This sort of problems can be handled with a DBMS engine.

Something like this?

#!/usr/bin/perl -w
use strict;
use DBI;

my @data = (
['2004-02-01 07:01',   82],
['2004-02-01 11:38',  172],
['2004-02-01 22:48',  154],
['2004-02-02 05:38',  107],
['2004-02-02 13:20',  117],
['2004-02-02 23:48',  188]);

my $dbh = DBI->connect("dbi:SQLite:averages","","")
    or die $DBI::errstr;

unless ( -s "averages") {
    $dbh->do(qq{ CREATE TABLE averages (
                d datetime not null,
                val INTEGER)})
                or die $DBI::errstr;
    for (@data) {
        $dbh->do( qq{INSERT INTO averages VALUES (?, ?)}, undef, @$_)
            or die $DBI::errstr
    }
}

my $query = qq{
 SELECT
    substr(d,1,10) as day,
    CASE
        WHEN substr(d,12,2) BETWEEN  4 AND 12 THEN "1 morning"
        WHEN substr(d,12,2) BETWEEN 13 AND 18 THEN "2 afternoon"
        WHEN substr(d,12,2) BETWEEN 19 AND 24 THEN "3 evening"
        ELSE "4 night"
    END AS tm,
    AVG(val)
FROM
    averages
GROUP BY
    day, tm
ORDER BY
    day, tm
};

my $averages = $dbh->selectall_arrayref($query)
    or die $DBI::errstr;
printf  "%-10s %-15s %5.2f\n", @$_ for @$averages;
__END__
output:
2004-02-01 1 morning       127.00
2004-02-01 3 evening       154.00
2004-02-02 1 morning       107.00
2004-02-02 2 afternoon     117.00
2004-02-02 3 evening       188.00
[download]

You can add/remove/modify the intervals by acting on the CASE statement inside the query.

If you want an average by time only, without the days, then use this query:

my $query = qq{
 SELECT
    CASE
        WHEN substr(d,12,2) BETWEEN  4 AND 12 THEN "1 morning"
        WHEN substr(d,12,2) BETWEEN 13 AND 18 THEN "2 afternoon"
        WHEN substr(d,12,2) BETWEEN 19 AND 24 THEN "3 evening"
        ELSE "4 night"
    END AS tm,
    AVG(val)
FROM
    averages
GROUP BY
    tm
ORDER BY
    tm
};

my $averages = $dbh->selectall_arrayref($query)
    or die $DBI::errstr;
printf  "%-15s %5.2f\n", @$_ for @$averages;

__END__
output:

1 morning       120.33
2 afternoon     117.00
3 evening       171.00
[download]

 _  _ _  _  
(_|| | |(_|><
 _|