Why does Benford’s Law work? This law says numbers in a data set are more likely to start with low digits (1, 2) than high digits (8, 9), but there are exactly as many numbers in existence beginning with each digit.

Question

Why does Benford’s Law work? This law says numbers in a data set are more likely to start with low digits (1, 2) than high digits (8, 9), but there are exactly as many numbers in existence beginning with each digit.

In: Mathematics

Answer 1

That’s a property of sets that grow multiplicatively. Benford’s law doesn’t apply to all kinds of datasets.

Imagine you start any number. And then you repeatedly increase your number by 10%.

So if I start from a number that starts with a small numeral like 100 it will stay with a small numberal for a while as it grows 110, 121, 133.1, …

But if you start with a large numeral like 900 the multiplicative nature will quickly flip to the next digit. Here it grows 990, 1089, …

So this effect happens for stuff like city population numbers where the growth rate is scaled by the population size, but it doesn’t work for other data sets like population age distribution