Real Numbers Start With 1 Too Often

Nine bars, tallest at the digit 1 and shortest at the digit 9, next to a pile of everyday numbers that lines up against them without anyone forcing it. This lesson checks that shape by hand, then shows the two ways people get it wrong: trusting the test on the wrong kind of numbers, and trusting a pass as proof of anything.

Before you start. You need to be comfortable turning a percentage into a count out of 100, and to know that multiplying a number by 10 just adds a zero. That is the whole of the maths below.

The idea in 30 seconds

Real-world numbers do not start with 1 through 9 equally often: about 30 in every 100 start with 1, fewer than 5 in every 100 start with 9.
Simon Newcomb spotted the skew in 1881 from worn pages in a book of number tables; Frank Benford confirmed it in 1938 by checking 20,229 numbers from unrelated sources.
The skew shows up because a number growing by a steady percentage lingers far longer on a leading 1 than on a leading 9.
The test only works on data spanning a wide range of sizes; squeeze it onto a narrow range - precinct vote counts, human heights - and it flags honest data as fraud.
Someone who knows the rule can build fake numbers that pass it easily, so a pass proves nothing and a fail is only a reason to look closer, never a verdict.

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