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## Algebra (all content)

### Unit 11: Lesson 31

Logarithmic scale (Algebra 2 level)

# Benford's law (with Vi Hart, 2 of 2)

Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Sal. Created by Sal Khan.

## Want to join the conversation?

• I heard about an indigenous culture somewhere (S. America?) in which the counting was logarithmic. Specifically, when asked what number was half-way between "1" and "9", the answer given was "3." Sorry I do not have a source, but it may have been Radiolab or something like that. Very interesting. • So powers of two follow Benford's distribution when expressed in decimal, but if I were to express them in binary (1, 10, 100, 1000, ...) everything falls in the '1' bucket. Question: do the "natural" data (like populations, financial data, physical constants) follow Benford's law under every number base? • Well, in base 2, powers of 2 become the special case, while powers of 10 would now follow Benford's Law, I think (but expressed in a different way). According to Wikipedia there's a more general version of the rule that applies to any/all bases.

It also has examples where the same data still follows the law even if you use different units.
• Usually, Sal is an excellent teacher for me, but I still have a few questions. I understand that on a logarithmic scale this makes sense, but it doesn't work on a linear scale. There are still the same number of numbers between 1 and 2 as there are between 2 and 3. So, why is there a greater probability for a number to land between 1 and 2 than there a chance to land between 2 and 3? Also, would Benford's distribution also apply to non-exponential sets? • You're on the right train of thought.
When you ask '[W]hy is there a greater probability for a number to land between 1 and 2 than there a chance to land between 2 and 3?', you need to consider that (assuming we are accepting any arbitrary set of numbers obtained from some real-world observations) the sub-range you refer to as 'between 1 and 2' refers just as much to those numbers between 10 and 20 (also 100 and 200 and so on, hence we are working with logarithmic scales), such that the sub-ranges of numbers (those numbers starting with the same digit) being examined at any given scale may always be considered as having an equal probability of 'being picked' as any of the other sub-ranges, but only when considering that individual scale. In this way you are correct, but your confusion stems from realising this truth, whilst not yet seeing the whole picture.
The next step involves thinking about how these probabilities vary with the scale being considered. As we might be talking about any set of numbers obtained from some real-world observations, the scale (think also about the range or upper limit) that such numbers span is, hopefully intuitively, not at all biased to a neat, round number like 1000. By this I mean that in the natural world there is no reason for, for example, the population of a town to tend more towards that tidy number of 1000 or 10,000; for it to do so would be as strange as saying that populations prefer to stop growing at nice round numbers, for the sake of making equal those probabilities of picking each of our sub-ranges (10 to 20, 200 to 300, etc.). Hopefully this is not too convoluted to follow. It is then this complete lack of bias of 'real-world data scales' toward nice round numbers that is responsible for changing up the probabilities of picking those sets of sub-ranges (I believe this is a direct answer to your question quoted above, which now, hopefully, in sufficient context should make more sense to you. Also, this is my first KA comment, so sorry if it's not so concise.)
• If we look at the behavior of the universe we will see a lot of exponential and logarithmic behavior in it. Physical constants are what determine the 'universe', so then there exist a relation Physical constants - exponential that would make it follow the Benford's Law? • Basically the reason seems obvious, to get to 2 u must pass through 1, to get to 3 you pass through 1 and 2...etc... sotherefore the chances of a random stop being on 1 are greater than 2 because everyone passes through or stops on one, the remainder (slightly less) all pass through or stop on 2. Cute relationship with Fibonnacci but hardly surprising methinks... • Uh. That reasoning is a bit sketchy. The very first `1` at the beginning of a logarithmic scale isn't the one you must magically pass through to get to everything else. In fact, if you go left on the scale the first thing you hit is `9`, which you must "pass though" to get down to `8`, etc etc all the way down to the first time you hit a one at `1/10th`.
• all natural increments follow logarithmic progression. so, a random series based on such a progression is likely to follow the benford's law. Increase in population is natural increment just like Fibonacci sequence or stock market data. Although, i am still confused why physical constants follow this law when the choice of scale lies entirely in our hand. would they follow this law for any choice of scales or is the present scale chosen something special? • Does the logarithmic scale have to be based on log_10? I suspect that plotting powers of eight on an octal logarithmic scale would break the Bensford's law.

Is there some non decimal numeric system that doesn't follow Bensford's law for any statistics? • What is Fibonacci sequence?? I have no clue about them...
Could someone help me with an easy explanation.....

Captain Leo
#PrayForBarcelona
(1 vote) • The Fibonacci sequence is the one that starts 1,1 and then proceeds with the recursive definition Fₙ = Fₙ₋₁ + Fₙ₋₂
So we have:
F₁ = 1
F₂ = 1
F₃ = F₂ + F₁ = 1 + 1 = 2
F₄ = F₃ + F₂ = 2 + 1 = 3
F₅ = F₄ + F₃ = 3 + 2 = 5
etc
Giving 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

The sequence has all sorts of interesting mathematical properties. Not all of them easy to understand. Use your favourite search engine, and you'll be able to find numerous articles on the Fibonacci numbers.
•  