1370 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 5, OCTOBER2004 The Random-Fuzzy Variables: A New Approach to the Expression of Uncertainty in Measurement Alessandro Ferrero, Fellow, IEEE, and Simona Salicone, Student Member, IEEE Abstract—The good measurement practice requires that the measurement uncertainty is estimated and provided together with the measurement result. The practice today, which is reflected in the reference standard provided by the IEC-ISO “Guide to the expression of uncertainty in measurement,” adopts a statistical ap- proach for the expression and estimation of the uncertainty, since the probability theory is the most known and used mathematical tool to deal with distributions of values. However, the probability theory is not the only tool to deal with distributions of values and is not the most suitable one when the values do not distribute in a totally random way. In this case, a more general theory, the theory of the evidence, should be considered. This paper recalls the fundamentals of the theory of the evidence and frames the random-fuzzy variables within this theory, showing how they can usefully be employed to represent the result of a measurement together with its associated uncertainty. The mathematics is de- fined on the random-fuzzy variables, so that the uncertainty can be processed, and simple examples are given. Index Terms—Fuzzy variables, metrology, uncertainty expres- sion. I. INTRODUCTION T HE correct estimation of the measurement uncertainty has represented a true challenge to the measurement experts since the early beginning of the measurement science and has steadily evolved as the state of the art of the measurement prac- tice has developed. Nowadays, state of the art reflects the scientific discussion that took place in the 1970s and 1980s and is perfectly expressed in the present reference Standard, that is the IEC-ISO “Guide to the expression of uncertainty in measurement” [1]. The def- inition of uncertainty of a measurement result provided by the Guide and the guidelines to estimate, express and process it, are not only a standard reference, but do represent concepts that are widely accepted by the scientific and technical community. For this reason, the definitions given by the Guide [1] will be shortly recalled here, since they appear to be the best definitions presently available, in order to frame the uncertainty concept in a broad context and to outline the limits of the present practice of uncertainty estimation. The measurement uncertainty is defined by the Guide as “a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand” [1]. The Guide recognizes that Manuscript received June 15, 2003; revised March 30, 2004. The authors are with the Dipartimento di Elettrotecnica, Politecnico di Mi- lano, 20133 Milano, Italy (e-mail: alessandro.ferrero@polimi.it). Digital Object Identifier 10.1109/TIM.2004.831506 “the word ‘uncertainty’ means ‘doubt,’ and, thus, in its broadest sense, ‘uncertainty of measurement’ means doubt about the ex- actness or accuracy of the result of a measurement” [1]. It also states that “the uncertainty of the result of a measurement re- flects 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 estimate of the value of the measurand because of the uncertainty arising from random ef- fects and from imperfect correction of the result for systematic effects” [1]. The guide takes also into account the concepts of ‘level of confidence’ and ‘confidence interval.’ In fact, it states that “in many industrial and commercial applications, as well as in the areas of health and safety, it is often necessary to provide an in- terval about the measurement result within which the values that could reasonably be attributed to the quantity subject to mea- surement may be expected to lie with a high level of confidence. Thus, the ideal method for evaluating and expressing measure- ment uncertainty should be capable of readily providing such a confidence interval, in particular, one that corresponds in a re- alistic way with the required level of confidence” [1]. The definitions and concepts above shortly recalled give clear evidence that a measurement result can no longer be expressed by a single scalar value, but it must be expressed by a distribu- tion of values over an interval within which the measurement result is expected to lie with a given level of confidence. One of the most known and used mathematical tools to deal with distributions of values is the well known probability theory, to which the present measurement practice and the IEC-ISO Guide refer almost totally. The followed probabilistic approach is somehow justified by the consideration that the result of a measurement after correction of recognized systematic effects is still affected by an uncertainty due to the imperfect correction of the systematic effects and to random effects. If the random effects are supposed to be the prevailing ones, then the prob- abilistic approach appears to be quite natural, since the proba- bility theory is the most suitable mathematical tool to deal with random phenomena. In recent years, however, the limitations of this probabilistic approach have been outlined, and the criticism has focused mainly on the following points. • In many practical applications, the random effects do not prevail over the systematic ones, especially when these latter ones are unknown, or the applied corrections are not totally effective. A probabilistic processing of non- negligible systematic effects may yield a wrong evaluation of the measurement uncertainty. 0018-9456/04$20.00 © 2004 IEEE