The awstats in cPanel is used to see visitor statistics. The script to update this stats runs as per the server settings. Mostly every week. If a customer wants to see his stats you can run the following command.

$ /scripts/runweblogs Username

The users can have an update button on there cPanel stats window.
To get this Update button.

Open the file >> tmp/awstats/domain-name.conf
You will find an entry:
AllowToUpdateStatsFromBrowser=0
Change it to:
AllowToUpdateStatsFromBrowser=1

Now reload your browser. You will find the Update button on top of the stats page.
Click on it to update your stats at any time.

Note:
The tmp folder resides in your home directory. If you are on a shared server, do not try to update it every now and then, as it might create load on the server.
Use this feature only if you are in real need. In other case every week end the stats will be updated automatically on the server.