I have set up Google Analytics on this site. This is a very sophisticated looking system for tracking web site visits. Google have set it up to help their advertisers count their money but it is free if your site has less than 5 million page views/month which should suffice for me.
The analytics site is still a bit flaky and overloaded and it is refusing to validate the code on this site. I'll give it till tomorrow to settle down and I'll try again.
It looks to be more biased towards grand overviews than statcounter which is good for analysing individual visits: especially the free version that only tracks the last 100 page views.
Apart from the current flakiness, the analytics reports are only updated every few hours (like awstats I suppose) whereas statcounter is real time.
UPDATE: although the site still tells me the tracker is not installed properly and the link to an example installation is broken, the tracker is working and I can see the stats. So far it looks very sophisticated for a free (to me) service: it looks professional and expensive. Here is the geographical chart:
This shows a tooltip of each town as you pass the mouse over it (the analytics page, not the screenshot above) whereas Statcounter just gives a list of countries. However, in Statcounter I can drill into the visits from each country to see which pages they have been looking at.
I now have a new metric to worry about: depth of visit, i.e. people looking at more than one page 
Analytics and Statcounter may complement each other: analytics for charts, Statcounter for the details of who is doing what. I could carry on using both, it's all a matter of how quickly I want pages to load and how much I trust both lots of servers to keep up with mine.
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