upper_bound

function upper_bound ($a)
{

  //Use this many standard deviations as upper bound
  $max_std_devs = 3;
  $max_val = 0;
  $removed_items = 1;

  while ($removed_items && count($a) > 0) { // Continue as long as 'outliers' are found
    $removed_items = 0;
    $the_variance = 0.0;
    $the_mean = 0.0;
    $the_array_sum = array_sum($a); //sum the elements
    $number_elements = count($a); //count the number of elements

    //calculate the mean
    $the_mean = $the_array_sum / $number_elements;

    //calculate the variance
    for ($i = 0; $i < $number_elements; $i++)
    {
        //sum the array
        $the_variance = $the_variance + ($a[$i] - $the_mean) * ($a[$i] - $the_mean);
    }

    $the_variance = $the_variance / $number_elements;

    $max_val = $the_mean + ($max_std_devs * sqrt($the_variance));

    foreach($a as $key => $value) {
        if ($value > $max_val) {
          unset($a[$key]);
          $removed_items = 1;
        }
    }
    $array = array_values($a);
  }

  return ceil($max_val);
}