You can install Joomla on your account via the automatic script installation system - Fantastico. Here is the way of installing Joomla from Fantastico:

cPanel --> Fantastico --> Joomla --> New Installation

The other way you can install Joomla on your web hosting account is to install in manually. If you want to perform a manual Joomla installation and you do not know how exactly to do it, you can follow our detailed tutorial here:

http://www.siteground.com/tutorials/joomla/joomla_manual_installation.htm

If you are a SiteGround customer, you can request the script to be installed on your account by SiteGround Technical Support Team. They would gladly assist you with the installation of any of the script/application that are compatible with your hosting account.

You can request the installation from your Customer's Area --> Get Support Section --> HelpDesk --> Get script installed -> Script Installation