NOTE: Most of the tests in DIEHARD return a p-value, which should be uniform on [0,1) if the input file contains truly independent random bits. Those p-values are obtained by p=F(X), where F is the assumed distribution of the sample random variable X---often normal. But that assumed F is just an asymptotic approximation, for which the fit will be worst in the tails. Thus you should not be surprised with occasional p-values near 0 or 1, such as .0012 or .9983. When a bit stream really FAILS BIG, you will get p's of 0 or 1 to six or more places. By all means, do not, as a Statistician might, think that a p < .025 or p> .975 means that the RNG has "failed the test at the .05 level". Such p's happen among the hundreds that DIEHARD produces, even with good RNG's. So keep in mind that " p happens". ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the BIRTHDAY SPACINGS TEST :: :: Choose m birthdays in a year of n days. List the spacings :: :: between the birthdays. If j is the number of values that :: :: occur more than once in that list, then j is asymptotically :: :: Poisson distributed with mean m^3/(4n). Experience shows n :: :: must be quite large, say n>=2^18, for comparing the results :: :: to the Poisson distribution with that mean. This test uses :: :: n=2^24 and m=2^9, so that the underlying distribution for j :: :: is taken to be Poisson with lambda=2^27/(2^26)=2. A sample :: :: of 500 j's is taken, and a chi-square goodness of fit test :: :: provides a p value. The first test uses bits 1-24 (counting :: :: from the left) from integers in the specified file. :: :: Then the file is closed and reopened. Next, bits 2-25 are :: :: used to provide birthdays, then 3-26 and so on to bits 9-32. :: :: Each set of bits provides a p-value, and the nine p-values :: :: provide a sample for a KSTEST. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA= 2.0000 Results for random_raw.pcm For a sample of size 500: mean random_raw.pcm using bits 1 to 24 2.006 duplicate number number spacings observed expected 0 71. 67.668 1 131. 135.335 2 138. 135.335 3 89. 90.224 4 43. 45.112 5 17. 18.045 6 to INF 11. 8.282 Chisquare with 6 d.o.f. = 1.42 p-value= .035588 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 2 to 25 1.936 duplicate number number spacings observed expected 0 83. 67.668 1 133. 135.335 2 127. 135.335 3 86. 90.224 4 44. 45.112 5 16. 18.045 6 to INF 11. 8.282 Chisquare with 6 d.o.f. = 5.38 p-value= .503513 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 3 to 26 1.998 duplicate number number spacings observed expected 0 69. 67.668 1 139. 135.335 2 141. 135.335 3 75. 90.224 4 46. 45.112 5 17. 18.045 6 to INF 13. 8.282 Chisquare with 6 d.o.f. = 5.70 p-value= .542051 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 4 to 27 2.138 duplicate number number spacings observed expected 0 60. 67.668 1 124. 135.335 2 145. 135.335 3 87. 90.224 4 48. 45.112 5 20. 18.045 6 to INF 16. 8.282 Chisquare with 6 d.o.f. = 10.21 p-value= .884049 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 5 to 28 1.956 duplicate number number spacings observed expected 0 66. 67.668 1 131. 135.335 2 158. 135.335 3 78. 90.224 4 43. 45.112 5 19. 18.045 6 to INF 5. 8.282 Chisquare with 6 d.o.f. = 7.08 p-value= .686633 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 6 to 29 1.968 duplicate number number spacings observed expected 0 63. 67.668 1 141. 135.335 2 143. 135.335 3 88. 90.224 4 46. 45.112 5 9. 18.045 6 to INF 10. 8.282 Chisquare with 6 d.o.f. = 5.96 p-value= .571804 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 7 to 30 1.960 duplicate number number spacings observed expected 0 61. 67.668 1 144. 135.335 2 146. 135.335 3 89. 90.224 4 37. 45.112 5 13. 18.045 6 to INF 10. 8.282 Chisquare with 6 d.o.f. = 5.29 p-value= .493324 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 8 to 31 1.936 duplicate number number spacings observed expected 0 74. 67.668 1 134. 135.335 2 141. 135.335 3 86. 90.224 4 40. 45.112 5 20. 18.045 6 to INF 5. 8.282 Chisquare with 6 d.o.f. = 3.13 p-value= .207913 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean random_raw.pcm using bits 9 to 32 2.040 duplicate number number spacings observed expected 0 64. 67.668 1 130. 135.335 2 144. 135.335 3 94. 90.224 4 35. 45.112 5 21. 18.045 6 to INF 12. 