I think that your range testing is suspect.

1. The test for the inverted range should at least be: ( $_->[1] < $_->[0] and $_->[1] >= $some_pos ) }

   That is, the test should be that the end is less than or equal to $some_pos

2. Also, whilst the test works for your current arbitrary chosen position, 36, it would fail when that arbitrary position is greater than 900.

   Ie. If $some_pos = 950; then a range of [0, 901] would fail:

   $some_pos = 950;  
   print +( $_->[1] < $_->[0] and $_->[1] >= $some_pos ) 
       ? 'contained' 
       : 'not contained' for [0, 901];;
   not contained
   [download]

However, neither of those affect the discrepancy you note.

I also thought that using a better rand() might change things, but as you note, Math::Random::MT shows the same discrepancy.

Then I thought perhaps the difference might be due to sample .v. population variance; and discovered that there appears to be an undocumented parameter to the variance() method that might be intended to affect the calculation:

sub variance {
  my $self = shift;  ##Myself
  my $div = @_ ? 0 : 1;
...
    $variance /= $count - $div;
[download]

Ie. If you pass any parameter(s) to the variance method, it appears (I think) to calculate the population rather than the sample variance? Update: the variations are the biased versus unbiased estimates of variance.

But as expected, whilst it does make some slight difference, it is very small:

[21:59:37.22] C:\test>874353
 10000: 10.003  9.138
10.0028 9.13839215999999

[21:59:44.34] C:\test>874353
 10000: 10.003  9.139
10.0028 9.13930609060905
[download]

At this point, I see two possibilities that I don't have answers to, or the knowledge to verify:

• Your understanding that the mean and variance should be the same might be wrong.

  I don't recall ever seeing it stated that variance and mean should coincide for such data?

• Your understanding is correct, but that it is not applicable to the particular mechanics of your test.

  Could the calculation of the variance be affected by a) the inclusive nature of the ranges; or b) the wrap-around?

Not much help, but it might trigger some thoughts somewhere.