Because you're both shifting and counting

Try this:

open SCORES, "a:/scores.txt" or die "Error: $! while opening scores fi
+le\n";
my $students = 0;
my $totalscore = 0;
while (<SCORES>) {
    chomp;
    my @line = split /:/;
    while (@line) {
        my $student = shift @line;
        my $score = shift @line;
        print "$student\n$score\n";
        $students++;
        $totalscore += $score;
    }
}
print "average: ", $totalscore/$students, "\n"  if $students;
[download]

Update: fixed embarrassing typos and minor errors