You can calculate the average and the variance for the 24h intervals centered around every hour (0, 1, 2,... 23) and pick the center which minimizes the variance: 
use POSIX qw(fmod);

my @data = (1, 1, 1, 0, 1, 2, 3, 4, 21, 22, 20, 22, 23, 20, 22);

sub circular_average {
    my ($pivot, $round, @data) = @_;

    my $sum = 0;
    my $sum2 = 0;

    my $displacement = 1.5 * $round - $pivot;

    for my $data (@data) {
        my $displaced = fmod($data + $displacement, $round);
        # printf "%2i->%3i,%2i ", $data, $displaced, $displaced - $dis
+placement;
        $sum += $displaced;
        $sum2 += $displaced * $displaced;
    }
    # print "\n";
    my $inv_n = 1.0 / @data;
    my $avg = fmod($inv_n * $sum + 0.5 * $round + $pivot, $round);
    wantarray ? ($avg, $inv_n * $sum2 - $inv_n * $inv_n * $sum * $sum)
+ : $avg;
}

my ($best_avg, $best_s2);
for my $time (0..23) {
    my ($avg, $s2) = circular_average $time, 24, @data;

    if (not defined $best_s2 or $best_s2 > $s2) {
        $best_avg = $avg;
        $best_s2 = $s2;
    }
}
printf "avg: %.1f, s2: %.1f\n", $best_avg, $best_s2;

[download]
Then, you can even iterate a few times centering the interval around the best average ($best_avg) found: 
for (0..5) {
    ($best_avg, $best_s2) = circular_average $best_avg, 24, @data;
    printf "avg: %.1f, s2: %.1f\n", $best_avg, $best_s2;
}

[download]

In reply to Re: Average start time handling midnight by salva
in thread Average start time handling midnight by chrisjej

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