BENFORD’S
LAW HAS ITS ORIGIN IN THE PARTITION OF INTEGERS
Donald
A. Windsor
Benford’s
Law states that 1 tends to be the most frequent number in
compilations of measured data (1).
In
the partition of any integer, 1 is the most frequent number. The
partitioning of integers is a basic property of our system of numbers
and could be the origin of Benford’s Law.
An
integer is partitioned by assembling all of the possible ways that
add up to it. For example, the partition of 5 consists of 7
partition sets.
5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
The
frequency distribution of the 20 numbers in the partition sets is:
5
4
3 3
2 2
2 2
1 1
1 1 1 1 1 1 1 1 1 1
This
highest frequency of 1s holds true for all integers, because the
lowest partition state of any integer is all 1s. Therefore, 1 is the
most frequent number; 2 is second, and so forth.
I
have been using the partitioning of integers as a non-probabilistic
standard for modeling phylogenetic, bibliometric, ecological, and
economic distributions (2). The reason is that partitions provide a
standard, immutable, frequency distribution for comparing against the
wavering frequency distributions found in nature and based on
randomness and probabilities.
I
can simulate many natural frequency distributions by using a simple
urn model driven by a random number generator. Some of these Monte
Carlo simulations resemble the partition distributions. The biggest
departure is at the top with the highest value numbers. The highest
value in the partition of 5 has to be 5. In a simulation the highest
value could be several times 5. this work continues, albeit at a
slow pace because of my advancing age and my obligations as caregiver
for my disabled wife.
I
did not appreciate the importance of relating partitions to Benford’s
Law until I read the article by Brooks (3) in which he asks, “Why
on earth should Benford’s law exist?” That was when I realized
that the partition of integers could be the origin of Benford’s
law, not just another example, because partitions are a basic
property of our system of numbers.
I
suspect that if Benford had known about my partition model, he may
have based his Law upon it.
References
cited:
1.
Benford, Frank. The law of anomalous numbers. Proceedings of the
American Philosophical Society 1938 March 31; 78(4): 551-572.
2.
Windsor, Donald A. Integer partitions result in skewed
rank-frequency distributions. Journal
of the American Society for Information Science and Technology
2002 December; 53(14): 1276.
3.
Brooks, Michael. Benford’s law. New Scientist 2017 August
26; 235(3140):38-39.
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