8.282 Chisquare with 6 d.o.f. = 5.54 p-value= .523582 ::::::::::::::::::::::::::::::::::::::::: The 9 p-values were .035588 .503513 .542051 .884049 .686633 .571804 .493324 .207913 .523582 A KSTEST for the 9 p-values yields .374271 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE OVERLAPPING 5-PERMUTATION TEST :: :: This is the OPERM5 test. It looks at a sequence of one mill- :: :: ion 32-bit random integers. Each set of five consecutive :: :: integers can be in one of 120 states, for the 5! possible or- :: :: derings of five numbers. Thus the 5th, 6th, 7th,...numbers :: :: each provide a state. As many thousands of state transitions :: :: are observed, cumulative counts are made of the number of :: :: occurences of each state. Then the quadratic form in the :: :: weak inverse of the 120x120 covariance matrix yields a test :: :: equivalent to the likelihood ratio test that the 120 cell :: :: counts came from the specified (asymptotically) normal dis- :: :: tribution with the specified 120x120 covariance matrix (with :: :: rank 99). This version uses 1,000,000 integers, twice. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: OPERM5 test for file random_raw.pcm For a sample of 1,000,000 consecutive 5-tuples, chisquare for 99 degrees of freedom=111.542; p-value= .816862 OPERM5 test for file random_raw.pcm For a sample of 1,000,000 consecutive 5-tuples, chisquare for 99 degrees of freedom= 89.837; p-value= .265977 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the BINARY RANK TEST for 31x31 matrices. The leftmost :: :: 31 bits of 31 random integers from the test sequence are used :: :: to form a 31x31 binary matrix over the field {0,1}. The rank :: :: is determined. That rank can be from 0 to 31, but ranks< 28 :: :: are rare, and their counts are pooled with those for rank 28. :: :: Ranks are found for 40,000 such random matrices and a chisqua-:: :: re test is performed on counts for ranks 31,30,29 and <=28. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Binary rank test for random_raw.pcm Rank test for 31x31 binary matrices: rows from leftmost 31 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 28 207 211.4 .092324 .092 29 5056 5134.0 1.185350 1.278 30 23105 23103.0 .000165 1.278 31 11632 11551.5 .560646 1.838 chisquare= 1.838 for 3 d. of f.; p-value= .480969 -------------------------------------------------------------- ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the BINARY RANK TEST for 32x32 matrices. A random 32x :: :: 32 binary matrix is formed, each row a 32-bit random integer. :: :: The rank is determined. That rank can be from 0 to 32, ranks :: :: less than 29 are rare, and their counts are pooled with those :: :: for rank 29. Ranks are found for 40,000 such random matrices :: :: and a chisquare test is performed on counts for ranks 32,31, :: :: 30 and <=29. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Binary rank test for random_raw.pcm Rank test for 32x32 binary matrices: rows from leftmost 32 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 29 214 211.4 .031533 .032 30 5093 5134.0 .327588 .359 31 23218 23103.0 .571969 .931 32 11475 11551.5 .506945 1.438 chisquare= 1.438 for 3 d. of f.; p-value= .418585 -------------------------------------------------------------- $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the BINARY RANK TEST for 6x8 matrices. From each of :: :: six random 32-bit integers from the generator under test, a :: :: specified byte is chosen, and the resulting six bytes form a :: :: 6x8 binary matrix whose rank is determined. That rank can be :: :: from 0 to 6, but ranks 0,1,2,3 are rare; their counts are :: :: pooled with those for rank 4. Ranks are found for 100,000 :: :: random matrices, and a chi-square test is performed on :: :: counts for ranks 6,5 and <=4. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Binary Rank Test for random_raw.pcm Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 1 to 8 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1031 944.3 7.960 7.960 r =5 22260 21743.9 12.250 20.210 r =6 76709 77311.8 4.700 24.910 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 2 to 9 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1036 944.3 8.905 8.905 r =5 22360 21743.9 17.457 26.361 r =6 76604 77311.8 6.480 32.842 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 3 to 10 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1027 944.3 7.242 7.242 r =5 22369 21743.9 17.971 25.213 r =6 76604 77311.8 6.480 31.693 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 4 to 11 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1031 944.3 7.960 7.960 r =5 22224 21743.9 10.600 18.561 r =6 76745 77311.8 4.155 22.716 p=1-exp(-SUM/2)= .99999 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 5 to 12 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1042 944.3 10.108 10.108 r =5 22223 21743.9 10.556 20.664 r =6 76735 77311.8 4.303 24.968 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 6 to 13 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1021 944.3 6.230 6.230 r =5 22219 21743.9 10.381 16.611 r =6 76760 77311.8 3.938 20.549 p=1-exp(-SUM/2)= .99997 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 7 to 14 