We consider the statistical properties over disordered samples of the overlap
distribution $P_{\cal J}(q)$ which plays the role of an order parameter in
spin-glasses. We show that near zero temperature (i) the {\it typical} overlap
distribution is exponentially small in the central region of $-1<q<1$: $P^{typ}(q) = e^{\bar{\ln P_{\cal J}(q)}} \sim e^{- \beta N^{\theta} \phi(q)}$,
where $\theta$ is the droplet exponent defined here with respect to the total
number $N$ of spins (in order to consider also fully connected models where the
notion of length does not exist); (ii) the rescaled variable $v = - (\ln
P_{\cal J}(q))/N^{\theta}$ remains an O(1) random positive variable describing
sample-to sample fluctuations; (iii) the averaged distribution $\bar{P_{\cal
J}(q)}$ is non-typical and dominated by rare anomalous samples. Similar
statements hold for the cumulative overlap distribution $I_{\cal J}(q_0) \equiv
\int_{0}^{q_0} dq P_{\cal J}(q)$. These results are derived explicitly for the
spherical mean-field model with $\theta=1/3$, $\phi(q)=1-q^2$, and the random
variable $v$ corresponds to the rescaled difference between the two largest
eigenvalues of GOE random matrices. Then we compare numerically the typical and
averaged overlap distributions for the long-ranged one-dimensional Ising
spin-glass with random couplings decaying as $J(r) \propto r^{-\sigma}$ for
various values of the exponent $\sigma$, corresponding to various droplet
exponents $\theta(\sigma)$, and for the mean-field SK-model (corresponding
formally to the $\sigma=0$ limit of the previous model). Our conclusion is that
future studies on spin-glasses should measure the {\it typical} values of the
overlap distribution or of the cumulative overlap distribution to obtain
clearer conclusions on the nature of the spin-glass phase.Comment: v2=final revised version (in particular new sections IIE,IIIC and
Appendix B w.r.t. v1