Some clues to understanding Benford's law - Eli Bendersky's website

Why Small Numbers Dominate Leading Digits

Benford's law is a well-known curiosity of many real-world datasets: the first digit is most likely to be 1, with each subsequent digit d appearing less often than d+1. The classic probability table from Wikipedia captures the expected distribution:

The catch is that Benford's law applies to data that spans multiple orders of magnitude. Heights of adults, for instance, cluster in a narrow band and show no such pattern. Random numbers also fail to conform; the datasets that exhibit the law are distinctly "non-random" in a structural sense.

Two immediate observations frame the discussion:

• Clue 1: The law holds only for real-world data.
• Clue 2: It's not merely about 1 being frequent; each digit is less common than the one before it.

A Concrete Example: City Populations

Populations of California's roughly 480 largest cities illustrate the effect well. The data ranges from Los Angeles at 3.9 million down to towns of a few hundred people. Plotting the distribution of the leading digit gives a near-perfect match to Benford's prediction:

What jumps out is that small cities vastly outnumber large ones. Of those 480 cities, only 74 exceed 100,000 residents. The same holds for stock prices of S&P 500 companies: prices span from roughly $1806 to $2, yet 90% trade below $182 and 65% below $100.

• Clue 3: In real-world data spread across orders of magnitude, smaller values are more common than larger ones.

This is the hallmark of the Pareto distribution — familiar through the "80-20 rule" — and its discrete cousin, Zipf's law.

Logarithmic Scales and Digit Spacing

Plotting such data on a logarithmic axis gives smaller numbers more visual "real estate." The scale between 1 and 2 is as wide as the scale between 10 and 20, yet wider than the gap between 2 and 3. Every time a data point lands on this axis, its leading digit is more likely to be 1 than any other.

Another way to see it: a number with a leading 1 needs a 100% increase to shift its leading digit to 2, while going from 2 to 3 requires only a 50% increase, and from 3 to 4 only 33%.

• Clue 4: On a logarithmic scale, the distance between numbers starting with 1 and those starting with 2 exceeds the distance between any other consecutive leading-digit pairs.

A practical visualization helps. Consider the ratio of numbers starting with 1 among all numbers up to a bound N. As N crosses each new order of magnitude, the ratio climbs to roughly 0.5 — because there are as many D-digit numbers starting with 1 as there are all numbers with D-1 digits. It then declines to about 0.1 just before the next decade, since each leading digit's share within a fixed digit count is one tenth. Averaged smoothly, this ratio settles near 0.3, consistent with Benford's law.

Putting the Pieces Together

The essential intuition boils down to the observation that smaller numbers occur more often than larger ones in relevant datasets — clue 3 — combined with an understanding of logarithmic spacing. Together, these facts explain why leading digits skew low.

This view also clarifies Benford's famous properties of scale invariance and base invariance. A law rooted in proportional growth and digit boundaries, rather than absolute magnitudes, naturally holds across measurement units and numeric bases.