First-Digit Distributions of Human Long-Bone Morphometric Parameters: An Exploratory X-Ray-Based Comparison with Benford's Law.

BACKGROUND: Benford's Law describes a theoretical first-digit distribution observed in many natural and scientific datasets. However, empirical datasets rarely match this distribution exactly, and the appearance of a Benford-like pattern may depend on the numerical range and data-generating process. OBJECTIVE: This exploratory study aimed to describe the first-digit distributions of X-ray-based human long-bone measurements and derived quantities, and to compare these observed distributions with the theoretical distribution predicted by Benford's Law. METHODS: Anteroposterior X-ray images of half of the long bones from three adult patients were retrospectively retrieved from the Picture Archiving and Communication System. The lengths, perimeters, and projected areas of each long bone were measured. The squares and cubes of bone lengths were subsequently calculated as exploratory dimension-based transformations. First-digit distributions were compared with the Benford-expected distribution using Pearson's chi-square goodness-of-fit test as an exploratory statistical comparison. Mean absolute deviation (MAD) between observed and expected first-digit proportions was additionally calculated as a descriptive measure of discrepancy. RESULTS: The first-digit distributions of raw lengths and perimeters showed statistically detectable discrepancies from the Benford-expected distribution (both P = 0.001). Projected areas and squared length values did not show statistically detectable evidence against the Benford-expected distribution in the chi-square comparison (P = 0.260 and P = 0.292, respectively). Cubed length values showed the smallest apparent discrepancy among the variables examined (P = 0.910). MAD values were 0.042 for length, 0.037 for perimeter, 0.024 for projected area, 0.026 for square of length, and 0.014 for cube of length. CONCLUSION: In this small exploratory dataset, projected areas and derived squared or cubed length values showed more Benford-like first-digit patterns than raw lengths and perimeters. These findings should be interpreted as descriptive evidence of digit-distribution characteristics in long-bone morphometric data rather than as categorical evidence establishing conformity or nonconformity to Benford's Law.