  The variance of the Lyapunov exponent is calculated exactly in the
one-dimensional Anderson model with random site energies distributed according
to the Cauchy distribution. We find a new significant scaling parameter in the
system, and derive an exact analytical criterion for single parameter scaling
which differs from the commonly used condition of phase randomization. The
results obtained are applied to the Kronig-Penney model with the potential in
the form of periodically positioned $\delta$-functions with random strength.
