Are there any statistics that indicate how many users an Android app has based on the number of reviews for it? Are there other ways to tell how many users are using an app?
On AppBrain, you can find the number of installs right at the beginning of the page for the app:
On Google Play, you will have to scroll a little further down, as with the relaunch the summary has moved to a less ideal position:
It's just a raw number of installs done via Google Play, and does not include other sources (such as AndroidPIT, Pdassi, or Amazon). I don't know how is dealt with users who installed an app and later uninstalled it, but that number should at least give you a raw idea.
I use many services for tracking the data related to my apps, for example: How many current active users and much more.
For checking the performance of a specific app:
And a very great performance online monitoring tool is New Relic. It can help you track down the realtime data just by including their small SDK in your app (if you are a developer).