I have a table "Register"

with columns:

  class_id bigint NOT NULL,
  disciple text,
  datelesson date NOT NULL,
  student_id integer NOT NULL,
  note character varying(2)

now I want to calculate the average score for each student_id and the number of absent

Select * from "Register" as m

Join

(SELECT AVG(average), COUNT(abs) FROM (SELECT
  CASE
      WHEN "note" ~ '[0-9]' THEN CAST("note" AS decimal)
  END AS average,
  CASE
      WHEN "note" ='a' THEN "note"
  END AS abs
FROM "Register" ) AS average)n
on class_id=0001
And datelesson between '01.01.2012' And  '06.06.2012'
And discipline='music' order by student_id

Result is this:

0001;"music";"2012-05-02";101;"6";7.6666666666666667;1
0001;"music";"2012-05-03";101;"a";7.6666666666666667;1
0001;"music";"2012-05-01";101;"10";7.6666666666666667;1
0001;"music";"2012-05-02";102;"7";7.6666666666666667;1
0001;"music";"2012-05-03";102;"";7.6666666666666667;1
0001;"music";"2012-05-01";102;"";7.6666666666666667;1

The result I receive is for the whole column but how do I calculate average marks for each student?

Could look like this:

SELECT student_id
     , AVG(CASE WHEN note ~ '^[0-9]*$' THEN note::numeric
                                       ELSE NULL END) AS average
     , COUNT(CASE WHEN note = 'a' THEN 1 ELSE NULL END) AS absent
FROM   "Register"
WHERE  class_id = 1
AND    datelesson BETWEEN '2012-01-01' AND  '2012-06-06'
AND    discipline = 'music'
GROUP  BY student_id
ORDER  BY student_id;

I added a couple of improvements.

• You don't need to double-quote lower case identifiers.
• If you want to make sure, there are only digits in note, your regular expression must be something like note ~ '^[0-9]*$'. What you have only checks if there is any digit in the string.
• It's best to use the ISO format for dates, which works the same with any locale: YYYY-MM-DD.
• The count for absence works, because NULL values do not count. Ypu could also use sum for that.
• As class_id is a number type, bigint to be precise, leading zeros are just noise.
  Use class_id = 1 instead of class_id = 0001.