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Article

Benford Behavior in Stick Fragmentation Problems

by
Bruce Fang
1,*,
Ava Irons
1,
Ella Lippelman
2 and
Steven J. Miller
1
1
Department of Mathematics, Williams College, Williamstown, MA 01267, USA
2
Department of Mathematics and Computer Science, Colorado College, Colorado Spring, CO 80903, USA
*
Author to whom correspondence should be addressed.
Stats 2025, 8(4), 91; https://doi.org/10.3390/stats8040091
Submission received: 23 August 2025 / Revised: 30 September 2025 / Accepted: 6 October 2025 / Published: 8 October 2025
(This article belongs to the Special Issue Benford's Law(s) and Applications (Second Edition))
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Abstract

Benford’s law states that in many real-world datasets, the probability that the leading digit is d equals log10((d+1)/d) for all 1≤d≤9. We call this weak Benford behavior. A dataset is said to follow strong Benford behavior if the probability that its significand (i.e., the significant digits in scientific notation) is at most s equals log10(s) for all s∈[1,10). We investigate Benford behavior in a multi-proportion stick fragmentation model, where a stick is split into m substicks according to fixed proportions at each stage. This generalizes previous work on the single proportion stick fragmentation model, where each stick is split into two substicks using one fixed proportion. We provide a necessary and sufficient condition under which the lengths of the stick fragments converge to strong Benford behavior in the multi-proportion model.
Keywords: Benford’s law; high-dimensional fragmentation; multinomial distribution Benford’s law; high-dimensional fragmentation; multinomial distribution

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MDPI and ACS Style

Fang, B.; Irons, A.; Lippelman, E.; Miller, S.J. Benford Behavior in Stick Fragmentation Problems. Stats 2025, 8, 91. https://doi.org/10.3390/stats8040091

AMA Style

Fang B, Irons A, Lippelman E, Miller SJ. Benford Behavior in Stick Fragmentation Problems. Stats. 2025; 8(4):91. https://doi.org/10.3390/stats8040091

Chicago/Turabian Style

Fang, Bruce, Ava Irons, Ella Lippelman, and Steven J. Miller. 2025. "Benford Behavior in Stick Fragmentation Problems" Stats 8, no. 4: 91. https://doi.org/10.3390/stats8040091

APA Style

Fang, B., Irons, A., Lippelman, E., & Miller, S. J. (2025). Benford Behavior in Stick Fragmentation Problems. Stats, 8(4), 91. https://doi.org/10.3390/stats8040091

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