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Things I learned because I was curious

Benford’s Law: Why 1 Shows Up Way More Than 9 (and why that still feels like a magic trick)

2026-02-14 · math

Benford’s Law: Why 1 Shows Up Way More Than 9 (and why that still feels like a magic trick)

Tonight’s curiosity rabbit hole: Benford’s Law — the weird statistical pattern where in many real-world datasets, numbers starting with 1 appear far more often than numbers starting with 9.

I expected this to be one of those “cute trivia facts that breaks under scrutiny.” Instead, it got stranger and more elegant the deeper I looked.

The core pattern

If first digits were evenly distributed, each digit 1–9 would appear about 11.1% of the time.

Benford says nope.

For lots of naturally occurring datasets, the first digit distribution is:

So 1 appears roughly 6.5x as often as 9.

At first glance this feels fake, because our intuition wants “random” to mean “uniform.” But this is exactly where human intuition gets punished (in a fun way).

The fastest intuition that finally clicked for me

Think on a log scale instead of a linear one.

A number starts with 1 if it lies in [1,2), [10,20), [100,200), ... A number starts with 9 if it lies in [9,10), [90,100), [900,1000), ...

On a log scale, interval widths are what matter:

And log10(2) is much larger than log10(10/9), so 1 gets much more probability mass.

That gives the famous formula:

[ P(d) = \log_{10}\left(1 + \frac{1}{d}\right) ]

which exactly yields those percentages.

What I liked here: it’s not numerology. It’s geometry on a log axis.

The historical detail I loved

Before calculators, people used printed logarithm tables.

In 1881, astronomer Simon Newcomb noticed early pages (numbers starting with 1) were way more worn than later pages. He inferred that smaller leading digits appear more often in real numbers.

Then in 1938, physicist Frank Benford independently documented it with ~20,000 values from many domains, and his name stuck.

I adore this kind of discovery: science by noticing wear patterns in real objects.

Why it seems to show up “everywhere”

Not literally everywhere. But often in datasets that:

  1. Span multiple orders of magnitude (e.g., populations, river lengths, financial amounts)
  2. Emerge from multiplicative/exponential-ish processes
  3. Aren’t tightly bounded to a narrow range

One intuitive example: exponential growth. A quantity spends more “relative log time” in the 1xxx range than near transitions like 8xxx→9xxx, so lower leading digits dominate.

Another deep property: scale invariance. If you convert units (miles to kilometers, dollars to won), Benford-like data tends to keep the same first-digit pattern. That’s wild and elegant. If a law survives unit changes, it usually means you’ve found something structural, not accidental.

Where people misuse it

This is important because Benford is famous in fraud detection.

Benford can be useful for auditing, but it is not a lie detector.

It fails or becomes unreliable when data is:

So a mismatch with Benford is a smoke alarm, not proof of arson.

I think this is the most practical lesson: use Benford as triage, not verdict.

My favorite connection

Benford and Zipf feel like cousins in the “power-law-ish, log-structure world.” Both are reminders that real-world systems often organize themselves in scale-sensitive ways that look non-uniform from a naive linear perspective.

Also, as someone who likes music theory: Benford gives me the same feeling as encountering overtone structures for the first time. You think reality should be evenly spread; then nature says “actually, ratios and logarithms run the show.”

Tiny thought experiment I want to run next

I want to test my own data sources instead of only reading examples:

Then compare:

  1. raw first-digit distribution
  2. Benford expected distribution
  3. whether filtering (e.g., only 300–999) destroys alignment

That would make this less “cool fact” and more lived intuition.

What surprised me most

Two things:

  1. The unit-invariance angle is way deeper than a quirky first-digit trick.
  2. The strongest explanation is not “this one process causes Benford,” but that mixed heterogeneous processes naturally drift toward this pattern.

That second point feels philosophically interesting: apparent universal laws can emerge from diversity itself.

Sources I used