I've updated this answer with a solution that avoids the join in favor of storing some data in a json object.
SOLUTION #1 (two selects and a join, one to get counts, one to get rankings)
This is a little lengthy, but it does work, and it's reasonably fast.
SELECT x.group_field,
avg(
if(
x.rank - y.vol/2 BETWEEN 0 AND 1,
value_field,
null
)
) as median
FROM (
SELECT group_field, value_field,
@r:= IF(@current=group_field, @r+1, 1) as rank,
@current:=group_field
FROM (
SELECT group_field, value_field
FROM table_name
ORDER BY group_field, value_field
) z, (SELECT @r:=0, @current:='') v
) x, (
SELECT group_field, count(*) as vol
FROM table_name
GROUP BY group_field
) y WHERE x.group_field = y.group_field
GROUP BY x.group_field
SOLUTION #2 (uses a json object to store the counts and avoids the join)
SELECT group_field,
avg(
if(
rank - json_extract(@vols, path)/2 BETWEEN 0 AND 1,
value_field,
null
)
) as median
FROM (
SELECT group_field, value_field, path,
@rnk := if(@curr = group_field, @rnk+1, 1) as rank,
@vols := json_set(
@vols,
path,
coalesce(json_extract(@vols, path), 0) + 1
) as vols,
@curr := group_field
FROM (
SELECT p.group_field, p.value_field, concat('$.', p.group_field) as path
FROM table_name
JOIN (SELECT @curr:='', @rnk:=1, @vols:=json_object()) v
ORDER BY group_field, value_field DESC
) z
) y GROUP BY group_field;