How can you measure your battery drain rate? Even turned off and with no background apps, my battery drains FAST. I would like to know how fast my drain rate is compared to other phones to find out if I should replace the battery.

What are some measurement methods?
Comparing your device battery drain to another device battery drain is pointless. There are too many variables at play like Android Verizon, how well the OS is optimised, apps loaded, usage conditions and so on. See the section at end of answer for reasons of Abnormal Battery Drain 
Drain rate could be high, even when not used due to Wakelocks. See wakelocks and greenify tags that can help in reducing battery drain. battery-life wiki has pointers as well
See How can I tell if I need to replace my battery? and in particular using the Vbatt app mentioned in my answer. This will not tell you whether you need to change your battery but helps you compare two batteries ( and is time consuming too)




See this study for an interesting infographic