  Different kinds of averaging of the wavefunctions/densities of the
two-electron atomic systems are investigated. Using the Pekeris-like method,
the ground state wave functions $\Psi$ of the helium-like atoms with nucleus
charge $1\leq Z\leq5$ are calculated in a few coordinate systems including the
hyperspherical coordinates $\left\{R,\alpha,\theta\right\}$. The wave functions
$\Psi_{av}(R)$ of the hyperspherical radius $R$ are calculated numerically by
averaging $\Psi$ over the hyperspherical angles $\alpha$ and $\theta$. The
exact analytic representations for the relative derivatives
$\Psi_{av}'(0)/\Psi_{av}(0)$ and $\Psi_{av}''(0)/\Psi_{av}(0)$ are derived.
Analytic approximations very close to the actual $\Psi_{av}(R)$ are obtained.
Using actual wave functions $\Psi$, the one-electron densities $\rho(r)$ are
calculated as functions of the electron-nucleus distance $r$. The relevant
derivatives $\rho'(0)/\rho(0)$ and $\rho''(0)/\rho(0)$ characterizing the
behavior of $\rho(r)$ near the nucleus are calculated numerically. Very
accurate analytical approximations, representing the actual one-electron
density both near the nucleus and far away from it, are derived. All the
analytical and numerical results are supplemented with tables and graphs.
