p Statistics: Variety of Digits
by JVSchmidt


General
When testing longer chains we gonna start with a simple question:
How many DIFFERENT DIGITS are contained in a chain of length L?
Let us call this number the Variety of digits VOD.

Here is an example for p with L=5.
14159 -> VOD = 4
26535 -> VOD = 4
89793 -> VOD = 4
23846 -> VOD = 5
26433 -> VOD = 4
83279 -> VOD = 5
50288 -> VOD = 4
41971 -> VOD = 4
69399 -> VOD = 3
37510 -> VOD = 5

The construction law for the VOD-distribution is easy to find.
Let w(L,d) be the probability that a chain of length L consists of exactly d different digits.
For L=1 we have even one digit:
w(1,d)=1 for d=1
w(1,d)=0 for d>1

For L>1 any w(L,d) can be calculated from its predecessor:
w(L+1,d) = w(L,d) x d/10 + w(L,d-1) x (10-d)/10

Finding a VOD=1 for L=5 is just the same as searching for a single digit run of length 5 (e.g. 11111).
But analyzing single digit runs is limited to L~7 due to the available data.
Thus testing variety of digits up to L=40 is a chance to get a better feeling for the regularity of single digit distribution. More than this the VOD can be an indicator for clusters.

Result's Overview
Digits analyzed: 4.2 * 10 9
Analysis started at digit: 1
Ellapsed computer time for each L: 5 min

Chi2-values for the distributions of chain distances for different length of chains L
Length of
chains
L
Chains analyzed Chi2
for the VOD
Number of statistical
relevant classes
10 420.000.000 6,1977 9
11 381.818.181 6,8173 9
12 350.000.000 8,2189 9
15 280.000.000 14,1435 8
20 210.0000 9,3921 7
40 105.0000 2,7799 5


Remarkable detail:
Chain 02220 22202 20202 (L=15,VOD=2) found at position 2.794.419.571
Chain 09905 55595 05009 59509 950 (L=23,VOD=3) found at position 273.876.641

Detailed results for this test you will find here: Details for Variety of Digits (EXCEL file)


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