Using the Hubble Space Telescope/Space Telescope Imaging Spectrograph, ultraviolet (UV) extinction curves have been measured in M31 along 13 new sight lines, increasing the M31 sample to 17. This sample covers a wide area of M31, having galactocentric distances of 516 kpc, enabling the analysis of UV extinction curve variations over a large region of an external galaxy similar to the Milky Way with global galactic characteristics such as metallicity for the first time. No correlation is found between the extinction parameters and galactocentric distance, which might be expected if there is a radial metallicity gradient in M31. Most of the new UV extinction curves presented here are significantly different from the average extinction curves of the Milky Way, Large Magellanic Cloud (LMC), and Small Magellanic Cloud (SMC), but the average M31 extinction curve is similar to the average extinction curve in the 30 Dor region of the LMC. The wide range of extinction curves seen in each individual Local Group galaxy suggests that global galactic properties such as metallicity may be less important than the local environmental conditions, such as density, UV radiation field, and shocks along each sight line. The combined behavior of the Milky Way, LMC, SMC, and now M31 UV extinction curves supports the idea that there is a family of curves in the Local Group with overlapping dust grain properties between different galaxies.