  We use dipole and quadrupole statistics to test the large-scale isotropy of
the first 1005 gamma-ray bursts observed by the Burst and Transient Source
Experiment (BATSE). In addition to the entire sample of 1005 gamma-ray bursts,
many subsets are examined. We use a variety of dipole and quadrupole statistics
to search for Galactic and other predicted anisotropies and for anisotropies in
a coordinate-system independent manner. We find the gamma-ray burst locations
to be consistent with isotropy, e.g., for the total sample the observed
Galactic dipole moment <cos theta> differs from the value predicted for
isotropy by 0.9 sigma and the observed Galactic quadrupole moment <sin^2 b -
1/3> by 0.3 sigma. We estimate for various models the anisotropies that could
have been detected. If one-half of the locations were within 86 degrees of the
Galactic center, or within 28 degrees of the Galactic plane, the ensuing dipole
or quadrupole moment would have typically been detected at the 99% confidence
level. We compare the observations with the dipole and quadrupole moments of
various Galactic models. Several Galactic gamma-ray bursts models have moments
within 2 sigma of the observations; most of the Galactic models proposed to
date are no longer in acceptable agreement with the data. Although a spherical
Dark Matter Halo distribution could be consistent with the data, the required
core radius is larger than the core radius of the Dark Matter Halo used to
explain the Galaxy's rotation curve. Gamma-ray bursts are much more isotropic
than any observed Galactic population, strongly favoring but not requiring an
origin at cosmological distances.
