weird
did you know one is often the most common first digit?
some collections of numbers have a surprisingly lopsided favourite.
30.1%
start with 1 in a Benford distribution
a logarithmic pattern for suitable datasets, not a rule for arbitrary numbers.
- 1: 30.1%
- 5: 7.9%
- 9: 4.6%
the idea
in many datasets spanning several powers of ten, roughly 30% of values begin with 1, while fewer than 5% begin with 9. this pattern is called Benford's law. it does not apply to every list of numbers.
how it works
a quantity growing at a steady percentage spends longer travelling from 100 to 200 than from 900 to 1,000. if you sample this kind of growth across many scales, the leading digit can favour 1.
go deeper
why would one get more space?
in many datasets spanning several powers of ten, roughly 30% of values begin with 1, while fewer than 5% begin with 9. this pattern is called Benford's law. it does not apply to every list of numbers.
on a logarithmic scale, the gap from 1 to 2 is much wider than the gap from 9 to 10. doubling takes a bigger proportional jump than growing by one-ninth.
watch a number grow by percentages
a quantity growing at a steady percentage spends longer travelling from 100 to 200 than from 900 to 1,000. if you sample this kind of growth across many scales, the leading digit can favour 1.
the same logarithmic intervals repeat between 10 and 100, 100 and 1,000, and beyond. that repeating scale structure explains the shape of the distribution.
how lopsided is the pattern?
for a Benford-distributed quantity, the probability that the first significant digit is d is the logarithmic width from d to d + 1.
P(first digit = d) = log₁₀(1 + 1/d); P(1) ≈ 30.1%; P(9) ≈ 4.6%
can it really catch suspicious numbers?
- deviations can prompt an investigation when the dataset should fit the model. they are not proof of manipulation.
- telephone numbers, assigned identifiers, and adult heights in centimetres need not follow Benford's law. their constraints break the assumptions.
sources & further reading
concepts: Benford's law · logarithms · scale invariance
2 minute read