In assessing fuzzy numbers to model imprecise data associated with random experiments, trapezoidal fuzzy numbers are often considered. Such an assessment is mainly due to easing both interpretation and computation. This becomes especially noticeable when those assessing fuzzy numbers to data have a weak knowledge, low background and little or no expertise in using fuzzy sets (as it happens when questionnaires whose responses involve a free fuzzy rating are conducted), since the required training to explain the meaning and use of trapezoidal fuzzy numbers is definitely lower than that associated with other shapes. Nevertheless, a question that constantly arises in connection with this trapezoidal assessment is whether it can importantly affect the conclusions of the study involving such data. This paper aims to answer the last question from a statistical perspective. More concretely, the analysis of the influence of the choice of trapezoidal fuzzy numbers to model data is to be based on the conclusions from statistical hypothesis testing about the mean values of the involved fuzzy datasets. For this purpose, the pvalues of tests have been compared for trapezoidal assessment vs. other frequently used ones, like some LU assessments. The analysis is first performed by developing simulation-based pairwise comparisons, and it is later illustrated and corroborated to some extent with a real-life example. The analysis indicates that the shape of the fuzzy assessment scarcely affects statistical conclusions.

In dealing with questionnaires concerning satisfaction, quality perception, attitude, judgement, etc., the fuzzy rating scale has been introduced as a flexible way to respond to questionnaires' items. Designs for this type of questionnaires are often based on Likert scales. This paper aims to examine three different real-life examples in which respondents have been allowed to doubly answer: in accordance with either a fuzzy rating scale or a Likert one. By considering a minimum distance-based criterion, each of the fuzzy rating scale answers is associated with one of the Likert scale labels. The percentages of coincidences between the two responses in the double answer are computed by the criterion-based association. Some empirical conclusions are drawn from the computation of such percentages.

Since Bertoluzza et al. 's metric between fuzzy numbers has been introduced, several studies involving it have been developed. Some of these studies concern equivalent expressions for the metric which are useful for either theoretical, practical or simulation purposes. Other studies refer to the potentiality of Bertoluzza et al.'s metric to establish statistical methods for the analysis of fuzzy data. This paper shortly reviews such studies and examine part of the scientific impact of the metric.

In previous papers, it has been empirically proved that descriptive (summary measures) and inferential conclusions (in particular, tests about means p-values) with imprecise-valued data are often affected by the scale considered to model such data. More concretely, conclusions from the numerical and fuzzy linguistic encodings of Likert-type data have been compared with those for fuzzy data obtained by using a totally free fuzzy assessment: the so-called fuzzy rating scale. These previous comparisons have been performed separately for each of the scales. This paper aims to perform a joint comparison in such a way that means of linked data (one associated with the fuzzy rating and the other one with the encoded Likert scale) are to be tested for equality. Two real-life examples, as well as several simulation-based synthetic ones, have unequivocally shown that the fuzzy rating scale means are significantly different from those for the encoded Likert scales.

Metrics between fuzzy values are a topic with interest for different purposes. Among them, statistics with fuzzy data is growing in modelling and techniques largely through the use of suitable distances between such data. This paper introduces a generalized (actually, parameterized) L 2 metric between fuzzy vectors which is based on their representation in terms of their support function and Steiner points. Consequently, the metric takes into account the deviation in 'central location' (represented by the Steiner points) and the deviation in 'shape' (represented by a deviation defined in terms of the support function and Steiner points). Then, sufficient conditions can be given for this representation to characterize fuzzy vectors, which is valuable for different aims, like optimization studies. Properties of the metric are analyzed and its application to quantify the mean (square) error of a fuzzy value in estimating the value of a fuzzy vector-valued random element is examined. Some immediate implications from this mean error are finally described.