如何根据月份条件选择列

Question

I have a mysql table like below,

Test : 

Id   start_Date    End Date      Points 
1     01-01-2017    15-01-2017     4 
2     01-02-2017    28-02-2017     5
3     16-01-2017    31-01-2017     5
4     01-03-2017    15-03-2017     3
5     16-03-2017    31-03-2017     4 
6     01-04-2017    30-04-2017     5

In this i want to select the Points based on Quatarwise ( Jan - Mar, Apr - Jun, Jul - Sep , Oct - Dec ).

I am trying the Following Query,

$q1 = "select sum(points) as point from Test where month(start_Date) Between '00' and '03' ";

    $q2 = "select sum(points) as point from Test where month(start_Date) Between '04' and '06' ";

    $q3 = "select sum(points) as point from Test where month(start_Date) Between '07' and '09' ";

    $q1 = "select sum(points) as point from Test where month(start_Date) Between '10' and '12' ";

i am getting the correct result . but what i want is ,

If any month having 2 rows of records then that particular month of records have to add and divide into 2 and sum into remaining 2 months.

Ex : Here jan month having 2 records ( 01 to 15 and 16 to 31 ) . so while getting the quarterwise result these 2 records want to add ( 4 + 5 = 9 ) and divide by 2 and take the points as jan month same for all months and finally want to get the quarterwise points.

How to do this in Mysql .