  We study the statistics of the conductance $g$ through one-dimensional
disordered systems where electron wavefunctions decay spatially as $|\psi| \sim
\exp (-\lambda r^{\alpha})$ for $0 <\alpha <1$, $\lambda$ being a constant. In
contrast to the conventional Anderson localization where $|\psi| \sim \exp
(-\lambda r)$ and the conductance statistics is determined by a single
parameter: the mean free path, here we show that when the wave function is
anomalously localized ($\alpha <1$) the full statistics of the conductance is
determined by the average $<\ln g>$ and the power $\alpha$. Our theoretical
predictions are verified numerically by using a random hopping tight-binding
model at zero energy, where due to the presence of chiral symmetry in the
lattice there exists anomalous localization; this case corresponds to the
particular value $\alpha =1/2$. To test our theory for other values of
$\alpha$, we introduce a statistical model for the random hopping in the tight
binding Hamiltonian.
