  Event-by-event analysis of heavy-ion collision events is an important tool
for the study of the QCD phase boundary and formation of a quark-gluon plasma.
A universal feature of phase boundaries is the appearance of increased
fluctuations of conserved measures as manifested by excess measure variance
compared to a reference. In this paper I consider a particular aspect of EbyE
analysis emphasizing global-variables variance comparisons and the central
limit theorem. I find that the central limit theorem is, in a broader
interpretation, a statement about the scale invariance of total variance for a
measure distribution, which in turn relates to the scale-dependent symmetry
properties of the distribution.. I further generalize this concept to the
relationship between the scale dependence of a covariance matrix for all
conserved measures defined on a dynamical system and a matrix of covariance
integrals defined on two-point measure spaces, which points the way to a
detailed description of the symmetry dynamics of a complex measure system.
Finally, I relate this generalized description to several recently proposed or
completed event-by-event analyses.
