  We calculate the cosmological evolution of the 1-point probability
distribution function (PDF), using an analytic approximation that combines
gravitational perturbation theory with the Edgeworth expansion of the PDF. Our
method applies directly to a smoothed mass density field or to the divergence
of a smoothed peculiar velocity field, provided that rms fluctuations are small
compared to unity on the smoothing scale, and that the primordial fluctuations
that seed the growth of structure are Gaussian. We use this `Edgeworth
approximation' to compute the evolution of $<\delta|\delta|>$ and $<|delta|>$;
these measures are similar to the skewness and kurtosis of the density field,
but they are less sensitive to tails of the probability distribution, so they
may be more accurately estimated from surveys of limited volume. We compare our
analytic calculations to cosmological N-body simulations in order to assess
their range of validity. When $\sigma \ll 1$, the numerical simulations and
perturbation theory agree precisely, demonstrating that the N-body method can
yield accurate results in the regime of weakly non-linear clustering. We show
analytically that `biased' galaxy formation preserves the relation $<\delta^3>
\propto <\delta^2>^2$ predicted by second-order perturbation theory, provided
that the galaxy density is a local function of the underlying mass density.