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1013 944.3 4.998 4.998 r =5 22165 21743.9 8.155 13.153 r =6 76822 77311.8 3.103 16.256 p=1-exp(-SUM/2)= .99970 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 8 to 15 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1034 944.3 8.520 8.520 r =5 22065 21743.9 4.742 13.262 r =6 76901 77311.8 2.183 15.445 p=1-exp(-SUM/2)= .99956 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 9 to 16 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1058 944.3 13.690 13.690 r =5 22385 21743.9 18.902 32.592 r =6 76557 77311.8 7.369 39.961 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 10 to 17 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1193 944.3 65.499 65.499 r =5 22267 21743.9 12.584 78.084 r =6 76540 77311.8 7.705 85.789 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 11 to 18 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1117 944.3 31.584 31.584 r =5 22424 21743.9 21.272 52.856 r =6 76459 77311.8 9.407 62.263 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 12 to 19 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1015 944.3 5.293 5.293 r =5 22212 21743.9 10.077 15.370 r =6 76773 77311.8 3.755 19.125 p=1-exp(-SUM/2)= .99993 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 13 to 20 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 995 944.3 2.722 2.722 r =5 22362 21743.9 17.570 20.292 r =6 76643 77311.8 5.786 26.078 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 14 to 21 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 999 944.3 3.168 3.168 r =5 22251 21743.9 11.826 14.995 r =6 76750 77311.8 4.082 19.077 p=1-exp(-SUM/2)= .99993 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 15 to 22 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1009 944.3 4.433 4.433 r =5 22193 21743.9 9.276 13.709 r =6 76798 77311.8 3.415 17.123 p=1-exp(-SUM/2)= .99981 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 16 to 23 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1035 944.3 8.711 8.711 r =5 22549 21743.9 29.810 38.521 r =6 76416 77311.8 10.380 48.901 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 17 to 24 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1120 944.3 32.691 32.691 r =5 22229 21743.9 10.822 43.513 r =6 76651 77311.8 5.648 49.161 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 18 to 25 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 973 944.3 .872 .872 r =5 22383 21743.9 18.785 19.657 r =6 76644 77311.8 5.768 25.425 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 19 to 26 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1026 944.3 7.068 7.068 r =5 22213 21743.9 10.120 17.189 r =6 76761 77311.8 3.924 21.113 p=1-exp(-SUM/2)= .99997 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 20 to 27 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1014 944.3 5.144 5.144 r =5 22207 21743.9 9.863 15.008 r =6 76779 77311.8 3.672 18.679 p=1-exp(-SUM/2)= .99991 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 21 to 28 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1046 944.3 10.953 10.953 r =5 22185 21743.9 8.948 19.901 r =6 76769 77311.8 3.811 23.712 p=1-exp(-SUM/2)= .99999 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 22 to 29 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 997 944.3 2.941 2.941 r =5 22273 21743.9 12.875 15.816 r =6 76730 77311.8 4.378 20.194 p=1-exp(-SUM/2)= .99996 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 23 to 30 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1052 944.3 12.283 12.283 r =5 22377 21743.9 18.433 30.717 r =6 76571 77311.8 7.098 37.815 p=1-exp(-SUM/2)=1.00000 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 24 to 31 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1014 944.3 5.144 5.144 r =5 22275 21743.9 12.972 18.117 r =6 76711 77311.8 4.669 22.786 p=1-exp(-SUM/2)= .99999 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG random_raw.pcm b-rank test for bits 25 to 32 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 1059 944.3 13.932 13.932 r =5 22266 21743.9 12.536 26.468 r =6 76675 77311.8 5.245 31.713 p=1-exp(-SUM/2)=1.00000 TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices These should be 25 uniform [0,1] random variables: .999996 1.000000 1.000000 .999988 .999996 .999965 .999705 .999557 1.000000 1.000000 1.000000 .999930 .999998 .999928 .999809 1.000000 1.000000 .999997 .999974 .999912 .999993 .999959 1.000000 .999989 1.000000 brank test summary for random_raw.pcm The KS test for those 25 supposed UNI's yields KS p-value=1.000000 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE BITSTREAM TEST :: :: The file under test is viewed as a stream of bits. Call them :: :: b1,b2,... . Consider an alphabet with two "letters", 0 and 1 :: :: and think of the stream of bits as a succession of 20-letter :: :: "words", overlapping. Thus the first word is b1b2...b20, the :: :: second is b2b3...b21, and so on. The bitstream test counts :: :: the number of missing 20-letter (20-bit) words in a string of :: :: 2^21 overlapping 20-letter words. There are 2^20 possible 20 :: :: letter words. For a truly random string of 2^21+19 bits, the :: :: number of missing words j should be (very close to) normally :: :: distributed with mean 141,909 and sigma 428. Thus :: :: (j-141909)/428 should be a standard normal variate (z score) :: :: that leads to a uniform [0,1) p value. The test is repeated :: :: twenty times. