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

Say you have values ranging from 0 to _n_. How many of those values start with a 1?

_n_ | % starting with 1
-:|-:
1 | 50%
2 | 33%
… | …
9 | 10%
10 | 18%
… | …
19 | 55%
… | …
99 | 11%
… | …
199 | 55.5%

So, for most values of _n_, numbers from 0 to _n_ are significantly more likely to start with a 1 than they are any other digit.

In general: you have just as many numbers shaped like _xx_ as you do _1xx_, so, as long as your values span multiple orders of magnitude, you’ll hit this effect.