Looking at Carole's battery screen compared to mine, my screen shows Swype, Facebook (even though I've killed the task still shows up on there), Media server, and Email usage. Her Screen is only 8%?! Mine is always on top and consistently showing 60-65% of the usage for that.
Just received my 2nd E4GT yesterday and this one came shipped with ICS on it...bummer. I'm coming from a Blackberry so the battery drain is a huge change for me.

But understandable with the difference in the size and power of this phone compared to the BB.
I had a Blackberry 9900 which had relativelyt poor battery performance. The SG3 does a lot more so I was expecting the usage to be higher, however like you my screen typically showed 55%-60% usage of battery power... seemingly even when it wasn't on for most of the time. I was concerned so I've been checking and testing I now have it down to
36%! I would love to know how to get it to 8%.
I've reduced battery drain quite a bit. My Samsung Galaxy S3 is not rooted.
Here are some of the things I've done (and am testing):
Limit Background Processes to no more than 4 (it's in settings > developer options> limit background processes. By default it's set at standard... and that has over 50 processes running.
I turned on power saving mode, but as I did not like the black screen in my email unchecked this option
I close tasks from task manager when I'm not using them (and clear RAM)
Turned off automatic brightness and haptic feedback.
Turn off smart stay... I rarely use it.
Set the screen shut-off time to after 1 minute of inactivity.
I also clear the memory...from task manager. It frees up ram as it closes stuff that really doesn't need to be running.
The most important thing I've done - is get Juice Defender (Ultimate) and I can see the improvement in battery life already. If you get it you will see why and you can see what saps the battery... seeking wi-fi/mobile signals is the worst battery sapper. Try the free versions of Juice Defender, you will probably see a marked improvement. JD also gives many tips for maximizing battery on the phone.
I also make the phone sleep when not in use (switch off the screen not power down) and I have my email synch every 15 mins. Widgets drain battery so any auto-synching I've disabled (including Facebook). Using social networking sites makes the screen use lots of battery. I manually synch my weather apps
What else? Oh yes, I removed Google Plus and in Battery saving mode limited max CPU.
Also be sure to turn off location/gps apps auto synching or altogether, as they eat battery.
Juice defender is still training my phone to spot wi-fi and connect/disconnect as needed. I believe when this is done the battery life will improve even more.
Note that if you limit background processes and then restart the phone the phone resets the background processes back to standard, which is rather annoying. And I have not noticed any degradation in performance so I ensure I limit background process.
Hope this helps.