Of course, most real biological dynamics are not uniformly mixing and so are non-ergodic, but we shall see that the ways they fail to be ergodic (and thus remain in the conceptual context of ergodic measures) are descriptively useful (Mandell and Selz, 1997a). The emergence of many statistical approaches to characterizing these motions have been accompanied by the expected controversies about which is best or correct (see below) and have been applied to the problem of diagnosis and clinical discrimination in a variety of neuroscience settings. In ideal abstract chaotic dynamical systems called Axiom A (Russians called them “C systems’), where most mathematical theorems are proven (Smale, 1967), all these measures, if properly computed, are equivalent. In real life, as in the related case of ergodicity, they are not, and since no single one is complete, the more (incomplete) measures we use in our studies along with interest in the way that they differ, supplies more useful information about the system. Though researching and elucidating the most reliable and valid ways of computing these measures are a valuable goal, the current debates focused on the superiority of a single particular measure, constructed in a particular way in relationship to issues of insoluble absolutes like “randomness” versus “deterministic chaos may not be particularly valuable for uncovering new characteristics and potential mechanisms underlying a specific set of real neurobiological observables. Emphasizing diversity and relevance to the clinical biological sciences, we note that quantifying patterns in ergodic (non-ergodic) measures have aided: the discrimination between normal and abnormal opticokinetic nystagmus in neurology patients (Aeson et al, 1997); localizing a two year old subcortical stroke in an EEG of a patient with no other signs or neurological findings (Molnar et al, 1997); the diagnosis of early (not late) multiple sclerosis, as a nonspecific long tract disorder, in patie