Mean and variance of non-normal fuzzy numbers and fuzzy quantities

The concept of mean is extended to non-normal fuzzy numbers and its properties (internality, zero-sum deviation, crispness, invariance, and associativity) are illustrated thoroughly. Analogously, the notion of variance is extended to non-normal fuzzy numbers and its properties (positivity, zero-variance, invariance, change of pole) are reported together with the relationships between two different definitions of variance and different poles. In particular, we demonstrate inequalities between variances of the same fuzzy set with respect to different poles. Finally, the variance is defined for the union of fuzzy sets.