We present results from four years of twice-weekly 15 GHz radio monitoring of about 1500 blazars with the Owens Valley Radio Observatory 40 m telescope. Using the intrinsic modulation index to measure variability amplitude, we find that, with $>\!6\sigma$ significance, the radio variability of radio-selected gamma-ray-loud blazars is stronger than that of gamma-ray-quiet blazars. Our extended data set also includes at least 21 months of data for all AGN with `clean' associations in the Fermi Large Area Telescope First AGN catalogue, 1LAC. With these additional data we examine the radio variability properties of a gamma-ray-selected blazar sample. Within this sample, we find no evidence for a connection between radio variability amplitude and optical classification. In contrast, for our radio-selected sample we find that the BL Lac object subpopulation is more variable than the flat spectrum radio quasar (FSRQ) subpopulation. Radio variability is found to correlate with the synchrotron peak frequency, with low- and intermediate-synchrotron-peaked blazars varying less than high-synchrotron-peaked ones. We find evidence for a significant negative correlation between redshift and radio variability among bright FSRQs.