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: THE OVERLAPPING 20-tuples BITSTREAM TEST, 20 BITS PER WORD, N words This test uses N=2^21 and samples the bitstream 20 times. No. missing words should average 141909. with sigma=428. --------------------------------------------------------- tst no 1: 147929 missing words, 14.06 sigmas from mean, p-value=1.00000 tst no 2: 148437 missing words, 15.25 sigmas from mean, p-value=1.00000 tst no 3: 147656 missing words, 13.43 sigmas from mean, p-value=1.00000 tst no 4: 147616 missing words, 13.33 sigmas from mean, p-value=1.00000 tst no 5: 147143 missing words, 12.23 sigmas from mean, p-value=1.00000 tst no 6: 147935 missing words, 14.08 sigmas from mean, p-value=1.00000 tst no 7: 148163 missing words, 14.61 sigmas from mean, p-value=1.00000 tst no 8: 148446 missing words, 15.27 sigmas from mean, p-value=1.00000 tst no 9: 148326 missing words, 14.99 sigmas from mean, p-value=1.00000 tst no 10: 148081 missing words, 14.42 sigmas from mean, p-value=1.00000 tst no 11: 147784 missing words, 13.73 sigmas from mean, p-value=1.00000 tst no 12: 147364 missing words, 12.74 sigmas from mean, p-value=1.00000 tst no 13: 147722 missing words, 13.58 sigmas from mean, p-value=1.00000 tst no 14: 147725 missing words, 13.59 sigmas from mean, p-value=1.00000 tst no 15: 147776 missing words, 13.71 sigmas from mean, p-value=1.00000 tst no 16: 146992 missing words, 11.88 sigmas from mean, p-value=1.00000 tst no 17: 147609 missing words, 13.32 sigmas from mean, p-value=1.00000 tst no 18: 147426 missing words, 12.89 sigmas from mean, p-value=1.00000 tst no 19: 147716 missing words, 13.57 sigmas from mean, p-value=1.00000 tst no 20: 147868 missing words, 13.92 sigmas from mean, p-value=1.00000 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: The tests OPSO, OQSO and DNA :: :: OPSO means Overlapping-Pairs-Sparse-Occupancy :: :: The OPSO test considers 2-letter words from an alphabet of :: :: 1024 letters. Each letter is determined by a specified ten :: :: bits from a 32-bit integer in the sequence to be tested. OPSO :: :: generates 2^21 (overlapping) 2-letter words (from 2^21+1 :: :: "keystrokes") and counts the number of missing words---that :: :: is 2-letter words which do not appear in the entire sequence. :: :: That count should be very close to normally distributed with :: :: mean 141,909, sigma 290. Thus (missingwrds-141909)/290 should :: :: be a standard normal variable. The OPSO test takes 32 bits at :: :: a time from the test file and uses a designated set of ten :: :: consecutive bits. It then restarts the file for the next de- :: :: signated 10 bits, and so on. :: :: :: :: OQSO means Overlapping-Quadruples-Sparse-Occupancy :: :: The test OQSO is similar, except that it considers 4-letter :: :: words from an alphabet of 32 letters, each letter determined :: :: by a designated string of 5 consecutive bits from the test :: :: file, elements of which are assumed 32-bit random integers. :: :: The mean number of missing words in a sequence of 2^21 four- :: :: letter words, (2^21+3 "keystrokes"), is again 141909, with :: :: sigma = 295. The mean is based on theory; sigma comes from :: :: extensive simulation. :: :: :: :: The DNA test considers an alphabet of 4 letters:: C,G,A,T,:: :: determined by two designated bits in the sequence of random :: :: integers being tested. It considers 10-letter words, so that :: :: as in OPSO and OQSO, there are 2^20 possible words, and the :: :: mean number of missing words from a string of 2^21 (over- :: :: lapping) 10-letter words (2^21+9 "keystrokes") is 141909. :: :: The standard deviation sigma=339 was determined as for OQSO :: :: by simulation. (Sigma for OPSO, 290, is the true value (to :: :: three places), not determined by simulation. