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Do the numerical results next to the keywords refer to the amount of searches done within a certain period of time or do they refer to the amount of competing pages for that keyword?
It's the (estimated) total number of results returned for the suggested keyword.
I'm unclear on how the ordering is achieved, although it seems like a worthwhile subject to look into further.
So it refers to the amount of pages that contain that keyword or which is optimized for that keyword?
The amount of pages deemed relevant to the query, for that particular search.
the "estimate" differs from what you see if you go to Google and actually type in the keyword and do the search
That's a separate topic with prior discussion in the Google forum [webmasterworld.com] - there's no authoritative number of results for a query: it depends on factors like location, the details of the query and which data is accessed to determine the result.
[edited by: Receptional_Andy at 10:52 pm (utc) on July 8, 2008]