## Test 3 Results

The number behind Norm Dist shows the expected "number of strings" of a certain length.
The number behind n = shows the observed "number of strings" of a certain length
```Example 1 Test -2 Seed 4 accuracy 2^24

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  1 power-0,683831174401323
i-9  000000000  Norm Dist 1,24501992031872  n =  2 power 0,316168825598677
i-8  00000000  Norm Dist 2,49003984063745  n =  6 power 1,31616882559868
i-7  0000000  Norm Dist 4,9800796812749  n =  6 power 2,31616882559868
i-6  000000  Norm Dist 9,9601593625498  n =  12 power 3,31616882559868
i-5  00000  Norm Dist 19,9203187250996  n =  12 power 4,31616882559868
i-4  0000  Norm Dist 39,8406374501992  n =  46 power 5,31616882559868
i-3  000  Norm Dist 79,6812749003984  n =  81 power 6,31616882559868
i-2  00  Norm Dist 159,362549800797  n =  136 power 7,31616882559868
i-1  0  Norm Dist 318,725099601593  n =  320 power 8,31616882559868
i 1  1  Norm Dist 318,725099601593  n =  333 power 8,31616882559868
i 2  11  Norm Dist 159,362549800797  n =  139 power 7,31616882559868
i 3  111  Norm Dist 79,6812749003984  n =  70 power 6,31616882559868
i 4  1111  Norm Dist 39,8406374501992  n =  33 power 5,31616882559868
i 5  11111  Norm Dist 19,9203187250996  n =  25 power 4,31616882559868
i 6  111111  Norm Dist 9,9601593625498  n =  11 power 3,31616882559868
i 7  1111111  Norm Dist 4,9800796812749  n =  5 power 2,31616882559868
i 8  11111111  Norm Dist 2,49003984063745  n =  2 power 1,31616882559868
i 9  111111111  Norm Dist 1,24501992031872  n =  1 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  1 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i 15 111111111111111  Norm Dist 1,94534362549801E-02  n =  1 power-5,68383117440132
Kappa2 = 71,7752005976096 igamc = p = 2,15268758374521E-09
Non Random

Example 2 Test -3 Seed 4 accuracy 2^14

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  1 power-0,683831174401323
i-9  000000000  Norm Dist 1,24501992031872  n =  1 power 0,316168825598677
i-8  00000000  Norm Dist 2,49003984063745  n =  2 power 1,31616882559868
i-7  0000000  Norm Dist 4,9800796812749  n =  3 power 2,31616882559868
i-6  000000  Norm Dist 9,9601593625498  n =  6 power 3,31616882559868
i-5  00000  Norm Dist 19,9203187250996  n =  20 power 4,31616882559868
i-4  0000  Norm Dist 39,8406374501992  n =  40 power 5,31616882559868
i-3  000  Norm Dist 79,6812749003984  n =  73 power 6,31616882559868
i-2  00  Norm Dist 159,362549800797  n =  157 power 7,31616882559868
i-1  0  Norm Dist 318,725099601593  n =  333 power 8,31616882559868
i 1  1  Norm Dist 318,725099601593  n =  323 power 8,31616882559868
i 2  11  Norm Dist 159,362549800797  n =  153 power 7,31616882559868
i 3  111  Norm Dist 79,6812749003984  n =  69 power 6,31616882559868
i 4  1111  Norm Dist 39,8406374501992  n =  44 power 5,31616882559868
i 5  11111  Norm Dist 19,9203187250996  n =  24 power 4,31616882559868
i 6  111111  Norm Dist 9,9601593625498  n =  15 power 3,31616882559868
i 7  1111111  Norm Dist 4,9800796812749  n =  4 power 2,31616882559868
i 8  11111111  Norm Dist 2,49003984063745  n =  4 power 1,31616882559868
i 9  111111111  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
Kappa2 = 13,6967630976096 igamc = p = 0,548640511658558
Random

Example 3 Test -4 Seed 4 accuracy 2^12

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-14 00000000000000  Norm Dist 3,89068725099601E-02  n =  1 power-4,68383117440132
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i-9  000000000  Norm Dist 1,24501992031872  n =  3 power 0,316168825598677
i-8  00000000  Norm Dist 2,49003984063745  n =  1 power 1,31616882559868
i-7  0000000  Norm Dist 4,9800796812749  n =  4 power 2,31616882559868
