  We have determined the amount of stellar mass segregation in over 50 globular
clusters and ultra-faint dwarf galaxy candidates based on deep HST and
ground-based photometry. We find that the amount of mass segregation in
globular clusters is strongly correlated with their relaxation time and that
all clusters with relaxation times of the order of their ages or longer have
little to no mass segregation. For each cluster, the amount of mass segregation
seen is fully compatible with the amount expected by dynamical evolution from
initially unsegregated clusters, showing that globular clusters formed without
primordial mass segregation among their low-mass stars. Ultra-faint dwarf
galaxy candidates split into two groups, star clusters which follow the same
trend between relaxation time and amount of mass segregation as globular
clusters and dark-matter dominated dwarf galaxies that are unsegregated despite
having relaxation times smaller than a Hubble time. Stellar abundance and
velocity dispersion data, where available, confirm our classification. After
classification of the ultra-faint dwarf galaxy candidates, we find that outer
halo star clusters have average densities inside their half-light radii of 0.03
M$_\odot$/pc$^3 \lesssim \rho_h \lesssim$ 1 M$_\odot$/pc$^3$, while dwarf
galaxies have stellar densities of 0.001 M$_\odot$/pc$^3 \lesssim \rho_h
\lesssim $ 0.03 M$_\odot$/pc$^3$. The reason for this separation in density is
most likely a combination of the initial conditions by which the systems formed
and the requirement to withstand external tidal forces.
