Number Theory

[mjd]

If you take a set of random numbers N and using only the final digits find the total number of 1s, 2s,3s, 9s. Then the percentage of 1s is always greater than the percentage of 2s etc. etc., and these precentages are always the same. Does anyone know what this theory/hypothesis is called?

Thanks mjd

[kujawski]

hold on ill just have to test that theory, ill write a little program to check and get back to you.

[kujawski]

well with 10000 random numbers i got
1 976
2 1032
3 975
4 1012
5 1057
6 1003
7 961
8 1002
9 1003

[kujawski]

and with a million i got
1 99973
2 100380
3 99971
4 99722
5 99689
6 99803
7 100100
8 99612
9 100413

so i dont know

[mjd]

Hi, Thanks for your trouble but I may have misled you by being inaccurate. I think that if you take a set of random numbers then each digit (1 - 0) has the same chance of being the final one. As your results tend to show. What I was trying to say, but badly, was that if you use real data then the final digits do show this characteristic
say 20% 1s, 15% 2s, 12% 3s, 10% 4s, 9% 5s etc.
And I think that this fact is actually used by accountants when checking say, expenses.

[Brugel]

An American physicist called Benford did a study of several thousand sets of numbers in economics and other fields. He discovered what is now known as Benford's Law. This states that approximately 30% of all random numbers will begin with a 1; 18% with a 2; 12.5% with a 3; and so on. The proportion of numbers is given by log (1 + 1/n) (where the logarithm is to the base 10). So, for example, the proportion of numbers beginning with a 9 will be log (1 + 1/9) which is approximately 0.046 (i.e. 4.6%).
Benford's Law is apparently used by accountants to detect fraud. When people make up numbers they tend to arrange it so that the numbers 1,2,...9 appear with roughly equal frequencies at the beginning of the numbers.

[Space]

mjd, are you sure you didn't mean "the FIRST digit" ?

[mjd]

Benford's Law sounds exactly what I was looking for. I'm sorry if I've confused anyone with incorrect information. Thanks to all.

mjd

[mibn2cweus]

Concerning fraud, pricing schemes seem to defy Benford's Law as the digit "9" seems to be the odds on favorite?

C�> ~ "Special today only �999.99!" ~ <�?

[OBonio]

You've got me writing a program to prove this aswell now...

Do pseudo-random numbers count?

[mjd]

Well I think probably that it should, see http://en.wikipedia.org/wiki/Benford's_law,

Seems to apply to share prices, population addresses etc.

[mibn2cweus]

Perhaps your program should generate 'random' numbers from an assortment of 'random' arbitrary limits?

[OBonio]

random arbitrary limits? Like between 4 and 11, 1 and 8, 563 and 13987? Could do... I'll add it as an option.

[mibn2cweus]

Sorry, I had only an upper limit in mind but it's only a suggestion . . .

[OBonio]

So far, the results are very much in favour of true randomness producing equal probabilties for each possible FSD. I think I may need to introduce an element of random limits too to possible 'simulate' real world data..