4 ms·

Same kind of thing works with counting the frequency of particular digits (n-digit analysis) in a given position for an integer or decimal. Especially for the h

Same kind of thing works with counting the frequency of particular digits (n-digit analysis) in a given position for an integer or decimal. Especially for the higher order decimal digits there's a far-from-random distribution that you can use to infer whether there's been any tampering, given enough values to study.

082349872349872 4y ago
To understand the mechanism, imagine we accounted in binary instead of decimal. What are the odds the first digit of an entry is a 1?
- thaumasiotes 4y ago
  That won't help you understand the mechanism. The reason the first digit of a binary number is 1 is just the anthropic principle. ("You can only observe things that exist.") That same observation will tell you that, just as the first nonzero digit of a binary number is zero 0% of the time, the first nonzero digit of a decimal number is zero 0% of the time. That is what it means to be a nonzero digit! But it won't tell you anything about why one possible digit might be more or less frequent than another possible digit. The mechanism is just multiplication. Suppose you have a quantity that increases by 20% every period. (Doesn't matter what the period is.) Let's start the quantity at 4 times 10 to the power of irrelevant. We'll consider the leading digit every period: 4 4 5 6 8 9 1 1 1 2 2 2 3 4 5 6 7 8 1 1 1 1 2 2 3 3 ... As you can see, we spend a lot more time with the amount of stuff we have being led by a 1 than we do with it being led by any other digit. Having our growth rate be less than one makes this easy to see by causing similar leading digits to occur next to each other, but the result will be the same for any growth rate. (You can see that it will be the same for any growth rate by considering that you can always match the observations of a system with the same observation period and a different growth rate by instead imagining a system with the same growth rate and a different observation period. But varying the observation period doesn't change the amount of time spent in any given quantity range.)
  - 082349872349872 4y ago
    those leading digits are suspiciously like looking at equal portions of the scale on a slide rule...
    - thaumasiotes 4y ago
      I assume you're aware of this, but that's exactly the same thing.
pdpi 4y ago
> Especially for the higher order decimal digits there's a far-from-random distribution that you can use to infer whether there's been any tampering, given enough values to study. Inversely, the lowest order decimals tend to be evenly distributed, which is also a useful way to detect tampering (trying to make the numbers look "random" does anything but).

082349872349872 4y ago

To understand the mechanism, imagine we accounted in binary instead of decimal. What are the odds the first digit of an entry is a 1?

thaumasiotes 4y ago
That won't help you understand the mechanism. The reason the first digit of a binary number is 1 is just the anthropic principle. ("You can only observe things that exist.") That same observation will tell you that, just as the first nonzero digit of a binary number is zero 0% of the time, the first nonzero digit of a decimal number is zero 0% of the time. That is what it means to be a nonzero digit! But it won't tell you anything about why one possible digit might be more or less frequent than another possible digit. The mechanism is just multiplication. Suppose you have a quantity that increases by 20% every period. (Doesn't matter what the period is.) Let's start the quantity at 4 times 10 to the power of irrelevant. We'll consider the leading digit every period: 4 4 5 6 8 9 1 1 1 2 2 2 3 4 5 6 7 8 1 1 1 1 2 2 3 3 ... As you can see, we spend a lot more time with the amount of stuff we have being led by a 1 than we do with it being led by any other digit. Having our growth rate be less than one makes this easy to see by causing similar leading digits to occur next to each other, but the result will be the same for any growth rate. (You can see that it will be the same for any growth rate by considering that you can always match the observations of a system with the same observation period and a different growth rate by instead imagining a system with the same growth rate and a different observation period. But varying the observation period doesn't change the amount of time spent in any given quantity range.)
- 082349872349872 4y ago
  those leading digits are suspiciously like looking at equal portions of the scale on a slide rule...
  - thaumasiotes 4y ago
    I assume you're aware of this, but that's exactly the same thing.

pdpi 4y ago

> Especially for the higher order decimal digits there's a far-from-random distribution that you can use to infer whether there's been any tampering, given enough values to study. Inversely, the lowest order decimals tend to be evenly distributed, which is also a useful way to detect tampering (trying to make the numbers look "random" does anything but).