  The distribution of finite time observable averages and transport in low
dimensional Hamiltonian systems is studied. Finite time observable average
distributions are computed, from which an exponent $\alpha$ characteristic of
how the maximum of the distributions scales with time is extracted. To link
this exponent to transport properties, the characteristic exponent $\mu(q)$ of
the time evolution of the different moments of order $q$ related to transport
are computed. As a testbed for our study the standard map is used. The
stochasticity parameter $K$ is chosen so that either phase space is mixed with
a chaotic sea and islands of stability or with only a chaotic sea. Our
observations lead to a proposition of a law relating the slope in $q=0$ of the
function $\mu(q)$ with the exponent $\alpha$.
