Do country populations really start with each digit 1 through 9 equally often?
Take every country's population and look only at the first digit. Intuition says 1 through 9 should each show up about 11% of the time.
Computing…
A number starting with 1 is about as likely as digits 5 through 9 combined. Benford's Law predicts about 30.1% for a leading 1, dropping to just 4.6% for 9. It holds for almost any dataset whose values span several orders of magnitude, from a few hundred people to 1.4 billion in this case.
"Random" doesn't have to mean "evenly spread." Auditors use this pattern to flag invented financial figures, because people who make up numbers tend to spread the first digits out more evenly than real data does. Before calling a pattern suspicious, ask what shape the real process should produce.
Where have you assumed “random” means “even”?
Not a quirk of population data
A number starting with 1 is about as likely as digits 5 through 9 combined. Benford's Law predicts the frequency of leading digit d as log₁₀(1 + 1/d) — about 30.1% for 1, dropping to just 4.6% for 9. It holds for river lengths, stock prices, physics constants, electricity bills, and yes, country populations — any dataset where values span several orders of magnitude (a few hundred people up to 1.4 billion, in this case) rather than clustering near one size.
The reason: growing from 100 to 200 means doubling, but growing from 900 to 1,000 takes only an 11% increase. So a number that grows steadily spends far longer with a leading 1 than a leading 9. Auditors use this pattern to flag possibly fabricated financial data, because people who invent numbers tend to spread the first digits out more evenly than real data does.
Explore another case where intuition misjudges probability
It belongs to the same family as our Base Rate Fallacy exhibit: in both, intuition guesses the odds without first asking how the numbers are actually spread out, and the gap only shows up once someone actually counts.
Related exhibits
Sources: population figures (POP_EST) for countries and territories, Natural Earth public-domain dataset, as already used elsewhere on this site. The chart above is computed directly from that data file in your browser when this page loads — nothing is precomputed or hardcoded. Benford's Law: Frank Benford, "The Law of Anomalous Numbers," 1938 (earlier noted by Simon Newcomb, 1881).