How to round an average to 2 decimal places in PostgreSQL?

        ((this is a Wiki! please edit to enhance!))

Try also the old syntax for casting,

SELECT ROUND( AVG(some_column)::numeric, 2 ) FROM table;

works with any version of PostgreSQL.
...But, as definitive solution, you can overload the ROUND function.

Overloading as casting strategy

CREATE FUNCTION ROUND(float,int) RETURNS NUMERIC AS $f$
  SELECT ROUND( CAST($1 AS numeric), $2 )
$f$ language SQL IMMUTABLE;

Now your instruction will works fine, try this complete comparison:

SELECT trunc(n,3), round(n,3) n_round, round(f,3) f_round,
       pg_typeof(n) n_type, pg_typeof(f) f_type, pg_typeof(round(f,3)) f_round_type
FROM (SELECT 2.0/3.0, 2/3::float) t(n,f);

The ROUND(float,int) function is f_round, it returns a (decimal) NUMERIC datatype, that is fine for some applications: problem solved!

In another applications we need a float also as result. An alternative is to use round(f,3)::float or to create a round_tofloat() function. Other alternative, overloading ROUND function again, and using all range of accuracy-precision of a floating point number, is to return a float when the accuracy is defined (see IanKenney's answer),

CREATE FUNCTION ROUND(
  input float,     -- the input number
  accuracy float   -- accuracy, the "counting unit"
) RETURNS float AS $f$
   SELECT ROUND($1/accuracy)*accuracy
$f$ language SQL IMMUTABLE;

Try

  SELECT round(21.04, 0.05);     -- 21.05 float!
  SELECT round(21.04, 5::float); -- 20
  SELECT round(1/3., 0.0001);    -- 0.3333
  SELECT round(2.8+1/3., 0.5);   -- 3.15
  SELECT round(pi(), 0.0001);    -- 3.1416

PS: the command \df round, on psql after overloadings, will show something like this table

 Schema     |  Name | Result  | Argument  
------------+-------+---------+------------------
 myschema   | round | numeric | float, int
 myschema   | round | float   | float, float
 pg_catalog | round | float   | float            
 pg_catalog | round | numeric | numeric   
 pg_catalog | round | numeric | numeric, int          

where float is synonymous of double precision and myschema is public when you not use a schema. The pg_catalog functions are the default ones, see at Guide the build-in math functions.

Rounding and formating

The to_char function apply internally the round procedure, so, when your aim is only to show a final result in the terminal, you can use the FM modifier as a prefix to a numeric format pattern:

SELECT round(x::numeric,2), trunc(x::numeric,2), to_char(x, 'FM99.99')
FROM (SELECT 2.0/3) t(x);

---

NOTES

Cause of the problem

There are a lack of overloads in some PostgreSQL functions, why (???): I think "it is a lack" (!), but @CraigRinger, @Catcall and the PostgreSQL team agree about "pg's historic rationale".

Note about performance and reuse

The build-in functions, such as ROUND of the pg_catalog, can be overloaded with no performance loss, when compared to direct cast encoding. Two precautions must be taken when implementing user-defined cast functions for high performance:

• The IMMUTABLE clause is very important for code snippets like this, because, as said in the Guide: "allows the optimizer to pre-evaluate the function when a query calls it with constant arguments"

• PLpgSQL is the preferred language, except for "pure SQL". For JIT optimizations (and sometimes for parallelism) language SQL can obtain better optimizations. Is something like copy/paste small piece of code instead of use a function call.

Conclusion: the above ROUND(float,int) function, after optimizations, is so fast than @CraigRinger's answer; it will compile to (exactly) the same internal representation. So, although it is not standard for PostgreSQL, it can be standard for your projects, by a centralized and reusable "library of snippets", like pg_pubLib.

---

Round to the nth bit or other numeric representation

Some people argue that it doesn't make sense for PostgreSQL to round a number of float datatype, because float is a binary representation, it requires rounding the number of bits or its hexadecimal representation.

Well, let's solve the problem, adding an exotic suggestion... The aim here is to return a float type in another overloaded function,
  ROUND(float, text, int) RETURNS float
The text is to offer a choice between

• 'dec' for "decimal representation",
• 'bin' for "binary" representation and
• 'hex' for hexadecimal representation.

So, in different representations we have a different interpretation about the number of digits to be rounded. Rounding a number x with an approximate shorter value, with less "fractionary digits" (tham its original d digits), will be shorter when d is couting binary digits instead decimal or hexadecimal.

