Benford's Law — the leading digit

What just happened?

In many natural data sets — city populations, invoice amounts, river lengths, physical constants — the leading digit is NOT uniformly distributed (not ~11% each). Instead the digit 1 appears at the start in as many as ~30% of cases, while the digit 9 only in ~5%. The distribution is logarithmic — this is Benford’s law.

The Benford formula:

where $d\in\{1,2,\dots,9\}$ is the leading digit. Hence $P(1)=\log_{10}2\approx 30.1\%$, $P(2)\approx 17.6\%$, …, $P(9)=\log_{10}\tfrac{10}{9}\approx 4.6\%$. The probabilities decrease in a “staircase” from 1 to 9 and sum to $100\%$.

Where does it come from?

Intuitively: for a number to grow from $100$ to $200$ (leading digit 1) it must double, but to grow from $900$ to $1000$ (again leading digit 1) an increase of only ~11% is enough. Data that grow exponentially (spanning many orders of magnitude) “spend more time” with a small leading digit. That is why even purely mathematical sequences such as the powers of two $2^n$ or the Fibonacci numbers obey the law — you can check this in the control above.

History

The phenomenon was noticed by the astronomer Simon Newcomb in 1881 — he observed that the first pages of printed logarithm tables (those with digit 1) were more worn than the later ones. The law was popularized by the physicist Frank Benford in 1938, analysing more than 20,000 numbers from very diverse sources (from river areas to atomic masses).

Fraud detection (the invariant in practice)

The most important application: artificially faked data — fictitious invoices in accounting, manipulated election results — usually do not obey Benford’s law, because a fraudster invents “uniformly random” digits. A departure of the empirical distribution from the curve $\log_{10}(1+\tfrac{1}{d})$ betrays manipulation. That is why Benford’s law is a real tool of auditors and investigators for detecting financial and electoral fraud (choose “Faked” above to see the flat, suspicious distribution).

Why this matters for your exam:

It is a beautiful combination of logarithms and probability with a real application. It shows that a distribution can be non-uniform against intuition, and it teaches a critical look at data. Remember the formula $P(d)=\log_{10}(1+\tfrac{1}{d})$ and that the base-10 logarithm here does the “weighting” of the orders of magnitude.

Remember: in natural data spread over many orders of magnitude the leading digit follows a logarithmic distribution $P(d)=\log_{10}(1+\tfrac{1}{d})$ — digit 1 dominates (~30%), digit 9 is the rarest (~5%). A distribution that is too “flat” betrays faked numbers.