i-6  000000  Norm Dist 9,9601593625498  n =  11 power 3,31616882559868
i-5  00000  Norm Dist 19,9203187250996  n =  14 power 4,31616882559868
i-4  0000  Norm Dist 39,8406374501992  n =  47 power 5,31616882559868
i-3  000  Norm Dist 79,6812749003984  n =  97 power 6,31616882559868
i-2  00  Norm Dist 159,362549800797  n =  149 power 7,31616882559868
i-1  0  Norm Dist 318,725099601593  n =  299 power 8,31616882559868
i 1  1  Norm Dist 318,725099601593  n =  330 power 8,31616882559868
i 2  11  Norm Dist 159,362549800797  n =  141 power 7,31616882559868
i 3  111  Norm Dist 79,6812749003984  n =  81 power 6,31616882559868
i 4  1111  Norm Dist 39,8406374501992  n =  35 power 5,31616882559868
i 5  11111  Norm Dist 19,9203187250996  n =  22 power 4,31616882559868
i 6  111111  Norm Dist 9,9601593625498  n =  8 power 3,31616882559868
i 7  1111111  Norm Dist 4,9800796812749  n =  6 power 2,31616882559868
i 8  11111111  Norm Dist 2,49003984063745  n =  2 power 1,31616882559868
i 9  111111111  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
Kappa2 = 43,7836755976096 igamc = p = 1,1887355927831E-04
Non Random

Example 4 Test -4 Seed 4 accuracy 2^10

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i-9  000000000  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i-8  00000000  Norm Dist 2,49003984063745  n =  4 power 1,31616882559868
i-7  0000000  Norm Dist 4,9800796812749  n =  9 power 2,31616882559868
i-6  000000  Norm Dist 9,9601593625498  n =  11 power 3,31616882559868
i-5  00000  Norm Dist 19,9203187250996  n =  19 power 4,31616882559868
i-4  0000  Norm Dist 39,8406374501992  n =  41 power 5,31616882559868
i-3  000  Norm Dist 79,6812749003984  n =  86 power 6,31616882559868
i-2  00  Norm Dist 159,362549800797  n =  155 power 7,31616882559868
i-1  0  Norm Dist 318,725099601593  n =  252 power 8,31616882559868
i 1  1  Norm Dist 318,725099601593  n =  240 power 8,31616882559868
i 2  11  Norm Dist 159,362549800797  n =  160 power 7,31616882559868
i 3  111  Norm Dist 79,6812749003984  n =  91 power 6,31616882559868
i 4  1111  Norm Dist 39,8406374501992  n =  43 power 5,31616882559868
i 5  11111  Norm Dist 19,9203187250996  n =  19 power 4,31616882559868
i 6  111111  Norm Dist 9,9601593625498  n =  7 power 3,31616882559868
i 7  1111111  Norm Dist 4,9800796812749  n =  10 power 2,31616882559868
i 8  11111111  Norm Dist 2,49003984063745  n =  6 power 1,31616882559868
i 9  111111111  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
Kappa2 = 56,0924630976094 igamc = p = 1,16895857871224E-06
Non Random

Example 5 Test -4 Seed 4 accuracy 2^8

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i-9 000000000  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i-8 00000000  Norm Dist 2,49003984063745  n =  0 power 1,31616882559868
i-7 0000000  Norm Dist 4,9800796812749  n =  0 power 2,31616882559868
i-6 000000  Norm Dist 9,9601593625498  n =  10 power 3,31616882559868
i-5 00000  Norm Dist 19,9203187250996  n =  20 power 4,31616882559868
i-4 0000  Norm Dist 39,8406374501992  n =  50 power 5,31616882559868
i-3 000  Norm Dist 79,6812749003984  n =  76 power 6,31616882559868
i-2 00  Norm Dist 159,362549800797  n =  195 power 7,31616882559868
i-1 0  Norm Dist 318,725099601593  n =  284 power 8,31616882559868
i 1 1  Norm Dist 318,725099601593  n =  304 power 8,31616882559868
i 2 11  Norm Dist 159,362549800797  n =  175 power 7,31616882559868
i 3 111  Norm Dist 79,6812749003984  n =  79 power 6,31616882559868
i 4 1111  Norm Dist 39,8406374501992  n =  48 power 5,31616882559868
i 5 11111  Norm Dist 19,9203187250996  n =  19 power 4,31616882559868
i 6 111111  Norm Dist 9,9601593625498  n =  10 power 3,31616882559868
