Question

I need to query for each minute the total count of rows up to that minute.

The best I could achieve so far doesn’t do the trick. It returns count per minute, not the total count up to each minute:

SELECT COUNT(id) AS count
     , EXTRACT(hour from "when") AS hour
     , EXTRACT(minute from "when") AS minute
  FROM mytable
 GROUP BY hour, minute

Answer 1

Return only minutes with activity

Shortest

SELECT DISTINCT
       date_trunc('minute', "when") AS minute
     , count(*) OVER (ORDER BY date_trunc('minute', "when")) AS running_ct
FROM   mytable
ORDER  BY 1;

Use date_trunc(), it returns exactly what you need.

Don’t include id in the query, since you want to GROUP BY minute slices.

count() is typically used as plain aggregate function. Appending an OVER clause makes it a window function. Omit PARTITION BY in the window definition – you want a running count over all rows. By default, that counts from the first row to the last peer of the current row as defined by ORDER BY. The manual:

The default framing option is RANGE UNBOUNDED PRECEDING, which is the
same as RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW. With ORDER BY,
this sets the frame to be all rows from the partition start up
through the current row’s last ORDER BY peer.

And that happens to be exactly what you need.

Use count(*) rather than count(id). It better fits your question ("count of rows"). It is generally slightly faster than count(id). And, while we might assume that id is NOT NULL, it has not been specified in the question, so count(id) is wrong, strictly speaking, because NULL values are not counted with count(id).

You can’t GROUP BY minute slices at the same query level. Aggregate functions are applied before window functions, the window function count(*) would only see 1 row per minute this way.
You can, however, SELECT DISTINCT, because DISTINCT is applied after window functions.

ORDER BY 1 is just shorthand for ORDER BY date_trunc('minute', "when") here.
1 is a positional reference reference to the 1st expression in the SELECT list.

Use to_char() if you need to format the result. Like:

SELECT DISTINCT
       to_char(date_trunc('minute', "when"), 'DD.MM.YYYY HH24:MI') AS minute
     , count(*) OVER (ORDER BY date_trunc('minute', "when")) AS running_ct
FROM   mytable
ORDER  BY date_trunc('minute', "when");

Fastest

SELECT minute, sum(minute_ct) OVER (ORDER BY minute) AS running_ct
FROM  (
   SELECT date_trunc('minute', "when") AS minute
        , count(*) AS minute_ct
   FROM   tbl
   GROUP  BY 1
   ) sub
ORDER  BY 1;

Much like the above, but:

I use a subquery to aggregate and count rows per minute. This way we get 1 row per minute without DISTINCT in the outer SELECT.

Use sum() as window aggregate function now to add up the counts from the subquery.

I found this to be substantially faster with many rows per minute.

Include minutes without activity

Shortest

@GabiMe asked in a comment how to get eone row for every minute in the time frame, including those where no event occured (no row in base table):

SELECT DISTINCT
       minute, count(c.minute) OVER (ORDER BY minute) AS running_ct
FROM  (
   SELECT generate_series(date_trunc('minute', min("when"))
                        ,                      max("when")
                        , interval '1 min')
   FROM   tbl
   ) m(minute)
LEFT   JOIN (SELECT date_trunc('minute', "when") FROM tbl) c(minute) USING (minute)
ORDER  BY 1;

Generate a row for every minute in the time frame between the first and the last event with generate_series() – here directly based on aggregated values from the subquery.

LEFT JOIN to all timestamps truncated to the minute and count. NULL values (where no row exists) do not add to the running count.

Fastest

With CTE:

WITH cte AS (
   SELECT date_trunc('minute', "when") AS minute, count(*) AS minute_ct
   FROM   tbl
   GROUP  BY 1
   ) 
SELECT m.minute
     , COALESCE(sum(cte.minute_ct) OVER (ORDER BY m.minute), 0) AS running_ct
FROM  (
   SELECT generate_series(min(minute), max(minute), interval '1 min')
   FROM   cte
   ) m(minute)
LEFT   JOIN cte USING (minute)
ORDER  BY 1;

Again, aggregate and count rows per minute in the first step, it omits the need for later DISTINCT.

Different from count(), sum() can return NULL. Default to 0 with COALESCE.

With many rows and an index on "when" this version with a subquery was fastest among a couple of variants I tested with Postgres 9.1 – 9.4:

SELECT m.minute
     , COALESCE(sum(c.minute_ct) OVER (ORDER BY m.minute), 0) AS running_ct
FROM  (
   SELECT generate_series(date_trunc('minute', min("when"))
                        ,                      max("when")
                        , interval '1 min')
   FROM   tbl
   ) m(minute)
LEFT   JOIN (
   SELECT date_trunc('minute', "when") AS minute
        , count(*) AS minute_ct
   FROM   tbl
   GROUP  BY 1
   ) c USING (minute)
ORDER  BY 1;