Postgres - Querying Min, Max and getting associated columns data specific to those result of those rows I have query in which returns averaged values by site and month.  What I want to know is the min and max of those monthly averages by site and (the difficult part) the month when each occurred.

Here is an example:

SQL Fiddle Stuff here

CREATE TABLE events (
    esite     integer NOT NULL,
    edate     timestamp with time zone NOT NULL,
    evalue    integer NOT NULL
);

INSERT INTO events Values
    (1, '2016-01-03', 11),
    (2, '2016-01-05', 90),
    (1, '2016-01-08', 7),
    (2, '2016-01-10', 40),
    (1, '2016-01-15', 12),
    (1, '2016-01-18', 66),
    (2, '2016-01-22', 54),
    (2, '2016-02-03', 70),
    (2, '2016-02-05', 56),
    (1, '2016-02-08', 61),
    (2, '2016-02-10', 23),
    (1, '2016-02-15', 30),
    (1, '2016-02-18', 15),
    (1, '2016-02-22', 41);


So I'm looking for a query that returns (by site) the min and max monthly average evalues and the months at which the min and max values occurred.  I can get this by using the query below: 

SQL Fiddle this

select esite, date_trunc('month', edate) as emonth, round(avg(evalue),2) as evalue_avg from events
  Group by esite, emonth

-- produces the following output:
esite | emonth                      | evalue_avg
2     | January, 01 2016 00:00:00   | 61.33
1     | January, 01 2016 00:00:00   | 24
1     | February, 01 2016 00:00:00  | 36.75
2     | February, 01 2016 00:00:00  | 49.67


Now for the part I'm having difficultly with, I need produce the following result: -- basically the min, max values and the date(month) at which each occurred by site -- (Desired Output)

-- (one row per site)
esite | avg_min | eval_avg_min_date          | avg_max | eval_avg_max
1     | 24.00   | January, 01 2016 00:00:00  | 36.75   | February, 01 2016 00:00:00
2     | 49.67   |February, 01 2016 00:00:00  | 61.33   | January, 01 2016 00:00:00


I've searched around and have seen some examples using windowing and lateral joins, but I haven't been successful in getting any of them to work.  This might be a pivot but since I don't have a fixed number of sites, this tends to be difficult with PostgresSQL.

I'm guessing if anyone has an easy way to do this using postgres 9.4+.  

Your help would be appreciated!
I would recommend you do something like this - and don't go down the route of having a row per month. 

What happens when you have 5 (10...15... x) years' data?

This is superior IMHO. You can always use the CROSSTAB table function, but I would recommend against it.

SELECT
  esite,
  MIN(evalue), 
  MAX(evalue),
  AVG(evalue)::INT,
  EXTRACT(MONTH FROM edate) AS emonth,
  CASE
    WHEN EXTRACT(MONTH FROM edate) = 1 THEN 'January'
    WHEN EXTRACT(MONTH FROM edate) = 2 THEN 'February'
    -- &lt;... fill in rest of months here ...&gt;
  END AS litMonth
FROM events
GROUP BY esite, emonth, litMonth
ORDER BY esite, emonth -- , min, max, avg;


This gives the result:

esite;     min;   max;   avg;    emonth;    litmonth
----------------------------------------------------
    1;       7;    66;    24;     1;         January
    1;      15;    61;    37;     2;        February
    2;      40;    90;    61;     1;         January
    2;      23;    70;    50;     2;        February


Note, that I have CAST the AVG as an INTEGER - you can of course ROUND this off to whatever you like. You can add EXTRACT(YEAR  FROM edate) AS eyear, after the EXTRACT(MONTH... for more clarity and a more elegant result.