Should I install programs from a source tarball (`.tar.gz`), from the Ubuntu Software Centre, or from elsewhere?

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http://askubuntu.com – There are several ways to install an application in Ubuntu: You can download a source tarball (generally a .tar.gz or a .tar.bz2 file) and install it manually. (See How to install a .tar.gz (or .tar.bz2) file?) You can download a .deb file and install it manually. You can search for the application in the Ubuntu Software Centre and install it manually. You can find a PPA or a third-party repo, and install it from there. What are the pros and cons of each method? Please discuss security implications, frequency of updates and program reliability of each method in your answer. (HowTos)