  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 derive an exact analytical criterion for the
validity of the single parameter scaling in this model. According to this
criterion, states with energies from the conduction band of the underlying
non-random system satisfy single parameter scaling when disorder is small
enough. At the same time, single parameter scaling is not valid for states
close to band boundaries and those outside of the original spectrum even in the
case of small disorder. The obtained results are applied to the Kronig-Penney
model with the potential in the form of periodically positioned
$\delta$-functions with random strengths. We show that an increase in the
disorder can restore single parameter scalingbehavior for states from band-gaps
of this model.
