Postgresql – Estimate average and median efficiently in Postgres

I think you see a false dichotomy that does not exist.

It can be useful to have connection pooling in place even if you expect a 1:1 mapping of clients to back-ends. If your connections are long-lived, you won't benefit from reducing backend setup/teardown overhead, as it's small and amortized across a long period. A pool like PgBouncer may remain useful for other reasons:

• Block until a connection is available rather than return an immediate max_connections exceeded error;
• You can switch the pool target server if you fail-over to a standby without having to reconfigure applications;
• You can limit application database workers to lower than max_connections, so you can still make reporting and maintenance connections as a non-superuser.

Additionally, if suitable for your application you can use transaction-level pooling to greatly increase the number of clients that can be served by your server.

I would not try to keep a strict 1:1 mapping from Apache workers to PostgreSQL workers, personally. If you've got (say) 16 cores and good I/O on your PostgreSQL box you might want something like 16-20 active PostgreSQL workers for optimal performance. You're almost certain to want more Apache workers than that, since they'll be kept busy by things like:

• persistent HTTP connections from idle clients;
• Unresponsive or very slow clients;
• Intentional DoS connections;
• Network interruptions between client and server; etc

If possible, consider a transaction-pooling design with short-lived transactions instead.

You only need the counts from the 2 tables and a division:

SELECT 
    (SELECT COUNT(*) FROM Lists_Users)  /  (SELECT COUNT(*) FROM `User`) 
    AS average_lists_per_user ;  

If you want more statistical information, you can write it using derived tables:

SELECT 
    total_users, active_users, total_assignments, 
    total_assignments / total_users                  -- this is
        AS average_lists_per_user,                   -- what you want
    total_assignments / active_users 
        AS average_lists_per_active_user
FROM
    ( SELECT COUNT(*) AS total_assignments,
             COUNT(DISTINCT user_id) AS active_users
      FROM Lists_Users
    ) AS lu
  CROSS JOIN
    ( SELECT COUNT(*) AS total_users 
      FROM `User`
    ) AS u ;