  We analyze the uncertainties in the amplitudes of the spatial correlation
functions estimated from angular correlations in a sample from the APM Galaxy
Survey, with $b_J=17-20$. We model the uncertainties in the selection function
and in the evolution of clustering. In particular we estimate $\sigma_8^{APM}$,
the rms galaxy number fluctuations in spheres of radius at $8 \Mpc$, from the
measured angular variance in the APM. The uncertainty in $\sigma_8^{APM}$ has
three main contributions: 8\% from sampling and selection function
uncertainties, 7\% from the uncertainty in the evolution of clustering and 3\%
from the uncertainty in the value of $\Omega_0$. Including all these
contributions, we find $\sigma_8^{APM}$ is in the range $0.78-1.08$. If the
galaxy clustering in the APM evolves as expected from gravitational clustering
of matter fluctuations, then $\sigma_8^{APM}=0.95 \pm 0.07$ ($1.00 \pm 0.08$)
for $\Omega_0 \simeq 1$ ($\Omega_0 \simeq 0$), close to the values for nearby
optical samples. On the other hand, if we assume that clustering evolution is
fixed in comoving coordinates $\sigma_8^{APM}=0.83 \pm 0.05$ ($0.87 \pm 0.06$),
closer to the results for nearby IRAS samples. The final uncertainty in the
range of values for the hierarchical amplitudes $S_J\equiv
\xibar_J/\xibar_2^{J-1}$ is typically twice the estimated sampling errors, with
the highest values for the case of less clustering evolution. We compare our
estimates with other results and discuss the implications for models of
structure formation.

Astrophysics