In this study, we elaborate on an important issue of membership function determination. The main point is that any membership estimation procedure should reconcile the semantics of a fuzzy set (regarded as an information granule arising at some level of information abstraction) with the experimental evidence conveyed by numeric data. This, in the sequel, calls for the development of the hybrid two-phase approach that starts from a rough specification of the support of the fuzzy set that is followed by detailed computations involving a specific type of membership function and an estimation of its parameters. The role of robust statistics in this setting is also raised. A number of experimental results are discussed.