I'm aware of Benford's law that covers the distribution of leading digits:

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data.

However, I'm wondering if there's a name for the distribution of digits throughout numbers. For example, the distribution of digits for the number 89112374633510 is:

1: 21%
3: 14%
0, 2, 4, 5, 6, 7, 8 and 9: 7% each

What is the distribution of digits, throughout a number called, and where can I learn more about it?

  • $\begingroup$ We don’t name everything, we name them as they become useful. I don’t see much use in this. It’s certainly a distribution, but is is unclear what use it might have. $\endgroup$ Sep 27, 2021 at 20:57
  • $\begingroup$ @ThomasAndrews I'm okay with that being the answer too. Just wanted to know if such a name existed. $\endgroup$ Sep 27, 2021 at 21:02
  • $\begingroup$ @ThomasAndrews Interesting. I am completely ignorant of the subject matter. Given that, when I skimmed the cited article, I got the impression that election fraud analysis, based on the leading (presumably leftmost) digit is inappropriate for small precincts. This leaves open the possibility that election fraud in small precincts might be detectable by examination of other than the leftmost digit. $\endgroup$ Sep 27, 2021 at 22:13
  • $\begingroup$ Yes, but the expected distribution for the rightmost digits are just uniform - $1/10$ for each digit. And the overall distribution of digits $n\mapsto (p_0,p_1,\dots,p_9)$ aren’t going to be very interesting, because there is nothing about the order of them, and the interesting anomalies occur in the earlier digits. @user2661923 $\endgroup$ Sep 27, 2021 at 22:18
  • $\begingroup$ You could certainly find weirdness in data using the average of these vectors over the dataset. Too many $3$s, not enough $4$s. But it won’t be because it varies too far from an unusual distribution, but rather because it varies too far from a uniform distribution. @user2661923 $\endgroup$ Sep 27, 2021 at 22:21


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