The RMS final distance of a random walk is sqrt(# of steps). Simple code to verify this:

#!/usr/bin/perl -w
# KR 2006

use strict;
my ($sum, $sos) = (0, 0); #sum of squares
my ($steps, $trials) = (1000, 1000);

for (1 .. $trials) {
	my $a = 0;
		for (1 .. $steps) { 
			$a += rand > 0.5 ? 1 : -1;
		}
		$sum += $a;
		$sos += $a ** 2;
}
print "mean = ", $sum/$trials, "\t";
print "rms = ", sqrt($sos/$trials), "\n";