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: OPSO test for generator random_raw.pcm Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p OPSO for random_raw.pcm using bits 23 to 32 147265 18.468 1.0000 OPSO for random_raw.pcm using bits 22 to 31 147508 19.306 1.0000 OPSO for random_raw.pcm using bits 21 to 30 147870 20.554 1.0000 OPSO for random_raw.pcm using bits 20 to 29 147760 20.175 1.0000 OPSO for random_raw.pcm using bits 19 to 28 148206 21.713 1.0000 OPSO for random_raw.pcm using bits 18 to 27 147925 20.744 1.0000 OPSO for random_raw.pcm using bits 17 to 26 147209 18.275 1.0000 OPSO for random_raw.pcm using bits 16 to 25 147289 18.551 1.0000 OPSO for random_raw.pcm using bits 15 to 24 147528 19.375 1.0000 OPSO for random_raw.pcm using bits 14 to 23 147354 18.775 1.0000 OPSO for random_raw.pcm using bits 13 to 22 147598 19.616 1.0000 OPSO for random_raw.pcm using bits 12 to 21 147700 19.968 1.0000 OPSO for random_raw.pcm using bits 11 to 20 148126 21.437 1.0000 OPSO for random_raw.pcm using bits 10 to 19 147875 20.571 1.0000 OPSO for random_raw.pcm using bits 9 to 18 147721 20.040 1.0000 OPSO for random_raw.pcm using bits 8 to 17 147567 19.509 1.0000 OPSO for random_raw.pcm using bits 7 to 16 147481 19.213 1.0000 OPSO for random_raw.pcm using bits 6 to 15 147858 20.513 1.0000 OPSO for random_raw.pcm using bits 5 to 14 147885 20.606 1.0000 OPSO for random_raw.pcm using bits 4 to 13 147591 19.592 1.0000 OPSO for random_raw.pcm using bits 3 to 12 148173 21.599 1.0000 OPSO for random_raw.pcm using bits 2 to 11 147621 19.695 1.0000 OPSO for random_raw.pcm using bits 1 to 10 147170 18.140 1.0000 OQSO test for generator random_raw.pcm Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p OQSO for random_raw.pcm using bits 28 to 32 147756 19.819 1.0000 OQSO for random_raw.pcm using bits 27 to 31 147416 18.667 1.0000 OQSO for random_raw.pcm using bits 26 to 30 147847 20.128 1.0000 OQSO for random_raw.pcm using bits 25 to 29 146581 15.836 1.0000 OQSO for random_raw.pcm using bits 24 to 28 148144 21.134 1.0000 OQSO for random_raw.pcm using bits 23 to 27 147218 17.995 1.0000 OQSO for random_raw.pcm using bits 22 to 26 147566 19.175 1.0000 OQSO for random_raw.pcm using bits 21 to 25 147123 17.673 1.0000 OQSO for random_raw.pcm using bits 20 to 24 148160 21.189 1.0000 OQSO for random_raw.pcm using bits 19 to 23 147522 19.026 1.0000 OQSO for random_raw.pcm using bits 18 to 22 147551 19.124 1.0000 OQSO for random_raw.pcm using bits 17 to 21 147948 20.470 1.0000 OQSO for random_raw.pcm using bits 16 to 20 147528 19.046 1.0000 OQSO for random_raw.pcm using bits 15 to 19 147770 19.867 1.0000 OQSO for random_raw.pcm using bits 14 to 18 147440 18.748 1.0000 OQSO for random_raw.pcm using bits 13 to 17 147546 19.107 1.0000 OQSO for random_raw.pcm using bits 12 to 16 147363 18.487 1.0000 OQSO for random_raw.pcm using bits 11 to 15 147518 19.012 1.0000 OQSO for random_raw.pcm using bits 10 to 14 147244 18.084 1.0000 OQSO for random_raw.pcm using bits 9 to 13 147274 18.185 1.0000 OQSO for random_raw.pcm using bits 8 to 12 147226 18.023 1.0000 OQSO for random_raw.pcm using bits 7 to 11 147427 18.704 1.0000 OQSO for random_raw.pcm using bits 6 to 10 147675 19.545 1.0000 OQSO for random_raw.pcm using bits 5 to 9 147064 17.473 1.0000 OQSO for random_raw.pcm using bits 4 to 8 146836 16.701 1.0000 OQSO for random_raw.pcm using bits 3 to 7 147324 18.355 1.0000 OQSO for random_raw.pcm using bits 2 to 6 147829 20.067 1.0000 OQSO for random_raw.pcm using bits 1 to 5 147731 19.734 1.0000 DNA test for generator random_raw.pcm Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p DNA for random_raw.pcm using bits 31 to 32 147446 16.332 1.0000 DNA for random_raw.pcm using bits 30 to 31 146762 14.315 1.0000 DNA for random_raw.pcm using bits 29 to 30 147518 16.545 1.0000 DNA for random_raw.pcm using bits 28 to 29 147586 16.745 1.0000 DNA for random_raw.pcm using bits 27 to 28 146371 13.161 1.0000 DNA for random_raw.pcm using bits 26 to 27 147293 15.881 1.0000 DNA for random_raw.pcm using bits 25 to 26 147220 15.666 1.0000 DNA for random_raw.pcm using bits 24 to 25 147548 16.633 1.0000 DNA for random_raw.pcm using bits 23 to 24 147616 16.834 1.0000 DNA for random_raw.pcm using bits 22 to 23 147374 16.120 1.0000 DNA for random_raw.pcm using bits 21 to 22 147498 16.486 1.0000 DNA for random_raw.pcm using bits 20 to 21 147764 17.270 1.0000 DNA for random_raw.pcm using bits 19 to 20 147617 16.837 1.0000 DNA for random_raw.pcm using bits 18 to 19 147405 16.211 1.0000 DNA for random_raw.pcm using bits 17 to 18 146890 14.692 1.0000 DNA for random_raw.pcm using bits 16 to 17 147311 15.934 1.0000 DNA for random_raw.pcm using bits 15 to 16 148379 19.085 1.0000 DNA for random_raw.pcm using bits 14 to 15 147857 17.545 1.0000 DNA for random_raw.pcm using bits 13 to 14 147534 16.592 1.0000 DNA for random_raw.pcm using bits 12 to 13 147810 17.406 1.0000 DNA for random_raw.pcm using bits 11 to 12 147832 17.471 1.0000 DNA for random_raw.pcm using bits 10 to 11 147388 16.161 1.0000 DNA for random_raw.pcm using bits 9 to 10 147102 15.318 1.0000 DNA for random_raw.pcm