Summary indicators of opinions expressed by the users of given service

In this paper we study the properties of a family of index, called CI. These indices have been proposed by Civardi, Zavarrone (2003) in order to evaluate the teaching quality in university disciplines. The most frequently used scales offer four or five points and the first two (or the last two) points on both scales are associated with negative evaluations and the last two (or the first two) are associated with symmetric positive evaluations. The empirical distribution of responses represents the starting point to compute the CI indices. Each index assumes values lying between ¿100 (in the case of maximum negative evaluation) to +100 (in the case of an evaluation of absolute excellence) and is obtained as the algebraic sum of two indices. The first expresses the score obtained in the semi-plane of positive evaluations while the second represents the score obtained in the semi-plane of negative evaluations. The CI index is characterized by the choice of the parameter of importance level k (0¿k¿1) on the degree of importance the ¿investigator/decision maker¿ wants to assign to the quota of very positive opinions and of the very negative ones. The construction of the universe of response models of N respondents (with 10¿N¿105) and, for each distribution, of the eleven CI indices (k=0, 0.1, ..., 0.9, 1)) allow to study the properties of effective distributions of the indices. The results highlight that all effective distributions, varying N and k, are symmetric with mean, mode and median equal to zero. The square mean error assumes values from 53 (when N=10 and k=0) to 31. The possibility of approximating CI distribution with a normal one offers interesting developments in an inferential framework.