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 $xi_D sim |mu|^{- u}$ ($ u$ 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 $z=3.2pm 0.1$, a new exponent. Critical path analysis is used in a generalized form to give form to give the macroscopic conductance for lattice above $p_c$.
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