Google gives one metric: The number of "active installs", which is a percentage of the number of downloads. For the paid app, it's 82%. For the free app, it's 59%. I presume that once the free app expires on a users phone, they uninstall the app, which partially explains the lower active install rate. In hindsight, this may not have been such a good idea, because it's possible (though I haven't confirmed this) that the active install rate is accounted for in my ranking in the app store.

For the free version, I also have the app ping my server whenever it is started. I think I may publish some numbers for a later post.