Why Does '1' Show
Up So Often?
Look at the first digit of almost any large collection of real numbers: populations, river lengths, stock prices, tax filings. It starts with 1 about 30% of the time, and starts with 9 less than 5% of the time. Nobody designed it that way.
Act one
Pick a sequence, count the first digits
These are computed live, right now, from real math, not cherry-picked. Choose a sequence and watch how its leading digits stack up against Benford's prediction, the dashed line on each bar.
Powers of 2 from 2^1 to 2^500. Even though nothing here was designed to follow any law, it lines up almost perfectly.
Act two
Does it matter what units you use?
Take the same 1,000 Fibonacci numbers and multiply every single one by an arbitrary constant, as if you'd switched to some unit nobody has ever heard of. Watch what happens to the pattern.
Multiplying every number by 1 changes nothing, obviously. Try the others.
Fabricated numbers don't know this law
In 1881, astronomer Simon Newcomb noticed something odd about the shared books of logarithm tables in his university library: the first pages, covering numbers that start with 1, were far more worn and dog-eared than the last pages, covering numbers that start with 9. People were looking up numbers starting with 1 far more often. Nobody made much of it until physicist Frank Benford rediscovered the same pattern in 1938 and tested it across more than twenty unrelated real-world datasets, from river lengths to baseball statistics, finding the same distribution every time.
The mechanism is the scale invariance you just saw in the second act. Numbers that arise from growth, multiplication, or measurements spanning many orders of magnitude naturally land on a logarithmic scale, and on a logarithmic scale the interval covered by numbers starting with 1 is simply wider than the interval covered by numbers starting with 9. That's the whole secret. It has nothing to do with the numbers themselves and everything to do with how spread out they are.
Genuine financial records, expense reports, tax filings, and election tallies tend to follow Benford's law closely, because they're full of naturally occurring quantities spanning many magnitudes. Fabricated numbers usually don't, because people inventing figures tend to spread their guesses too evenly across all nine digits. Forensic accountants and auditors run exactly this kind of digit analysis on real filings, and a leading-digit distribution that looks suspiciously flat instead of logarithmic is often enough to trigger a closer look.