Just takes the dates, turn them into timestamps, add them up, divide by the number of birthdays, and the print out the date of that final timestamp.
$birthdays = array('2014-05-14','2014-05-15','2014-05-16','2014-05-17','2014-05-18');
$total_timstamp = 0;
foreach ($birthdays as $birthday) {
$total_timstamp = bcadd($total_timstamp, (new DateTime($birthday))->format('U'));
}
$average_timestamp = bcdiv($total_timstamp, count($birthdays));
echo date('Y-m-d', $average_timestamp);
This should also work with dates in MM/DD/YYYY
format.