How do I see current average battery statistics on Android 5.1.1 & Nexus 5 I want to see what my current, calculated standby and daily battery life statistics are before Android M is available on my device, so I can see if I've actually an improvement.

Is it possible for me to see such stats on the phone itself?
The simplest method of doing this would be to install GSam Battery Monitor. It gives a clear representation of what functions are draining the battery very precisely and does not "package" battery consumers as the stock monitor does. (Packaging being taking separate processes and representing them as a single entity in it's consumption.

Google play services is one of the big offenders of this. Stock will usually give a 15x larger burden to GP services when in fact battery drain may just be pinned on it from the kernel or system.

Here's a nice snapshot comparison. 

Stock: 

(Click image to enlarge)



Standby is not even listed as an entity on GSam, but is instead delegated to each offending process.

(Both screenshots are taken at 80% charge. Be aware that GSam just scales the used amount to 100% while the stock does not.)

GSam Battery Monitor (Google Play Store):




Personally, the stock monitor seems to be a bit buggy and unreliable anyway. 1% for screen? Really?