using bits 8 to 9 147619 16.843 1.0000 DNA for random_raw.pcm using bits 7 to 8 146884 14.675 1.0000 DNA for random_raw.pcm using bits 6 to 7 147204 15.619 1.0000 DNA for random_raw.pcm using bits 5 to 6 147298 15.896 1.0000 DNA for random_raw.pcm using bits 4 to 5 147394 16.179 1.0000 DNA for random_raw.pcm using bits 3 to 4 148235 18.660 1.0000 DNA for random_raw.pcm using bits 2 to 3 147818 17.430 1.0000 DNA for random_raw.pcm using bits 1 to 2 147159 15.486 1.0000 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the COUNT-THE-1's TEST on a stream of bytes. :: :: Consider the file under test as a stream of bytes (four per :: :: 32 bit integer). Each byte can contain from 0 to 8 1's, :: :: with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let :: :: the stream of bytes provide a string of overlapping 5-letter :: :: words, each "letter" taking values A,B,C,D,E. The letters are :: :: determined by the number of 1's in a byte:: 0,1,or 2 yield A,:: :: 3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus :: :: we have a monkey at a typewriter hitting five keys with vari- :: :: ous probabilities (37,56,70,56,37 over 256). There are 5^5 :: :: possible 5-letter words, and from a string of 256,000 (over- :: :: lapping) 5-letter words, counts are made on the frequencies :: :: for each word. The quadratic form in the weak inverse of :: :: the covariance matrix of the cell counts provides a chisquare :: :: test:: Q5-Q4, the difference of the naive Pearson sums of :: :: (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Test results for random_raw.pcm Chi-square with 5^5-5^4=2500 d.of f. for sample size:2560000 chisquare equiv normal p-value Results fo COUNT-THE-1's in successive bytes: byte stream for random_raw.pcm 23998.31 304.032 1.000000 byte stream for random_raw.pcm 22421.49 281.732 1.000000 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the COUNT-THE-1's TEST for specific bytes. :: :: Consider the file under test as a stream of 32-bit integers. :: :: From each integer, a specific byte is chosen , say the left- :: :: most:: bits 1 to 8. Each byte can contain from 0 to 8 1's, :: :: with probabilitie 1,8,28,56,70,56,28,8,1 over 256. Now let :: :: the specified bytes from successive integers provide a string :: :: of (overlapping) 5-letter words, each "letter" taking values :: :: A,B,C,D,E. The letters are determined by the number of 1's, :: :: in that byte:: 0,1,or 2 ---> A, 3 ---> B, 4 ---> C, 5 ---> D,:: :: and 6,7 or 8 ---> E. Thus we have a monkey at a typewriter :: :: hitting five keys with with various probabilities:: 37,56,70,:: :: 56,37 over 256. There are 5^5 possible 5-letter words, and :: :: from a string of 256,000 (overlapping) 5-letter words, counts :: :: are made on the frequencies for each word. The quadratic form :: :: in the weak inverse of the covariance matrix of the cell :: :: counts provides a chisquare test:: Q5-Q4, the difference of :: :: the naive Pearson sums of (OBS-EXP)^2/EXP on counts for 5- :: :: and 4-letter cell counts. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Chi-square with 5^5-5^4=2500 d.of f. for sample size: 256000 chisquare equiv normal p value Results for COUNT-THE-1's in specified bytes: bits 1 to 8 4549.68 28.987 1.000000 bits 2 to 9 4502.63 28.321 1.000000 bits 3 to 10 4469.61 27.855 1.000000 bits 4 to 11 4464.35 27.780 1.000000 bits 5 to 12 4497.20 28.245 1.000000 bits 6 to 13 4545.28 28.925 1.000000 bits 7 to 14 4547.99 28.963 1.000000 bits 8 to 15 4586.48 29.507 1.000000 bits 9 to 16 4701.96 31.140 1.000000 bits 10 to 17 4713.94 31.310 1.000000 bits 11 to 18 4708.57 31.234 1.000000 bits 12 to 19 4719.78 31.392 1.000000 bits 13 to 20 4667.90 30.659 1.000000 bits 14 to 21 4538.90 28.834 1.000000 bits 15 to 22 4761.28 31.979 1.000000 bits 16 to 23 4814.86 32.737 1.000000 bits 17 to 24 4846.61 33.186 1.000000 bits 18 to 25 4741.67 31.702 1.000000 bits 19 to 26 4926.06 34.310 1.000000 bits 20 to 27 4728.53 31.516 1.000000 bits 21 to 28 4912.04 34.111 1.000000 bits 22 to 29 4789.56 32.379 1.000000 bits 23 to 30 4614.04 29.897 1.000000 bits 24 to 31 4809.15 32.656 1.000000 bits 25 to 32 4696.82 31.068 1.000000 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THIS IS A PARKING LOT TEST :: :: In a square of side 100, randomly "park" a car---a circle of :: :: radius 1. Then try to park a 2nd, a 3rd, and so on, each :: :: time parking "by ear". That is, if an attempt to park a car :: :: causes a crash with one already parked, try again at a new :: :: random location. (To avoid path problems, consider parking :: :: helicopters rather than cars.) Each attempt leads to either :: :: a crash or a success, the latter followed by an increment to :: :: the list of cars already parked. If we plot n: the number of :: :: attempts, versus k:: the number successfully parked, we get a:: :: curve that should be similar to those provided by a perfect :: :: random number generator. Theory for the behavior of such a :: :: random curve seems beyond reach, and as graphics displays are :: :: not available for this battery of tests, a simple characteriz :: :: ation of the random experiment is used: k, the number of cars :: :: successfully parked after n=12,000 attempts. Simulation shows :: :: that k should average 3523 with sigma 21.9 and is very close :: :: to normally distributed. Thus (k-3523)/21.9 should be a st- :: :: andard normal variable, which, converted to a uniform varia- :: :: ble, provides input to a KSTEST based on a sample of 10. