Can I access directly Awstats without going through the cPanel?

Yes, it is possible to access your AWStats directly through your domain name.

However, doing this requires you to add a script to your site and configure it. In other words you need to manually install AWStats for your domain.

In order to install AWStats on your domain you can follow the steps below:

Download Awstats from their website here.
Extract the contents of the file to your computer.
Use an FTP client (or File Manager in cPanel) to upload the contents of both the cgi-bin and icon folders into the root directory of your domain.
Change the permissions of the awredir.pl and awstats.pl files to 755 (in the cgi-bin directory). You can do that with your FTP client.
The next step is to download the AWStats configuration file from your account to your local computer from /tmp/awstats/awstats.yourdomain.tld.conf
Open this file with a text editor and go to the “DirIcons” line. Change the value to: DirIcons=”/icon/“.
Upload the modified file to the cgi-bin folder.

Now you can access your statistics at http://www.yourdomain.tld/cgi-bin/awstats.pl?config=yourdomain.tld

To access HTTPS statistics open your browser the following URL: http://www.yourdomain.tld/cgi-bin/awstats.pl?config=yourdomain.tld&ssl

Note: Please password protect the /cgi-bin folder so that only authorized users can access it (found under cPanel -> Files -> Directory Privacy).