  For a positive measure set of nonuniformly expanding quadratic maps on the
interval we effect a multifractal formalism, i.e., decompose the phase space
into level sets of time averages of a given observable and consider the
associated {\it Birkhoff spectrum} which encodes this decomposition. We derive
a formula which relates the Hausdorff dimension of level sets to entropies and
Lyapunov exponents of invariant probability measures, and then use this formula
to show that the spectrum is continuous. In order to estimate the Hausdorff
dimension from above, one has to "see" sufficiently many points. To this end,
we construct a family of towers. Using these towers we establish a large
deviation principle for empirical distributions, with Lebesgue as a reference
measure.
