Average Time That an App's Client is connected to a Database?
MySQL does not keep any metric like that.

I would instrument the client in order to get the info.  (In fact, I have done such.)  At the beginning of the client (or when the client connects), capture the time to sufficient precision to get milliseconds.  At the end of the client (or connection), capture the time again.  Save the result in a table for such.  Later do SELECT AVG(...)... to get the information you desire.  It is a more complex mathematical problem to say "Most connections are between 50ms and 70ms."  For that, you might want to deduce the 10th and 90th percentiles.  (This is best done in client code, not SQL.)