Joomla! Documentation™

Also remembering that the Web Server generally doesn't run as your own user or in the same group. When you run the Web Installer from a browser, it is the Web Server trying to access the files, thus it is the "Other" permissions that will apply to it. If the "Other" permissions do not allow the Web Server to Read, Write or Execute commands in the Joomla! directories, you will receive the message saying that the directories are not writable.