  The pair-weighted relative velocity dispersion of galaxies provides a measure
of the thermal energy of fluctuations of the observed galaxy distribution, but
the measure is difficult to interpret and is very sensitive to the existence of
rare, rich clusters of galaxies. Several alternative statistical procedures
have recently been suggested to relieve these problems. We apply a variant of
the object-weighted statistical method of Davis, Miller, & White (1997) to the
Las Campanas Redshift Survey (LCRS), which is the largest and deepest existing
redshift survey that is nearly fully sampled. The derived one-dimensional
dispersion on scales ~ 1 Mpc/h is quite low: sigma_1 = 126 +/- 10 km/s, with a
modest decrease at larger scales. The statistic is very stable; the six
independent slices of the LCRS all yield consistent results. We apply the same
statistical procedure to halos in numerical simulations of an open cosmological
model and flat models with and without a cosmological constant. In contrast to
the LCRS, all the models show a dispersion which increases for scales > 1
Mpc/h; it is uncertain whether this is a numerical artifact or a real physical
effect. The standard cluster-normalized Cold Dark Matter model with Omega_m = 1
as well as a tilted variant with n = 0.8 yield dispersions substantially hotter
than the LCRS value, while models with low matter density (Omega_m = 0.3) are
broadly consistent with the LCRS data. Using a filtered cosmic energy equation,
we measure Omega_m ~ 0.2, with small-scale bias factors b = 1.0--1.5 for
high-density models and b = 0.7--1.1 for low-density models.
