  We estimate the variance $\xibar_2$, the skewness $\xibar_3$ and the kurtosis
$\xibar_4$ in the distribution of density fluctuations in a complete sample
from the APM Cluster Redshift Survey with 339 clusters and a mean depth $ \sim
250\Mpc$. We are able to measure the statistics of fluctuations in spheres of
radius $R \simeq 5-80 \Mpc$, with reasonable errorbars. The statistics in the
cluster distribution follow the hierarchical pattern
$\xibar_J=S_J~\xibar_2^{J-1}$ with $S_J$ roughly constant, $S_3 \simeq 2$ and
$S_4 \sim 8$. We analyse the distribution of clusters taken from N-body
simulations of different dark matter models. The results are compared with an
alternative method of simulating clusters which uses the truncated Zel'dovich
approximation. We argue that this alternative method is not reliable enough for
making quantitative predictions of $\xibar$. The N-body simulation results
follow similar hierarchical relations to the observations, with $S_J$ almost
unaffected by redshift distortions from peculiar motions. The standard
$\Omega=1$ Cold Dark Matter (CDM) model is inconsistent with either the second,
third or fourth order statistics at all scales. However both a hybrid Mixed
Dark Matter model and a low density CDM variant agree with the $\xibar_J$
observations.
