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For a normal distribution, the probability density (x) i s e v erywhere positive, so in principle, all real numbers are possible. In reality, the probability that a random va r i a b l e i s f a r a way from the mean is so small that this possibility can be often safely ignored. Usually, a small real number k is picked (e.g., 2 or 3) then, with a probability P0(k) 1 (depending on k), the normally distributed random variable with mean a and standard deviation belongs to the interval a = a ; k a + k ]. The actual error distribution may be non-Gaussian hence, the probability P (k) that a random variable belongs to a di ers from P0(k). It is desirable to select k for which the dependence of P0(k) on the distribution is the smallest possible. Empirically, this dependence is the smallest for k 2 1:5 2:5]. In this paper, we g i v e a theoretical explanation for this empirical result.

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In this paper, we discuss the possibility of using the formalism of fuzzy intervals combined with automatic differentiation technique as a basis for numerical software selfverification in metrology. The natural domain of such approach is calculating indirect measurements results using the inaccurate results of direct measurements as the initial data. We propose to support software for such computations with tools that allow us to receive simultaneously calculated results and their error characteristics. Only such software can be put to metrological validation in full.In many practical situations, the inaccurate results of direct measurements are used for calculations of indirect measurements results. Final data are also uncertain. Characteristics of this uncertainty should be expressed in quantitative form and presented together with indirect measurement result. The main purpose of this paper is to discuss ways to provide software for measured data processing with tools of automatic calculation of final result uncertainty. Only software that is supported in such manner can pass the metrological certification in full.To achieve this purpose, we propose to use combination of two formalisms: fuzzy intervals approach -to represent inaccuracy of initial data for calculations, and formalism of software automatic differentiation -to compute how initial data uncertainty transforms to inherited uncertainty of final result.There are many approaches for representing inaccuracy of measured data that act as initial information for subsequent calculations. Modern approaches take into account different information about the initial data inaccuracy. Some of them use random variables [1][2][3] for uncertainty representing and handling with it, other ones use bounds on possible values of initial data [4][5][6]. Interval representation of data inaccuracy was firstly mentioned by Wiener [7] and Kantorovich [8]. With the development of the fuzzy set theory, its formalism became actual tool for uncertainty expressing in metrology [9,10]. Natural evolution of ideas of interval and fuzzy frameworks is the concept of fuzzy