  The self averaging properties of conductance $g$ are explored in random
resistor networks with a broad distribution of bond strengths
$P(g)\simg^{\mu-1}$. Distributions of equivalent conductances are estimated
numerically on hierarchical lattices as a function of size $L$ and distribution
tail parameter $\mu$. For networks above the percolation threshold, convergence
to a Gaussian basin is always the case, except in the limit $\mu$ --> 0. A {\it
disorder length} $\xi_D$ is identified beyond which the system is effectively
homogeneous. This length diverges as $\xi_D \sim |\mu|^{-\nu}$ ($\nu$ is the
regular percolation correlation length exponent) as $\mu$-->0. This suggest
that exactly the same critical behavior can be induced by geometrical disorder
and bu strong bond disorder with the bond occupation probability $p$<-->$\mu$.
Only lattices at the percolation threshold have renormalized probability
distribution in a {\it Levy-like} basin. At the threshold the disorder length
diverges at a vritical tail strength $\mu_c$ as $|\mu-\mu_c|^{-z}$, with
$z=3.2\pm 0.1$, a new exponent. Critical path analysis is used in a generalized
form to give form to give the macroscopic conductance for lattice above $p_c$.
