p Statistics: Gap Test
by JVSchmidt


General considerations
The Gap Test (GT) measures the distance between two neighboared appearances of the same digit and divides the whole sequence into separate chains of various length.
Here is the example for digit "1":

1 4 >> gap = 2
1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 >> gap = 34
1 9 7 >> gap = 3
1 6 9 3 9 9 3 7 5 >> gap = 9
and so on ...

A gap of length 1 is simply a pair of identically digits. In general the probability to find a chain of length L with nonrepeating starting digit is

w = 1/10 x (9/10)L-1

From there we expect an average chainlength of L = 10. So running the test on N digits we will
get approximativly N/10 chains. We can estimate the maximum chain length from
N ~ 1/w = 10*(10/9)L-1 which gives a estimated value Lmax = 189.


Results
Table shows the maximum gap for each digit and the position where this gap occurs in the p sequence. A remarkable result is the early max gap for "8" at pos. 18.522.937.
Measured number of chains and the average gap distance don't give any information about "abnormal" digit clusters or holes. By the way: this test uncovered an error we got when burning downloaded original data to cd (an unmotivated sequence of 0's inside a file).

Digit Maximum gap Found at Position Number of chains
0 184 412.877.541 420.003.528
1 188 1.334.846.310 420.006.394
2 200 1.016.765.663 419.971.749
3 206 1.105.670.181 419.978.657
4 193 1.322.440.993 420.007.057
5 198 2.383.747.141 419.989.094
6 194 380.007.622 420.033.987
7 203 1.715.735.518 419.996.867
8 196 18.522.937 420.001.483
9 190 362.361.794 420.011.183


Download GAPDETAILS-File in Excel format for more detailed results of the test.

Graphic shows the increase of the longest gap while stepping through the p digits.
The gap length L follows log(N).



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