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Hidden order: Benford's law

If you pull a digit from a hat, every digit from 0 to 9 has the same chance of appearing. But many lists of real-world measurements behave differently.

Think of city populations, river lengths, company values, or numbers in a newspaper. Look only at the first non-zero digit of each number. You might expect 1 through 9 to appear equally often. Surprisingly, 1 appears most often.

This pattern is called Benford’s law:

\begin{aligned} P(d)&=\log_{10}\!\left(1+\frac{1}{d}\right) \\ &=\log_{10}(d+1)-\log_{10}(d) \end{aligned}

Here, $d$ is the first non-zero digit, from 1 to 9. For example, the first non-zero digit of $0.025$ is 2. $P(d)$ means “the probability that this first digit is $d$ .”

What does $\log_{10}$ mean?

The expression $\log_{10}(x)$ asks:

What power must we raise 10 to in order to get $x$ ?

For example,

\log_{10}(1)=0,\qquad \log_{10}(10)=1,\qquad \log_{10}(100)=2

These work because

10^0=1,\qquad 10^1=10,\qquad 10^2=100

The answer does not have to be a whole number. For example,

\log_{10}(2)\approx0.301

because $10^{0.301}$ is approximately 2.

What is the chance of starting with 2?

Imagine a large list that follows Benford’s law. If we choose one number at random, what is the probability that its first digit is 2?

Substitute $d=2$ into the formula:

\begin{aligned} P(2) &=\log_{10}\!\left(1+\frac{1}{2}\right) \\ &=\log_{10}\!\left(\frac{3}{2}\right) \end{aligned}

The logarithm of a fraction can be found by subtracting two logarithms:

\begin{aligned} P(2) &=\log_{10}(3)-\log_{10}(2) \\ &\approx0.47712-0.30103 \\ &\approx0.17609 \end{aligned}

To turn a decimal probability into a percentage, multiply it by 100. Therefore, the chance of the first digit being 2 is about 17.6%.

If the digits 1 through 9 were equally likely, each would appear $1/9$ , or about 11.1%, of the time. Benford’s law says that they are not equally likely in suitable real-world datasets.

The full pattern

First digit	Chance
1	30.1%
2	17.6%
3	12.5%
4	9.7%
5	7.9%
6	6.7%
7	5.8%
8	5.1%
9	4.6%

A number in one of these lists begins with 1 about 30 times out of 100. It begins with 9 only about 5 times out of 100. The digit 1 is therefore about six times as common as 9.

Benford’s law has a deeper mathematical explanation involving logarithms and how numbers spread across different scales. The result may feel counterintuitive, yet one short formula predicts this surprising pattern in many real-world datasets. Isn’t that beautiful?

What should you remember?

Benford’s law is about the first non-zero digit.
The first digits are not always equally likely.
In suitable datasets, smaller first digits appear more often.
A short logarithmic formula predicts how often each digit appears.

Next: How escaping numbers draw the Mandelbrot set →

Back to the Hidden order series