We consider Birkhoff averages of an
observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$
with respect to a non-invariant measure $m$. In the simple case
where the averages converge $m$-almost everywhere, one may discuss
the distribution of values of the average in a natural way. We
extend this analysis to the case where convergence does not hold
$m$-almost everywhere. The case that the averages converge
$m$-almost everywhere is shown to be related to the recently
defined notion of "predictable" behavior, which is a condition on
the existence of pointwise asymptotic measures (SRB measures). A
heteroclinic attractor is an example of a system which is not
predictable. We define a more general notion called
"statistically predictable" behavior which is weaker than
predictability, but is strong enough to allow meaningful
statistical properties, i.e. distribution of Birkhoff averages,
to be analyzed. Statistical predictability is shown to imply the
existence of an asymptotic measure, but not vice versa. We
investigate the relationship between the various notions of
asymptotic measures and distributions of Birkhoff average.
Analysis of the heteroclinic attractor is used to illustrate the
applicability of the concepts.