Where should I place a javascript code in the WordPress installation for capturing website statistics?

Usually the header or footer file of the theme that you use.
That way every page will have it when they load.

I did that, but when I went to the home page and viewed the source code it was not included. Is this right?

If you do not see the javascript for the analytics in the source code, it isn't there.

For Google Analytics, you will have a file named footer.php. You can paste the analytics code right in that. You can also use a plugin for google analytics for wordpress, but I see no reason for using plugins for something of this nature.

Or you could try looking into this plugin:
http://wordpress.org/extend/plugins/stats/

If you have an API key, of course.