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: CDPARK: result of ten tests on file random_raw.pcm Of 12,000 tries, the average no. of successes should be 3523 with sigma=21.9 Successes: 3516 z-score: -.320 p-value: .374623 Successes: 3527 z-score: .183 p-value: .572463 Successes: 3512 z-score: -.502 p-value: .307734 Successes: 3551 z-score: 1.279 p-value: .899470 Successes: 3522 z-score: -.046 p-value: .481790 Successes: 3493 z-score: -1.370 p-value: .085365 Successes: 3506 z-score: -.776 p-value: .218799 Successes: 3505 z-score: -.822 p-value: .205562 Successes: 3561 z-score: 1.735 p-value: .958644 Successes: 3503 z-score: -.913 p-value: .180558 square size avg. no. parked sample sigma 100. 3519.600 20.510 KSTEST for the above 10: p= .437412 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE MINIMUM DISTANCE TEST :: :: It does this 100 times:: choose n=8000 random points in a :: :: square of side 10000. Find d, the minimum distance between :: :: the (n^2-n)/2 pairs of points. If the points are truly inde- :: :: pendent uniform, then d^2, the square of the minimum distance :: :: should be (very close to) exponentially distributed with mean :: :: .995 . Thus 1-exp(-d^2/.995) should be uniform on [0,1) and :: :: a KSTEST on the resulting 100 values serves as a test of uni- :: :: formity for random points in the square. Test numbers=0 mod 5 :: :: are printed but the KSTEST is based on the full set of 100 :: :: random choices of 8000 points in the 10000x10000 square. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: This is the MINIMUM DISTANCE test for random integers in the file random_raw.pcm Sample no. d^2 avg equiv uni 5 .0281 1.2468 .027835 10 4.1457 1.1627 .984494 15 .4803 1.0934 .382923 20 2.3144 1.3173 .902315 25 .5568 1.1822 .428584 30 1.7251 1.1254 .823387 35 .3115 1.0487 .268822 40 .2411 .9783 .215203 45 1.5946 .9691 .798624 50 1.8307 .9578 .841166 55 .5152 1.0113 .404161 60 .9967 1.0670 .632763 65 .4069 1.0366 .335676 70 .0174 .9995 .017359 75 .1114 .9689 .105885 80 1.0532 1.0188 .653036 85 .0123 .9995 .012272 90 1.8656 1.0401 .846639 95 1.1214 1.0544 .675997 100 .4859 1.0481 .386361 MINIMUM DISTANCE TEST for random_raw.pcm Result of KS test on 20 transformed mindist^2's: p-value= .334672 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE 3DSPHERES TEST :: :: Choose 4000 random points in a cube of edge 1000. At each :: :: point, center a sphere large enough to reach the next closest :: :: point. Then the volume of the smallest such sphere is (very :: :: close to) exponentially distributed with mean 120pi/3. Thus :: :: the radius cubed is exponential with mean 30. (The mean is :: :: obtained by extensive simulation). The 3DSPHERES test gener- :: :: ates 4000 such spheres 20 times. Each min radius cubed leads :: :: to a uniform variable by means of 1-exp(-r^3/30.), then a :: :: KSTEST is done on the 20 p-values. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The 3DSPHERES test for file random_raw.pcm sample no: 1 r^3= 3.436 p-value= .10823 sample no: 2 r^3= 48.254 p-value= .79980 sample no: 3 r^3= 8.379 p-value= .24370 sample no: 4 r^3= 8.926 p-value= .25734 sample no: 5 r^3= 1.922 p-value= .06205 sample no: 6 r^3= 19.247 p-value= .47353 sample no: 7 r^3= 18.588 p-value= .46184 sample no: 8 r^3= 13.408 p-value= .36041 sample no: 9 r^3= 10.160 p-value= .28728 sample no: 10 r^3= 49.102 p-value= .80538 sample no: 11 r^3= 1.329 p-value= .04333 sample no: 12 r^3= 72.128 p-value= .90967 sample no: 13 r^3= 6.136 p-value= .18498 sample no: 14 r^3= 5.795 p-value= .17566 sample no: 15 r^3= 29.489 p-value= .62581 sample no: 16 r^3= 5.670 p-value= .17223 sample no: 17 r^3= 25.512 p-value= .57275 sample no: 18 r^3= 22.657 p-value= .53010 sample no: 19 r^3= 22.946 p-value= .53461 sample no: 20 r^3= 22.585 p-value= .52898 A KS test is applied to those 20 p-values. --------------------------------------------------------- 3DSPHERES test for file random_raw.pcm p-value= .765160 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the SQEEZE test :: :: Random integers are floated to get uniforms on [0,1). Start- :: :: ing with k=2^31=2147483647, the test finds j, the number of :: :: iterations necessary to reduce k to 1, using the reduction :: :: k=ceiling(k*U), with U provided by floating integers from :: :: the file being tested. Such j's are found 100,000 times, :: :: then counts for the number of times j was <=6,7,...,47,>=48 :: :: are used to provide a chi-square test for cell frequencies. