mean of the *n* observations:

The individual observations *qk *differ in value because of random variations in the influence quantities, or random effects. The experimental variance of the observations, which estimates the variance σ2 of the probability distribution of *q*, is given by

This estimate of variance and its positive square root *s*(*qk*), termed the **experimental standard deviation **characterize the variability of the observed values *qk *, or more specifically, their dispersion about their mean *q*

The best estímate of σ σ²(q)=σ²/n , the variance of the mean, is given by:

that, together with the **experimental standard deviation of the mean ***s*(*q*), quantify how well *q *estimates the mathematical expectation of *q*

Thus, for an input quantity *Xi *determined from *n *independent repeated observations *Xi*,*k*, the standard uncertainty *u*(*xi*) of its estimate *xi *= *Xi *is u(xi)=s(Xi)=s(xi/sqrt(n)) called a *Type A variance *and a *Type A standard uncertainty*, respectively.

In some situations other statistical methods may be used, such as the method of least squares or analysis of variance. For example, the use of calibration models based on the least squares method is useful for evaluating the uncertainties arising from random variations in the short and long term of the results of comparisons of materialized patterns of unknown value such as blocks or masses Pattern, with reference standards of known value. The components of uncertainty can be evaluated in these cases by statistical analysis of the data obtained using experimental designs consisting of sequences of measurements of the measurand for a certain number of values different from the quantities on which it depends.

**4.2.- Type B evaluation of standard uncertainty**

For an estimate *xi *of an input quantity *Xi *that has not been obtained from repeated observations, the associated estimated variance *u*²(*xi*) or the standard uncertainty *u*(*xi*) is evaluated by scientific judgement based on all of the available information on the possible variability of *Xi *. The pool of information may include:

⎯ previous measurement data;

⎯ experience with or general knowledge of the behaviour and properties of relevant materials and instruments;

⎯ manufacturer’s specifications;

⎯ data provided in calibration and other certificates;

⎯ uncertainties assigned to reference data taken from handbooks.

For convenience, *u*²(*xi*) and *u*(*xi*) evaluated in this way are sometimes called a *Type B variance *and a *Type B standard uncertainty*, respectively.

According to the source from which this type B uncertainty is obtained, it will be estimated differently. Some examples of type B evaluation are:

**4.2.1.- Uncertainty due to calibrated standard or instrument **

The typical uncertainty is obtained by dividing the expanded uncertainty given in the calibration certificate of the standard by the indicated coverage factor:

**4.2.2.- Uncertainty due to resolution **

One of the sources of uncertainty of an instrument is the resolution of its indicating device, if it is a digital instrument, or the uncertainty due to the read resolution, if it is an analog instrument. In the case of the analogue instrument, the resolution depends on the operator or the media used in the reading (optical amplification, for instance)

If the resolution of the indicating device is *δx*, the input signal value producing a given indication *X* can be placed with equal probability at any point within the range from (*X-δx/*2) to (*X+δx/*2). The input signal can then be described by means of a rectangular distribution of range *δx* and variance *u*2*=*(*δx*)²*/*12, which represents a typical uncertainty for any indication of:

**4.2.3.- Uncertainty due to working measurement standard drift **

The drift of a measurement standard is not easy to determine in many cases, and is an independent and characteristic parameter of each measurement standard. Its value depends, among other factors, on the conditions of use and maintenance, the frequency of use, the accuracy of the measuring instrument, the period between calibrations, etc.

For its calculation, it is possible to start from the history of successive calibrations of the working standard and to estimate a variation of the certified value* δp*. For the evaluation of the uncertainty we will be able to apply a type of rectangular or triangular distribution according to the knowledge that we have of the history of the working measurement standard.

**Bibliography:**

- Vocabulario Internacional de Metrología VIM, 3ª edición 2008 (español).
- Metrología Abreviada, traducción al español de 3ª edición. Edición digital.
- Evaluación de datos de medición. Guía para la expresión de la incertidumbre de medida. Edición digital.
- Evaluación de datos de medición. Suplemento 1 de la Guía para la expresión de la incertidumbre de medida. Propagación de incertidumbres utilizando el método de Monte Carlo.
- UNE-EN ISO 14253-1:1999: “Especificación geométrica de productos (GPS). Inspección mediante medición de piezas y equipos de medida. Parte 1: Reglas de decisión para probar la conformidad o no conformidad con las especificaciones”.
- LIRA Ignacio. “Evaluating the Measurement Uncertainty”, Bristol, IoP Publishing Ltd., 2002
- JGCM 102:2011. Evaluation of measurement data – Supplement 2 to the “Guide to the expression of uncertainty in measurement” – Extension to any number of output quantities.
- JCGM 106:2012. Evaluation of measurement data – The role of measurement uncertainty in conformity assessment.

**NEXT: Uncertainty Estimation (II): Combined & Expanded Uncertainties. Cover Factor, k**