I have probably already lost the most maths-allergic among you. But really, you don’t need to be Cédric Villani, France’s Fields Medal-winning mathematician, to see why, in the end, it all makes sense.

Sure, our tidy, rational minds expect an even distribution: each digit should show up about 11% of the time (counting the 0). Yet reality is something else entirely.

The explanation is the logarithmic scale of growth. The words are scary, but it isn’t that complicated.

To understand the distribution, picture a company growing by 10% a year.

• When it starts out with €100K, its revenue will begin with the digit 1;
• With 10% annual growth, it will take 8 years⁴ for its revenue to reach €200K and begin with the digit 2;
• But once past that threshold, it will take only 4 years to reach €300K (and so a figure beginning with 3);
• And only 3 years to reach €400K. And so on.

In other words, if we follow this company’s growth over 30 years, its revenue will start with 1 for 8 years, with 2 for 4 years, with 3 for 3 years, and so on. When it hits €900K, there will be only one year left before it passes €1M and the cycle starts again with 1 as the first digit.

That is the principle of the logarithmic scale:

• To go from 1 to 2, you have to double (a 100% increase);
• To go from 2 to 3, you need a 50% increase;
• To go from 3 to 4, you need a 33% increase;
• To go from 8 to 9, you need an increase of just 12.5% ;

This property shows up in any sequence of numbers that grows proportionally. Take a simple sequence where each number is double the previous one: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...

Of these 11 numbers, 4 start with 1 (that is 36%), 2 start with 2 (that is 18%), 1 starts with 3, 1 with 4, 1 with 5, 1 with 6, and 1 with 8. Roughly Benford’s distribution.

The law holds in a rather astonishing number of fields:

• Financial data: balance sheets, share prices, transaction amounts;
• Demographic data: city populations, birth rates;
• Physical data: river flows, lake areas, mountain heights;
• Scientific constants: atomic masses, physical constants;
• Economic data: national GDP, inflation, stock market indices.

That said, it does require a few rules, and comes with limits for some exceptions:

• Arbitrarily assigned numbers: phone numbers, postcodes, social security numbers;
• Data constrained by human limits: adult heights and weights (which vary within narrow ranges);
• Data that is too homogeneous or too narrow in range.

And it is precisely humans’ inability to generate true randomness, or their tendency to stick too closely to an even spread, that makes it possible to detect fraud.

Take a dishonest person making up figures: they tend to spread the first digits more evenly than nature would. Instead of 30% of figures starting with 1 and only 4.6% with 9, faked data often shows a more balanced distribution.

Likewise, fraudsters often avoid round numbers and favour figures like 53, 87 or 29, which look more random to the human mind. By that very attempt to make their numbers more believable, they create detectable statistical anomalies, which a simple Benford test picks up without even needing to understand what the figures are about.

And the authorities eventually caught on.