1 per cent of redshift z~0.1 active galactic nuclei (AGNs) show velocity splitting of a few hundred km/s in the narrow emission lines in spatially integrated spectra. Such line profiles have been found to arise from the bulk motion of ionized gas clouds associated with galactic-scale outflows, merging pairs of galaxies each harbouring a supermassive black hole (SMBH), and/or galactic-scale disc rotation. It remains unclear, however, how the frequency of narrow-line velocity splitting may depend on AGN luminosity. Here we study the correlation between the fraction of Type 2 AGNs with double-peaked narrow emission lines and AGN luminosity as indicated by [OIII] {lambda}5007 emission-line luminosity L_[OIII]_. We combine the sample of Liu et al. (2010. Cat. J/ApJ/708/427) at z~0.1 with a new sample of 178 Type 2 AGNs with double-peaked [OIII] emission lines at z~0.5. We select the new sample from a parent sample of 2089 Type 2 AGNs from the SDSS-III/Baryon Oscillation Spectroscopic Survey. We find a statistically significant (~4.2{sigma}) correlation between L_[OIII}]_ and the fraction of objects that exhibit double-peaked narrow emission lines among all Type 2 AGNs, corrected for selection bias and incompleteness due to [OIII] line width, equivalent width, splitting velocity, and/or equivalent width ratio between the two velocity components. Our result suggests that galactic-scale outflows and/or merging pairs of SMBHs are more prevalent in more powerful AGNs, although spatially resolved follow-up observations are needed to resolve the origin(s) for the narrow-line velocity splitting for individual AGNs.