The R-chart blog explains how to read a weblog file into R, so you can analyze traffic to a website. For example, here's a page request chart created with R:
Now, charts like this are stock-in-trade for tools like Google Analytics, but this is still useful if you want to look at the performance of a site that hasn't been instrumented for analytics. It also allows you to analyze other data in the weblog, such as frequency of HTTP status codes:
(Looks like this particular site is experiencing a lot of page-not-found 404 errors.) With access to the raw weblog data in R, you could subject it to any kind of statistical analysis.
You can find the R code and lots of tips for dealing with weblog data in R at the link below.