Three obvious places to look:

1. Is perl at /usr/bin/perl?
2. Does your cgi file have a Ctrl-M tacked onto the shebang line?
3. Is the winner, I think. Try an absolute path to statistics.txt.

Additionally, you aren't printing the http headers.

Update: ++Ovid's reply, warnings, strict and taint mode will help you a lot.

Update²: RuneK, minimally, print "Content-type: text/html\n\n"; will do, but the CGI::header() call Ovid suggests is better. Ctrl-M sometimes creeps in at the end of the line when documents are produced on DOS/Windows, and transferred in binary mode to *nix servers. For a Windows user, the simplest way to fix that is to upload again in text mode.

After Compline,
Zaxo