My s3 used to be really laggy and the battery life was really bad. Barely got 1.5hrs of SoT. I read online that FB uses a lot of ram and battery so I deleted it and disabled most of google apps like Google+, chrome, maps etc. I also deleted all this useless apps like clean master and anti virus apps(except for battery doctor which I find quite useful) cause they really dont help at all and now my s3 runs really smooth even on power saving mode( unless im using intensive apps then I disable it) As for the battery, I found out that my battery got swollen a bit and it was around 1.5 yrs old. Replaced it with an original samsung battery and Im pretty satisfied with the battery life.
On a good day I get over6hrs of SoT if im watching local videos and light browsing on Wi-Fi and on average I usually get around 3.5 to 5 if Im using 3G browsing and playing ganes like candy crush, subway surfers and real racing 3. Heres a screenshot of what SoT Ive gotten so far today thanks