Why growth changes the first digit
The pattern is known as Benford’s law: in many datasets, the first nonzero digit is 1 more often than 2, 2 more often than 3, and so on. It appears when numbers cover a wide range of sizes and their distribution is shaped more by proportional change than by a fixed amount of change.
Imagine a quantity that grows by multiplying—say, a balance that repeatedly increases by a percentage. It can begin with 1, grow through values such as 1.4 and 1.8, then switch to 2. The same process later carries it from 9 to 10, but that final step is brief relative to the full journey across the digits. The key is that equal percentage increases, not equal numerical increments, govern the movement.
Why the pattern can survive a change of units
Multiplying every value by the same conversion factor—turning miles into kilometers, for example—often leaves the leading-digit pattern broadly similar. That is a clue that the effect is tied to relative scale rather than to a particular unit or a special preference for the number 1.
Benford’s law is not a rule for every collection of numbers. It is more likely to fit data that span several orders of magnitude, such as values ranging from tens to millions, and less likely to fit numbers constrained to a narrow band, assigned as ID codes, or shaped by a fixed minimum or maximum.
That distinction matters when people use the pattern to examine records. A mismatch can suggest that a dataset deserves a closer look, but it does not by itself prove anything was falsified: the way the numbers were generated may simply make Benford’s law a poor fit.
