I've recently replaced the battery on my iPhone 5S (running latest iOS 10.2.1) and the battery percentage has been wrong ever since.
It originally showed around 50% (which I believe is what the original battery was at when I did the replacement), I then discharged the phone (which turned off with the battery still at 50%), then let it charge fully to 100% (when it came back up it showed 50% instead of the expected 1% or so).
I then let it drain again and it shut down at 90% (though battery life was great as it lasted several hours of full brightness + graphics intensive gaming), when it came back up it showed 90% again instead of around 1%. I let it charge fully.
Finally today I once again used the phone (expecting the battery meter to be calibrated) and despite great battery life the lowest the meter ever showed was 90% and it shut down. It's now on charge and came back up at around 90%.
One odd thing I noticed is that when it dies it shows the low battery screen when plugged in but only for a very short time before going straight to the Apple logo, where I would expect it to stick around on the charging screen for a few minutes at least.
I haven't restored it since the battery swap, I am not sure whether I should do that (I'd like to avoid it as it doesn't remember any of the e-mail nor wireless passwords and it's a pain to set them up again).