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I'm trying to wrap my mind around Benford's Law.

According to wikipedia Benford's law applies in many "naturally occurring collections of numbers." It "applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants,[3] and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude."

I think I understand it and have plenty of examples of how it works. And it makes sense. However, I tried out this website to see more examples to show my class.

http://www.testingbenfordslaw.com/most-common-iphone-passcodes

One of the examples, is iPhone passcodes which are 4 digits and I believe can start with any digits 0,1,2,3....9. According to the website, they are suggesting Benford's Law applies here.

To me, this doesn't seem like Benford's law applies here. It seems to me that generally speaking a person that chooses a random 4 digit number has an equally likely chance at choosing numbers that start with 0,1,2,3,4...9. All passcodes are 4 digits so there doesn't seem to be a span of "multiple order of magnitude". In addition, this doesn't seem to be "naturally occurring" to me, in the same sense of listing all the populations of cities in the world. In one sense it is naturally occurring because you are not instructing people to do anything but choose a number, but I can't figure out why I think its not naturally occurring in the same sense? Am I wrong?

What are your thoughts?

Ethan Bolker
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B flat
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    I would be surprised if Benford's law applied here. Generally speaking it will only apply to numbers that are allowed to range over many orders of magnitude, as in Wikipedia, and iPhone passcodes don't; you'd expect those to be more or less evenly distributed. – Qiaochu Yuan Sep 18 '17 at 18:06
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    "Naturally occurring" is not a sufficient condition for something to follow Benford's law. Roughly speaking you want numbers which "grow multiplicatively" as opposed to additively. – Qiaochu Yuan Sep 18 '17 at 18:07
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    I'm sure it has occurred to you that $4$-digit passcodes are not "distributed across multiple orders of magnitude." In particular it must be obvious that thelinked sample distribution allows that the first digit in a passcode might be zero, but this is not a leading digit in measured values. – hardmath Sep 18 '17 at 18:08
  • I see. Even if self selected passcodes were allowed to span multiple orders of magnitude, I don't think benfords law applies right? – B flat Sep 18 '17 at 18:10
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    Your reasoning sounds correct to me. I think this particular case is more likely due to other psychological factors and not Benford's law. For instance, many people use a calendar date for their PIN, and those are more likely to start with a 1 or 2 (because months go up to 12 and days go up to 31). There are other documented psychological factors such as people favoring ending digits like 3 and 7, presumably because they feel "more random". – Erick Wong Sep 18 '17 at 18:11
  • Qiaochu Yuan - Yes. "grow multiplicatively" I think is what my intuition was telling me. But I'm having a hard time knowing which examples do this and which don't. I wonder if there is another way to classify it. – B flat Sep 18 '17 at 18:11
  • Erick Wong - Yes. Psychological reasons makes sense! I guess technically one could say the example of the iPhones "satisfies" Bedford's law even though it doesn't apply. Seems silly and misleading to me though. – B flat Sep 18 '17 at 18:13
  • This just occurred to me... Is it true to say that Bedford's law applies in "naturally occurring phenomena that span multiple orders of magnitude and grow multiplicatively" because our number system is a place value number system based on powers of base number? – B flat Sep 18 '17 at 18:20
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    @MichaelMcCain There are certainly some questionable choices on that website: I suspect the data sets are not chosen purely for their applicability to Benford's Law. For instance, the Fibonacci number test is superfluous because it can provably be shown to obey Benford's law with a fairly precise rate of convergence: but it does provide a tangible way to observe the ideal case where the agreement is essentially exact. – Erick Wong Sep 18 '17 at 18:24
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    No matter how illogical man can seem, he's never perfectly random. Consider that many people choose the year of a relative's birth as a $4$-digit ATM PIN. With that assumption, numbers between $1920$ and $2016$ would be good guesses. – Mr. Brooks Sep 18 '17 at 21:35
  • I agree with @Mr.Brooks. In the example about pins, even if leading 0s are discarded you are not observing random data. I think that mmdd combinations (and ddmm ones for us Europeans) are really common and skew the distribution towards 1 and in part 2; indeed, the first digits from 6 to 9 are roughly equal in probability. – mau Sep 20 '17 at 05:09

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