el encabezado de la tabla se repite desde un bucle for en php

Eliminar encabezado del bucle.

Haz esto:

<?php
require_once 'header.php';
    // Send variables for the MySQL database class.
    $database = mysql_connect('localhost', 'root', 'password') or die('Could not connect: ' . mysql_error());
    mysql_select_db('robinsnest') or die('Could not select database');

    $query = "SELECT * FROM `members` ORDER by `quiz_score` DESC LIMIT 10";
    $result = mysql_query($query) or die('Query failed: ' . mysql_error());

    $num_results = mysql_num_rows($result);  

    echo "<table>
          <tr>
          <th>Position</th>
          <th>User Name</th>      
           <th>Score</th>
           </tr>";

    for($i = 1; $i <= $num_results; $i++)
    {
         $row = mysql_fetch_array($result);

         echo "<tr>
              <td>".$i."</td>
              <td>".$row['user']."</td>       
              <td>".$row['quiz_score']."</td>
              </tr>";
    }

    echo "</table>";

    echo '<footer>
                <p class="pull-right"><a href="#">Back to top</a></p>
                <p>&copy; 2014 Company, Inc. &middot; <a href="#">Privacy</a> &middot; <a href="#">Terms</a></p>
            </footer>';
?>

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