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 L_{max} = 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).
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). <= Back to Main |