03. Units and Bases — Settling the Argument

The claim that is too strong

A common statement, repeated by textbooks and by chatbots, is that river lengths obey Benford's law in any unit and any base. The objection is immediate: converting kilometres to feet multiplies every number by 3280.84, so the digits must move. The objection is correct, and the popular statement is false as usually phrased. What Roger Pinkham proved in 1961 runs the other way: if a digit law is to be independent of the unit, it must be Benford's, and no other law has that property. That is a conditional about laws, not a promise about datasets, and the difference is the whole confusion. Benford's law is unit-independent; a pile of numbers is not, unless it already happens to be Benford.

What the experiment shows

Running the test on real data settles it, and the book ships the script. The first 300 Fibonacci numbers span 62.35 orders of magnitude and conform closely to Benford under every multiplier tested and in base 8 and base 16, with a mean absolute deviation of 0.0023 to 0.0040. That is scale invariance: the unit changes and the answer does not. The 132 longest rivers span only 0.84 orders of magnitude, because such a list has a floor built into it, and they fail Benford in every unit tested. Their digits also move: 65.2 per cent lead with 1 in kilometres, 37.1 per cent with 3 in feet, 31.1 per cent with 5 in furlongs. Mean adult height across 195 countries spans 0.06 orders of magnitude and leads with 1 for 100 per cent of countries in metres and with 5 for 99.0 per cent in feet.

Why units rotate the circle

Keep only the fractional part of a number's base-ten logarithm, and every number becomes a point on a circle of circumference one. Multiplying everything by a constant adds that constant's logarithm to every point, rotating the circle and changing nothing else. Benford's law is exactly the statement that those points are spread uniformly, and a uniform distribution is the only one a rotation leaves unchanged. So the question is never which unit, but how uniformly the data is spread on the log scale: Fibonacci sits 0.0106 from uniform, the rivers 0.3764, the heights 0.7357, and the digit tables follow those three numbers exactly.

Example: Rotating a plain ring leaves it identical; rotating a ring set with one diamond moves the diamond into view — only the featureless ring survives the turn.

Verdict. Benford's law is unit-invariant; datasets are not, and a dataset is Benford in every unit only if it was already Benford in one.

Overturned if: a dataset far from uniform on the log-mantissa circle keeps the same leading-digit frequencies under a change of unit, breaking Pinkham's uniqueness result.

Sources. 1961 · Roger S. Pinkham, "On the Distribution of First Significant Digits," Annals of Mathematical Statistics 32(4) — JSTOR · 1995 · Theodore P. Hill, "Base-Invariance Implies Benford's Law," Proceedings of the AMS 123(3) — AMS · 2026 · AICE, "The units experiment" — source/experiment.py