  The spin state of an atomic ensemble can be viewed as two bosonic modes,
i.e., a quantum signal mode and a $c$-numbered ``local oscillator'' mode when
large numbers of spin-1/2 atoms are spin-polarized along a certain axis and
collectively manipulated within the vicinity of the axis. We present a concrete
procedure which determines the spin-excitation-number distribution, i.e., the
diagonal elements of the density matrix in the Dicke basis for the collective
spin state. By seeing the collective spin state as a statistical mixture of the
inherently-entangled Dicke states, the physical picture of its multi-particle
entanglement is made clear.
