First of all, Perl is list-oriented and handles passed in arguments by reference. Therefore the Perlish way to do this is to just take a list of numbers and work with that. But I will play off the rule as given even though it is not the API I would choose.

Also the stricture to avoid having any variables should not be looked at as just a bizarre restriction. People who program in a functional style often do that voluntarily. With that in mind I offer the following version that follows the restrictions as a matter of intentional style and just happens to solve a more complex problem. The function is avg_nested which extracts the numbers from a nested set of arrays of arrays and computes the average of the entire list:

sub avg {
  sum(@_) / @_;
}

sub avg_nested {
  avg(flatten_arrays(@_));
}
  
sub flatten_arrays {
  map {ref($_) ? flatten_arrays(@$_) : $_} @_;
}

sub sum {
  @_ ? shift(@_) + sum(@_) : 0;
}
[download]

UPDATE
jynx caught me. I wrote it, then changed a function name at the last minute and didn't test properly. I forgot to change one flatten to flatten_arrays...