853 Statistical Analysis of Uncertain Measurements in Modeling of Water and Wastewater Systems Olgierd Hryniewicz 1 , Jan Studzinski 1 Abstract Statistical data used in the analysis of environmental problems are very often presented in an imprecise way. We make a distinction between random uncertainty that can be described by stochastic models and linguistic lack of pre- cision that may be modelled by fuzzy sets or possibilistic distributions. In the paper we present a short introduction to fuzzy statistics which may be useful for the analysis of such kind of data. This methodology is illustrated with the example of the communal water supply prediction problem when the statistical data include both the results of meas- urements and imprecise expert opinions. 1. Introduction Statistical methods have been widely used in all areas related to environmental sciences. A collection of statistical methods that are typical for solving problems encountered in environmental sciences has been given a specific name – environmetrics. This special name makes it clear that these methods are some- times somewhat different from the methods used in other fields of application. A good description of the most frequently used methods is given, for example, in (Barnett 2004). A much more comprehensive de- scription of statistical methods used for solving environmental problems can be found in (El- Shaarawi/Piegorsch 2002). However, what does not distinguish environmetrics from other fields of statis- tics is its usage of purely stochastic models for the description of uncertainties of all kind. Statistical data analysed while solving environmental problems may be of a different kind. If the uncer- tainty of statistical data is purely of a random character the usage of classical statistical methods does not raise any questions. However, in many cases statistical data is not only random but also imprecise. This happens especially often when the data contains information which is reported by humans using an impre- cise plain language. Consider, for example, the information given by an expert that a certain parameter has a value "about five". For many years such imprecise information has been modelled by probability distri- butions. However, probabilistic description of the so called "linguistic uncertainty" has raised many ques- tions. Some authors, especially from the area of computer data analysis (data mining, knowledge discov- ery) have claimed that other theories, such as fuzzy set theory and possibility theory, are more suitable for the description of uncertainties of that type. It must be stressed however, that the question mentioned above has not been answered yet. For more detailed discussion of the problem the readers are encouraged to read a special issue of the journal "Reliability Engineering and System Safety", and its editorial (Helton/Oberkampf 2004). In this paper we present the methodology which can be useful for solving statistical problems where both type of uncertainty, random and linguistic, are present in the data. Following (Grzego- rzewski/Hryniewicz 1999), in the second section we introduce the notion of a fuzzy random variable that will be used for the formal description of imprecise random data. We recall some basic notions of fuzzy statistics, and discuss the most important features of the proposed approach. In the third section of the pa- 1 Systems Research Institute, Newelska 6, 01-447 Warszawa, Poland EnviroInfo 2005 (Brno) Informatics for Environmental Protection - Networking Environmental Information Copyright © Masaryk University Brno, Brno 2005. ISBN: 80-210-3780-6