The fuzzy rating method has been introduced in psychometric studies as a tool, which allows the capture of and accurate reflection of the diversity, subjectivity, and imprecision inherent in human responses to many questionnaires. The lack of statistical techniques for in-depth analysis of these responses has been, for years, the appearance of an important barrier. At present, this barrier is being overcome thanks to new statistical techniques. In this way, the information from fuzzy rating method-based responses can be suitably explored and exploited. This paper aims to formally endorse some of the main statistical benefits of using free-response format fuzzy rating scale-based questionnaires instead of using the closed-response format involving fuzzy linguistic representations.Index Terms-Fuzzy linguistic representation, fuzzy numbers, fuzzy rating method, questionnaires, random fuzzy sets, statistical analysis of fuzzy data.

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The fuzzy rating scale was introduced as a tool to measure intrinsically ill-defined/ imprecisely-valued attributes in a free way. Thus, users do not have to choose a value from a class of prefixed ones (like it happens when a fuzzy semantic representation of a linguistic term set is considered), but just to draw the fuzzy number that better represents their valuation or measurement. The freedom inherent to the fuzzy rating scale process allows users to collect data with a high level of richness, accuracy, expressiveness, diversity and subjectivity, what is especially valuable for statistical purposes. This paper presents an inferential approach to analyze data obtained by using the fuzzy rating scale. More concretely, the paper is to be focussed on testing different hypothesis about means, on the basis of a sound methodology which has been stated during the last years. All the procedures that have been developed to this aim will be presented in an algorithmic way adapted to the usual generic fuzzy rating scale-based data, and they will be illustrated by means of a real-life example.

In evaluating aspects like quality perception, satisfaction or attitude which are intrinsically imprecise, the fuzzy rating scale has been introduced as a psychometric tool that allows evaluators to give flexible and quite accurate, albeit non numerical, ratings. The fuzzy rating scale integrates the skills associated with the visual analogue scale, because of the total freedom in assessing ratings, with the ability of fuzzy linguistic variables to capture the natural imprecision in evaluating such aspects. Thanks to a recent methodology, the descriptive analysis of the responses to a fuzzy rating scale-based questionnaire can be now carried out. This paper aims to illustrate such an analysis through a real-life example, as well as to show that statistical conclusions can often be rather different from the conclusions one could get from either Likert scale-based responses or their fuzzy linguistic encoding. This difference encourages the use of the fuzzy rating scale when statistical conclusions are important, similarly to the use of exact real-valued data instead of grouping them.