Did you use
their uninstaller tool PRIOR to uninstalling it or did you just drag it to the trash bin?
Here is a link that may help you get rid of traces of Sophos, which others of had to deal with as well:
http://forums.macrumors.com/showthread.php?t=780504. Reboot your machine after doing this. You may have to uninstall Avast and reinstall since you had 2 antiviruses running on your machine at the same time and this can create conflicts. If you do this, use the Avast Uninstaller tool
http://www.avast.com/uninstall-utility - works for Macs as well, then reboot and do a fresh install.
In real stubborn cases, there are third party app. uninstallers for Macs that you can look up and use as a last resort, but I do not recommend it for Avast.
Let us know if you have any questions. Thank you.