It is a specific case of PERCENTILE_CONT, with an argument of 0.5 and the
ORDER BY column the one in MEDIAN's argument.

MEDIAN(<median-arg>) OVER ( [ PARTITION BY partition_expression] )

Is equivalent to:

PERCENTILE_CONT(0.5) WITHIN 
 GROUP (ORDER BY <median-arg>) OVER ( [ PARTITION BY partition_expression ])

Examples
--------

CREATE TABLE book_rating (name CHAR(30), star_rating TINYINT);

INSERT INTO book_rating VALUES ('Lord of the Ladybirds', 5);
INSERT INTO book_rating VALUES ('Lord of the Ladybirds', 3);
INSERT INTO book_rating VALUES ('Lady of the Flies', 1);
INSERT INTO book_rating VALUES ('Lady of the Flies', 2);
INSERT INTO book_rating VALUES ('Lady of the Flies', 5);

SELECT name, median(star_rating) OVER (PARTITION BY name) FROM book_rating;
+-----------------------+----------------------------------------------+
| name                  | median(star_rating) OVER (PARTITION BY name) |
+-----------------------+----------------------------------------------+
| Lord of the Ladybirds |                                 4.0000000000 |
| Lord of the Ladybirds |                                 4.0000000000 |
| Lady of the Flies     |                                 2.0000000000 |
| Lady of the Flies     |                                 2.0000000000 |
| Lady of the Flies     |                                 2.0000000000 |
+-----------------------+----------------------------------------------+

URL: https://mariadb.com/kb/en/median/https://mariadb.com/kb/en/median/