  The H\beta\ emission line in a typical Type I quasar is composed of a broad
base and a narrow core, with the core velocity typical of narrow-line region
emission, and line-fitting routines typically assume this picture. We test the
effects of removing this constraint, and find a substantial group of Type I
quasars in the Sloan Digital Sky Survey catalog with H\beta\ emission line
cores broader than 1200 km/s, above the velocity believed possible for gas in
the quasar narrow-line region. We identify this group of "anomalous H\beta\
quasars" (AHQs) as a distinct population because of a variety of spectral and
photometric signatures common to these AHQs but atypical of other quasars.
These features are similar to some aspects of narrow-line Seyfert 1s and
correlations identified by Eigenvector 1, but also contain distinct features
that make AHQs difficult to classify. We demonstrate that AHQs comprise at
least 11% and most likely approximately one quarter of the SDSS Type I quasar
population at 0.2 < z < 0.8. For AHQs, the [O III]\lambda 4959,5007 profile is
often better fit by de-linking it from the H\beta\ core, while a more standard
linked fit produces a tight correlation between narrow- and broad-line
velocities. We find that [O III] in AHQs sometimes has a standard narrow-line
profile and other times matches the H\beta core, but is rarely in between the
two, implying that the broadened core emission arises from a distinct physical
region. Another feature of AHQs is a diminished [O II] line, which might
indicate a connection between AHQs and the interstellar mediums of their host
galaxies, through reduced photoionization or star formation. We find that it is
difficult to produce AHQs using the current quasar standard model.
