# practical statistics

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Benford's Law shows that a plurality of numbers start with 1:

In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time

What sets obey the law?

Benford's law tends to apply most accurately to data that span [sic] several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's law applies.

Well, the set of data with the *most* orders of magnitude is "subsets of numerals in existing data"

Let's take, for the sake of simplicty, the number twelve thousand, three hundred and forty-five: 12345

What subsets of numbers exist within this number? There's 14 of them: 1, 2, 3, 4, 5, 12, 23, 34, 45, 123, 234, 345, 1234, 2345. As predicted, a plurality of numbers in this set start with 1 (1, 12, 123, 1234), only two of them start with 4 (4, 45) and only one of them starts with 5 (5).

"But wait," you say, "you just arbitrarily picked a number that has numerals in ascending order of rank. What if you picked 54321?" And yes, Benford's law wouldn't apply to that particular number. But 54321 starts with a 5, which according to Benford's law only happens about 8% of the time. But 12345 starts with a 1, which happens 30% of the time. What this means is, across *all* subsets of data, there's going to be almost 4x as many numbers like 12345 as there are 54321.

Additionally, since there are logarithmically more numbers with an *n*th digit than numbers with an (*n*+1)th digit, that means that, holistically speaking, most numbers only have one digit.

Therefore, we can extrapolate to conclude that, across the whole corpus of human-generated data—which practically speaking is *all numbers*—not only do a plurality of numbers **start** with 1, 30% of all numbers **are** 1.

### conclusion

There's just no way "one is the loneliest number" was meant to be taken literally. The data doesn't back it up. Instead, I believe the song is about the kinship fallacy: just because most of the people around you superficially resemble you, that doesn't inherently mean you're going to be any less lonely. In fact, this superficial resemblance is an ersatz substitution for real friendship, and relying on it will make you *less* likely to form bonds than if you were surrounded by people who aren't like you.

Sure, there may be six times as many 1s in the universe as 9s, but when a 9 and a 1 come together you get the rarest and most special of all numerals, the 0. No number starts with a 0, but without it, all the other numbers are meaningless, and I think that's beautiful 🦝