Benford's Law

The probability that the leading digit $d$ (for $d \in \{1, 2, \ldots, 9\}$) of a naturally occurring number equals $d$ is:

$P(\text{leading digit} = d) = \log_{10}\!\left(1 + \frac{1}{d}\right)$

This gives:

More generally, for the first two digits $d_1 d_2$ (where $d_1 d_2$ ranges from 10 to 99):

$P(\text{first two digits} = d_1 d_2) = \log_{10}\!\left(1 + \frac{1}{d_1 d_2}\right)$

On a logarithmic scale, the interval from 1 to 2 is the same width as the interval from 2 to 4, or from 5 to 10. The interval $[1, 2)$ covers $\log_{10}(2) - \log_{10}(1) = 0.301$ of the unit interval on the log scale. Numbers whose leading digit is 1 correspond to this interval (and its translations by integer powers of 10). Since 0.301 is the largest such interval, digit 1 appears most often. Each successive digit gets a smaller slice of the logarithmic scale.

Benford's law is one of the cleanest examples of a distributional regularity that arises from structural properties of the data (scale invariance) rather than from any specific generative mechanism. It connects to information theory (the Benford distribution maximizes entropy subject to scale-invariance constraints), to hypothesis testing (you can test whether data conforms to Benford's law using chi-squared or KS tests), and to practical fraud detection.

Benford's law fails when the data does not span multiple orders of magnitude. Human heights (roughly 150 to 200 cm) will not follow the law because the leading digit is always 1 or 2. Lottery numbers, telephone numbers, and data sampled from a uniform distribution on a narrow range will not follow it either. The law requires the data to be "scale-rich": drawn from a process where the log of the values is spread broadly.

If a random variable $X > 0$ has the property that the leading-digit distribution of $cX$ is the same as that of $X$ for all $c > 0$, then the leading digits of $X$ follow Benford's law.

Equivalently, $\log_{10}(X) \mod 1$ is uniformly distributed on $[0, 1)$.

If the leading-digit distribution is unchanged by multiplying all values by any constant (changing the currency, the units, the scale), then the only possible distribution is the Benford distribution. This is because multiplication by $c$ shifts $\log_{10}(X)$ by $\log_{10}(c)$, and the only distribution on $[0, 1)$ that is invariant under arbitrary shifts (modulo 1) is the uniform distribution.

This explains why Benford's law appears in so many unrelated datasets. Any data-generating process that is independent of the unit of measurement (populations don't "know" whether they're measured in ones, thousands, or millions) will tend to produce Benford-distributed leading digits. The law is a consequence of scale invariance, which is a property of the measurement process, not of the specific phenomenon being measured.

Strict scale invariance is impossible for a distribution with bounded support. Real data only approximately satisfies it. Data that spans 3 or more orders of magnitude typically shows good agreement with Benford's law. Data spanning less than 2 orders of magnitude typically does not.