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 ($z\sim0.03$) obscured AGN from the 70-month Swift/BAT catalogue. 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 anticorrelation 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 [O iii] λ5007 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.