what part of numbers has more entropy?

Question

Given the sequence pf numbers N₁, N₂, N₃... from some source, not a PRNG but say sensor or logging data of some kind, is it safe to assume that processing it like this

Nn / B = Q_n  Rem Mn

will result in the sequence Q haveing less entropy than the sequence M?

Note: assume that B is such that both Q and M has the same sized range.

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This is related to the observation that most real world data sets, regardless or there source, have a logarithmic distribution; numbers starting in 1 are much more common than numbers starting in 9. But this says little about the low order parts.

for a fun way to test this (and piss off you sys admin by bogging down his computer) run this in bash:

 ll -R 2>/dev/null | grep -v -e "^\./" | sed "s/[-rdwxlp]*\W*[0-9]*\W*[a-z]*\W*[a-z]*\W*\([0-9]\).*/\1/" | sort | uniq -c

and get the histogram of the first digit of files sizes.

Answer 1

This depends on the sequence. For example, take [1 * 7 = 7, 3 * 7 = 21, 6 * 7 = 42 ... (2 * N - 1) * 7] and B = 7. Qn will be [1, 3, 6, ... 2 * N - 1] and Mn will be 0 always. Usually, entropy for Q will be less as it's like shifting some bits off, but it's not always like this.

And of course this won't work especially for data coming from a (P)RNG, as the range for Qn will be the same as the range for Mn and for both, numbers are (almost) equally distributed.