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: RESULTS OF SQUEEZE TEST FOR random_raw.pcm Table of standardized frequency counts ( (obs-exp)/sqrt(exp) )^2 for j taking values <=6,7,8,...,47,>=48: 3.4 .1 3.4 4.4 4.1 1.1 5.0 7.3 5.6 5.0 4.0 1.3 4.3 1.1 1.1 .1 .1 -2.4 -2.4 -2.4 -2.1 -3.4 -3.2 -1.4 -1.6 -3.0 -1.0 -.8 -.6 -.2 -.4 .6 -.9 .2 -.9 2.4 .0 .2 .5 1.5 .1 -1.0 -1.1 Chi-square with 42 degrees of freedom:307.905 z-score= 29.013 p-value=1.000000 ______________________________________________________________ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: The OVERLAPPING SUMS test :: :: Integers are floated to get a sequence U(1),U(2),... of uni- :: :: form [0,1) variables. Then overlapping sums, :: :: S(1)=U(1)+...+U(100), S2=U(2)+...+U(101),... are formed. :: :: The S's are virtually normal with a certain covariance mat- :: :: rix. A linear transformation of the S's converts them to a :: :: sequence of independent standard normals, which are converted :: :: to uniform variables for a KSTEST. The p-values from ten :: :: KSTESTs are given still another KSTEST. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Test no. 1 p-value .739503 Test no. 2 p-value .618917 Test no. 3 p-value .881534 Test no. 4 p-value .078026 Test no. 5 p-value .119003 Test no. 6 p-value .375996 Test no. 7 p-value .575687 Test no. 8 p-value .258520 Test no. 9 p-value .225410 Test no. 10 p-value .862048 Results of the OSUM test for random_raw.pcm KSTEST on the above 10 p-values: .019674 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the RUNS test. It counts runs up, and runs down, :: :: in a sequence of uniform [0,1) variables, obtained by float- :: :: ing the 32-bit integers in the specified file. This example :: :: shows how runs are counted: .123,.357,.789,.425,.224,.416,.95:: :: contains an up-run of length 3, a down-run of length 2 and an :: :: up-run of (at least) 2, depending on the next values. The :: :: covariance matrices for the runs-up and runs-down are well :: :: known, leading to chisquare tests for quadratic forms in the :: :: weak inverses of the covariance matrices. Runs are counted :: :: for sequences of length 10,000. This is done ten times. Then :: :: repeated. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The RUNS test for file random_raw.pcm Up and down runs in a sample of 10000 _________________________________________________ Run test for random_raw.pcm : runs up; ks test for 10 p's: .988984 runs down; ks test for 10 p's: .867113 Run test for random_raw.pcm : runs up; ks test for 10 p's: .182708 runs down; ks test for 10 p's: .673891 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the CRAPS TEST. It plays 200,000 games of craps, finds:: :: the number of wins and the number of throws necessary to end :: :: each game. The number of wins should be (very close to) a :: :: normal with mean 200000p and variance 200000p(1-p), with :: :: p=244/495. Throws necessary to complete the game can vary :: :: from 1 to infinity, but counts for all>21 are lumped with 21. :: :: A chi-square test is made on the no.-of-throws cell counts. :: :: Each 32-bit integer from the test file provides the value for :: :: the throw of a die, by floating to [0,1), multiplying by 6 :: :: and taking 1 plus the integer part of the result. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Results of craps test for random_raw.pcm No. of wins: Observed Expected 98629 98585.86 98629= No. of wins, z-score= .193 pvalue= .57650 Analysis of Throws-per-Game: Chisq= 14.47 for 20 degrees of freedom, p= .19400 Throws Observed Expected Chisq Sum 1 66298 66666.7 2.039 2.039 2 37818 37654.3 .711 2.750 3 26919 26954.7 .047 2.798 4 19292 19313.5 .024 2.821 5 13807 13851.4 .142 2.964 6 10057 9943.5 1.295 4.258 7 7298 7145.0 3.275 7.534 8 5176 5139.1 .265 7.799 9 3697 3699.9 .002 7.801 10 2634 2666.3 .391 8.192 11 1919 1923.3 .010 8.202 12 1358 1388.7 .680 8.883 13 987 1003.7 .278 9.161 14 753 726.1 .994 10.155 15 542 525.8 .497 10.651 16 404 381.2 1.370 12.021 17 282 276.5 .108 12.129 18 200 200.8 .003 12.132 19 143 146.0 .061 12.193 20 121 106.2 2.058 14.251 21 295 287.1 .217 14.468 SUMMARY FOR random_raw.pcm p-value for no. of wins: .576501 p-value for throws/game: .194002 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Results of DIEHARD battery of tests sent to file random_raw.txt