It is not easy without C++, using "pure SQL", but this code snippets will illustrate and can be used as workaround:

-- Looking for a round_bin() function! this is only a workaround:
CREATE FUNCTION trunc_bin(x bigint, t int) RETURNS bigint AS $f$
    SELECT ((x::bit(64) >> t) << t)::bigint;
$f$ language SQL IMMUTABLE;
 
CREATE FUNCTION ROUND(
   x float, 
   xtype text,  -- 'bin', 'dec' or 'hex'
   xdigits int DEFAULT 0
) 
RETURNS FLOAT AS $f$
    SELECT CASE
       WHEN xtype NOT IN ('dec','bin','hex') THEN 'NaN'::float
       WHEN xdigits=0 THEN ROUND(x)
       WHEN xtype='dec' THEN ROUND(x::numeric,xdigits)
       ELSE (s1 ||'.'|| s2)::float
      END
    FROM (
      SELECT s1,
             lpad( 
               trunc_bin( s2::bigint, CASE WHEN xd<bin_bits THEN bin_bits - xd ELSE 0 END )::text,
               l2,
               '0'
             ) AS s2
      FROM (
        SELECT *, 
             (floor( log(2,s2::numeric) ) +1)::int AS bin_bits, -- most significant bit position
             CASE WHEN xtype='hex' THEN xdigits*4 ELSE xdigits END AS xd
        FROM (
          SELECT s[1] AS s1, s[2] AS s2, length(s[2]) AS l2
          FROM (SELECT regexp_split_to_array(x::text,'\.')) t1a(s)
        ) t1b
      ) t1c
    ) t2
$f$ language SQL IMMUTABLE;

Try

 SELECT round(1/3.,'dec',4);     -- 0.3333 float!
 SELECT round(2.8+1/3.,'dec',1); -- 3.1 float!
 SELECT round(2.8+1/3.,'dec');   -- ERROR, need to cast string 
 SELECT round(2.8+1/3.,'dec'::text); -- 3 float
 SELECT round(2.8+1/3.,'dec',0); -- 3 float

 SELECT round(2.8+1/3.,'hex',0); -- 3 float (no change)
 SELECT round(2.8+1/3.,'hex',1); -- 3.1266
 SELECT round(2.8+1/3.,'hex',3); -- 3.13331578486784

 SELECT round(2.8+1/3.,'bin',1);  -- 3.1125899906842625
 SELECT round(2.8+1/3.,'bin',6);  -- 3.1301821767286784
 SELECT round(2.8+1/3.,'bin',12); -- 3.13331578486784

And \df round have also:

 Schema     |  Name | Result  | Argument  
------------+-------+---------+---------------
 myschema   | round | float   | x float, xtype text, xdigits int DEFAULT 0

Try with this:

SELECT to_char (2/3::float, 'FM999999990.00');
-- RESULT: 0.67

Or simply:

SELECT round (2/3::DECIMAL, 2)::TEXT
-- RESULT: 0.67

PostgreSQL does not define round(double precision, integer). For reasons @Mike Sherrill 'Cat Recall' explains in the comments, the version of round that takes a precision is only available for numeric.

regress=> SELECT round( float8 '3.1415927', 2 );
ERROR:  function round(double precision, integer) does not exist

regress=> \df *round*
                           List of functions
   Schema   |  Name  | Result data type | Argument data types |  Type  
------------+--------+------------------+---------------------+--------
 pg_catalog | dround | double precision | double precision    | normal
 pg_catalog | round  | double precision | double precision    | normal
 pg_catalog | round  | numeric          | numeric             | normal
 pg_catalog | round  | numeric          | numeric, integer    | normal
(4 rows)

regress=> SELECT round( CAST(float8 '3.1415927' as numeric), 2);
 round 
-------
  3.14
(1 row)

(In the above, note that float8 is just a shorthand alias for double precision. You can see that PostgreSQL is expanding it in the output).

You must cast the value to be rounded to numeric to use the two-argument form of round. Just append ::numeric for the shorthand cast, like round(val::numeric,2).

---

If you're formatting for display to the user, don't use round. Use to_char (see: data type formatting functions in the manual), which lets you specify a format and gives you a text result that isn't affected by whatever weirdness your client language might do with numeric values. For example:

regress=> SELECT to_char(float8 '3.1415927', 'FM999999999.00');
    to_char    
---------------
 3.14
(1 row)

to_char will round numbers for you as part of formatting. The FM prefix tells to_char that you don't want any padding with leading spaces.