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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