So, 'scores' is your table name AND column name we're working with here? Just making sure. Maybe try altering your statement as:
You can even echo the statement to see if it prints out in a manner you are aiming for. And just a note as a few may toss in, in your learning of databases, try to no longer use the mysql_ items. MySQLi and PDO are the newer thing, and Prepared Statements for security. If you're still learning rather from the start, go PDO. And perhaps look into some OOP, just to toss something out there.
$query = "SELECT COUNT(*) FROM `scores` WHERE `score` > " . $score;
EDIT: I just noted the scores/score. My bad.
EDIT2: And I suppose I should ask. What is a score? i.e. The format held within that column. If a straight number, it won't need to be quoted within your statement. Otherwise, it may be.