number crunching

[Guest guest]

Kurt, SC, Mr Mutt and any other numerophiles (my word) around the fire...

found this snipped from the Newsletter of Science Fiction & Linguistics most

intriguing...

.... If all of you are already familiar with *Benford's Law* -- a law " so

unexpected that at first many people simply refuse to believe it can be

true " -- and haven't mentioned it to me, shame on you. Otherwise, please

do read " The power of one, " by s, pages 27-30 of the 7/10/99

*New Scientist.* The Benford in question is physicist Bedford. On

page 27: " Using more than 20,000 numbers culled from everything from

listings of the drainage areas of rivers to numbers appearing in old

magazine articles, Benford showed that they all followed the same basic

law: around 30 per cent began with the digit 1, 18 per cent with 2, and so

on. " ( " And so on " turns out to end with 4-6 % starting with a 9.) You can

use Benford's law to look for fraud in research data. You can use it to

check accounts -- think of royalty statements! If the data doesn't conform

to the 30% starting with 1, 18% starting with 2, and so on, something's

wrong. On page 29: " In a ground-breaking doctoral thesis published in 1992,

[Mark] Nigrini showed that many key features of accounts, from sales

figures to expenses claims, follow Benford's law -- and that deviations

from the law can be quickly detected using standard statistical tests.

Nigrini calls the fraud-busting technique 'digital analysis', and its

successes are starting to attract interest in the corporate world and

beyond. " I'll bet they are; I can also imagine a strong interest arising in

ways to keep Benford's law from leaking out to The People. A sidebar on

page 30 says: " In a nice little twist, it turns out that the Fibonacci

sequence, the Golden Mean and Benford's law are all linked. The ratio of

successive terms in a Fibonacci sequence tends toward the golden mean,

while the digits of all the numbers making up the Fibonacci sequence tend

to conform to Benford's law. " I love this. I'm even willing to put up with

being told that Benford's law is the mathematical form of the Distribution

Of Distributions, which sounds like only a qualified wizard ought to be

allowed to say it. [Fibonacci sequence....just in case, that's every number

being the sum of the two numbers before it, as in 1-1-2-3-5-8; as in the

seeds on sunflower heads, for example.] There's lots more of this, like the

fact that the most common second digit in a batch of non-random numbers is

zero and the least common is 9, so that there should be roughly 10 times

more that begin with 10 than begin with 99. Thanks to Frances Green for the

article.

[Guest guest]

Interesting post, Phoebe. The following is from Ron Knott's authoritative

pages on the Fibonacci sequence, which, BTW has 2 links to my home pages :^)

---

So the question is, why does this all-digits-equally-likely property not

apply to the first digits of each of the following:

- the Fibonacci numbers,

- the Lucas numbers,

- populations of countries or towns

- sizes of lakes

- prices of shares on the Stock Exchange

Whether we measure the size of a country or a lake in square kilometres or

square miles (or square anything), does not matter - Benford's Law will still

apply.

So when is a number random? We often meant that we cannot predict the next

value. If we toss a coin, we can never predict if it will be Heads or Tails

if we give it a reasonably high flip in the air. Similarly, with throwing a

dice - " 1 " is as likely as " 6 " . Physical methods such as tossing coins or

throwing dice or picking numbered balls from a rotating drum as in Lottery

games are always unpredictable.

The answer is that the Fibonacci and Lucas Numbers are governed by a Power

Law.

We have seen that Fib(i) is round(Phi^i/5) and Lucas(i) is round(Phi^i).

Dividing by sqrt(5) will merely adjust the scale - which does not matter.

Similarly, rounding will not affect the overall distribution of the digits in

a large sample.

Basically, Fibonacci and Lucas numbers are powers of Phi. Many natural

statistics are also governed by a power law - the values are related to B^i

for some base value B. Such data would seem to include the sizes of lakes and

populations of towns as well as non-natural data such as the collection of

prices of stocks and shares at any one time.

---

I wouldn't phib to you...

--Kurt

[Guest guest]

<< Similarly, rounding will not affect the overall distribution of the digits

in

a large sample. >>

Interesting stuff, Kurt. My first response was: Oh, no! Now I'll have to

start imagining a Number-Cruncher with a long white beard (or, heh heh, long

white hair and red nails) sitting at the edge of the Cosmos. Smiling here,

of course.

I've come to the gematria and the study of numbers late, so I am still in the

state of amazed excitement about the elegance of it. Never had much commerce

with numbers until a part of my brain creaked open and I saw them as a

language. And I am still struggling with some of the concepts. I should

have seen this a long time ago... about 30 years ago I signed up for MathI at

a local college just because I felt I didn't understand mathematics (loved

geometry when I was in high school, back in the dark ages)... I remember my

euphoria the day I PROVED 2 + 2 = 4. That could have sent me off on this

search for meaning. It didn't. Think I got a tour or something and got

other-directed.

However... I've poured over some of the numbers games you several on the list

have posted, saved and savored them. Most of the time without understanding

them... but I'm working at it.

Happy New Year to all!

01010101010101

phoebe