  Accreting supermassive black holes (SMBHs), also known as active galactic
nuclei (AGN), are generally surrounded by large amounts of gas and dust. This
surrounding material reprocesses the primary X-ray emission produced close to
the SMBH and gives rise to several components in the broadband X-ray spectra of
AGN, including a power-law possibly associated with Thomson-scattered
radiation. In this work, we study the properties of this scattered component
for a sample of 386 hard-X-ray-selected, nearby ($z \sim 0.03$) obscured AGN
from the 70-month Swift/BAT catalog. We investigate how the fraction of
Thomson-scattered radiation correlates with different physical properties of
AGN, such as line-of-sight column density, X-ray luminosity, black hole mass,
and Eddington ratio. We find a significant negative correlation between the
scattering fraction and the column density. Based on a large number of spectral
simulations, we exclude the possibility that this anti-correlation is due to
degeneracies between the parameters. The negative correlation also persists
when considering different ranges of luminosity, black hole mass, and Eddington
ratio. We discuss how this correlation might be either due to the angle
dependence of the Thomson cross-section or to more obscured sources having a
higher covering factor of the torus. We also find a positive correlation
between the scattering fraction and the ratio of [OIII] $\lambda$5007 to X-ray
luminosity. This result is consistent with previous studies and suggests that
the Thomson-scattered component is associated with the narrow-line region.