i 7 1111111  Norm Dist 4,9800796812749  n =  0 power 2,31616882559868
i 8 11111111  Norm Dist 2,49003984063745  n =  0 power 1,31616882559868
i 9 111111111  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
Kappa2 = 38,3101380976096 igamc = p = 8,11782132752681E-04
Non Random

Example 6 Test -4 Seed 4 accuracy 2^6

Test 3 Special Normal Distribution Test
sum 3,921875 power nx 8,31616882559868 length n 2500
i-11 00000000000  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
i-10 0000000000  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i-9  000000000  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i-8  00000000  Norm Dist 2,49003984063745  n =  0 power 1,31616882559868
i-7  0000000  Norm Dist 4,9800796812749  n =  0 power 2,31616882559868
i-6  000000  Norm Dist 9,9601593625498  n =  0 power 3,31616882559868
i-5  00000  Norm Dist 19,9203187250996  n =  39 power 4,31616882559868
i-4  0000  Norm Dist 39,8406374501992  n =  39 power 5,31616882559868
i-3  000  Norm Dist 79,6812749003984  n =  39 power 6,31616882559868
i-2  00  Norm Dist 159,362549800797  n =  156 power 7,31616882559868
i-1  0  Norm Dist 318,725099601593  n =  509 power 8,31616882559868
i 1  1  Norm Dist 318,725099601593  n =  509 power 8,31616882559868
i 2  11  Norm Dist 159,362549800797  n =  156 power 7,31616882559868
i 3  111  Norm Dist 79,6812749003984  n =  78 power 6,31616882559868
i 4  1111  Norm Dist 39,8406374501992  n =  39 power 5,31616882559868
i 5  11111  Norm Dist 19,9203187250996  n =  0 power 4,31616882559868
i 6  111111  Norm Dist 9,9601593625498  n =  0 power 3,31616882559868
i 7  1111111  Norm Dist 4,9800796812749  n =  0 power 2,31616882559868
i 8  11111111  Norm Dist 2,49003984063745  n =  0 power 1,31616882559868
i 9  111111111  Norm Dist 1,24501992031872  n =  0 power 0,316168825598677
i 10 1111111111  Norm Dist 0,622509960159362  n =  0 power-0,683831174401323
i 11 11111111111  Norm Dist 0,311254980079681  n =  0 power-1,68383117440132
Kappa2 = 326,14326309761 igamc = p = 0
Non Random
```
The above results show 6 examples for the same Seed (i.e. 4) but with different accuracies of the Random Number Generator RND. The function RND is part of Visual Basic.
Each example is divided in two parts:
• The top part shows the results of the o bits i.e. the combinations 0, 00, 000, 0000 etc
• the bottom part shows the results for the 1 bits i.e. the combinations 1, 11, 111, 1111 etc
If you study example 6 (accuracy 2^6) than you will see that there are no bit strings with 6 zero's or more detected and no bit strings with 5 one's or more. There should be resp 10 and 20. The p-value is 0 i.e non random.
If you study example 5 (accuracy 2^8) than you will see that there are no bit strings with 7 zero's or more detected and no bit strings with 7 one's or more. There should be resp 5 and 5. The p-value is 0.0008 i.e non random.
If you study example 1 (accuracy 2^24) than you will see that almost all the zero and one bit combinations are detected, however the quantities are not as expected. The p value is allmost zero i.e. shows non random behavior.
If you study example 2 (accuracy 2^14) than you will see that almost all the zero and one bit combinations are detected. The quantities are as expected. The p value is 0.54 and the string is declared as random.

The Seed 4 is specially selected because the result of test 3 (example 1 accuracy 2^24) is negatif. The following Seeds give the same result: 4,5,7,15,16,18,20,21,22,25,26 and 29. In total 12. This type of behavior is normal for strings generated by physical systems i.e. the tests -1 and -2.

### Feedback

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Created: 10 June 2010

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