Pick any country's population. It starts with a 1 31% of the time, not 11%.
Nine possible first digits, so you would expect about one in nine each. Real data disagrees. First draw the shape you expect, then see 217 countries mark it.
Step 1 / Guess first
Of all countries, what share of populations start with each digit?
Set the nine sliders to your guess. They do not need to add up to 100. We scale them.
your guessreal, 2023 populationBenford (computed)
Step 2 / Not a fluke
Try other datasets, then break the pattern
Same pattern in land area and in GDP. The curve is log10(1 + 1/d), called Benford's law. Now squeeze the data into a narrow window and watch it fade.
share in the kept dataBenford
Predict, then test
Keep only populations from 20 million up to just under 200 million (one power of ten wide). Will the digit 1 still take roughly 30%?
Step 3 / Why
Why a 1 lasts so long
Something growing at a steady percentage takes as long to go from 1,000 to 2,000 as from 2,000 to 4,000. It then rushes through the 9s. Pick a growth rate and see how many years each first digit lasts on the way from 1,000 to 10,000 (computed here).
Step 4 / Be the auditor
One list is real GDP data. One is invented.
Both lists have 18 numbers with the same lengths. The real one is drawn at random from 2023 GDP in current US$ millions. In the invented one, every first digit is picked with equal chance (computed in your browser, new each round). Which list is real?