The only useful way to check you rank is to add you site to Google webmaster tools and to look at "Search queries". Then, look at "Average position" for each query and click on specific query to see distribution of positions. You can additionally apply filters by country and type of search at the top of "Search queries" page.
There are some free tools you can use to disable personalized search
For more info read this article - ... www.seomoz.org/blog/google-personalized-search
or you can try this one
- ... www.scroogle.org/cgi-bin/scraper.htm
Are you asking for which page your site appears on, or what your page rank is? There are a lot of free tools to check both, though I'm not sure if they provide the demographic options that you're asking about.
Try marketsamurai and other similar apps
Also depends how you are getting your visibility. If you are using Google Adwords then do NOT search your own keywords in the normal way. Two reasons for this, first is that you won't get an accurate picture of your visibility for several reasons (budget, geo-targeting, etc). Secondly, you will damage your Quality Score and therefore damage your campaign.
If you need any help with Adwords, that's what we do, so please don't hesitate to contact me to discuss.