------------------------------------------------------------------------ XR232USB RANDOM DATA SAMPLE AND TEST REPORTS ------------------------------------------------------------------------ Date: 20140725 HW: 20130725 SW: 20140720 Baud: 115.200 bps Filename: xr232utest.dat Size: 11468800 Bytes ( = 11200 KB pursuant to DIEHARD recommendation) ------------------------------------------------------------------------ RESULTS FROM ENTITLE ENTROPY-CHECK (http://www.fourmilab.ch/random) ------------------------------------------------------------------------ Entropy = 7.999985 bits per byte. Optimum compression would reduce the size of this 11468800 byte file by 0 percent. Chi square distribution for 11468800 samples is 235.83, and randomly would exceed this value 79.99 percent of the times. Arithmetic mean value of data bytes is 127.5015 (127.5 = random). Monte Carlo value for Pi is 3.141571966 (error 0.00 percent). Serial correlation coefficient is 0.000119 (totally uncorrelated = 0.0). ------------------------------------------------------------------------ RESULTS FROM DIEHARD (http://www.stat.fsu.edu/pub/diehard) ------------------------------------------------------------------------ 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 xr232utest.dat For a sample of size 500: mean xr232utest.dat using bits 1 to 24 1.938 duplicate number number spacings observed expected 0 72. 67.668 1 138. 135.335 2 136. 135.335 3 90. 90.224 4 38. 45.112 5 21. 18.045 6 to INF 5. 8.282 Chisquare with 6 d.o.f. = 3.24 p-value= .221731 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 2 to 25 1.860 duplicate number number spacings observed expected 0 93. 67.668 1 125. 135.335 2 143. 135.335 3 72. 90.224 4 44. 45.112 5 13. 18.045 6 to INF 10. 8.282 Chisquare with 6 d.o.f. = 16.18 p-value= .987190 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 3 to 26 1.998 duplicate number number spacings observed expected 0 75. 67.668 1 135. 135.335 2 120. 135.335 3 98. 90.224 4 41. 45.112 5 22. 18.045 6 to INF 9. 8.282 Chisquare with 6 d.o.f. = 4.51 p-value= .391640 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 4 to 27 1.994 duplicate number number spacings observed expected 0 70. 67.668 1 132. 135.335 2 142. 135.335 3 81. 90.224 4 50. 45.112 5 17. 18.045 6 to INF 8. 8.282 Chisquare with 6 d.o.f. = 2.03 p-value= .083405 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 5 to 28 1.978 duplicate number number spacings observed expected 0 64. 67.668 1 143. 135.335 2 139. 135.335 3 81. 90.224 4 45. 45.112 5 23. 18.045 6 to INF 5. 8.282 Chisquare with 6 d.o.f. = 4.34 p-value= .368770 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 6 to 29 1.956 duplicate number number spacings observed expected 0 69. 67.668 1 132. 135.335 2 147. 135.335 3 91. 90.224 4 42. 45.112 5 10. 18.045 6 to INF 9. 8.282 Chisquare with 6 d.o.f. = 4.98 p-value= .454125 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 7 to 30 1.944 duplicate number number spacings observed expected 0 81. 67.668 1 126. 135.335 2 150. 135.335 3 73. 90.224 4 35. 45.112 5 26. 18.045 6 to INF 9. 8.282 Chisquare with 6 d.o.f. = 13.98 p-value= .970182 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 8 to 31 1.966 duplicate number number spacings observed expected 0 70. 67.668 1 142. 135.335 2 136. 135.335 3 79. 90.224 4 49. 45.112 5 14. 18.045 6 to INF 10. 8.282 Chisquare with 6 d.o.f. = 3.41 p-value= .243602 ::::::::::::::::::::::::::::::::::::::::: For a sample of size 500: mean xr232utest.dat using bits 9 to 32 2.042 duplicate number number spacings observed expected 0 67. 67.668 1 133. 135.335 2 133. 135.335 3 91. 90.224 4 43. 45.112 5 25. 18.045 6 to INF 8. 8.282 Chisquare with 6 d.o.f. = 2.88 p-value= .176644 ::::::::::::::::::::::::::::::::::::::::: The 9 p-values were .221731 .987190 .391640 .083405 .368770 .454125 .970182 .243602 .176644 A KSTEST for the 9 p-values yields .723713 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat For a sample of 1,000,000 consecutive 5-tuples, chisquare for 99 degrees of freedom= 65.764; p-value= .004109 OPERM5 test for file xr232utest.dat For a sample of 1,000,000 consecutive 5-tuples, chisquare for 99 degrees of freedom=176.297; p-value= .999997 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat Rank test for 31x31 binary matrices: rows from leftmost 31 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 28 199 211.4 .729394 .729 29 5045 5134.0 1.543204 2.273 30 23169 23103.0 .188279 2.461 31 11587 11551.5 .108948 2.570 chisquare= 2.570 for 3 d. of f.; p-value= .591656 -------------------------------------------------------------- ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat Rank test for 32x32 binary matrices: rows from leftmost 32 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 29 199 211.4 .729394 .729 30 5140 5134.0 .006988 .736 31 23181 23103.0 .263025 .999 32 11480 11551.5 .442863 1.442 chisquare= 1.442 for 3 d. of f.; p-value= .419230 -------------------------------------------------------------- $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 1 to 8 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 897 944.3 2.369 2.369 r =5 21917 21743.9 1.378 3.747 r =6 77186 77311.8 .205 3.952 p=1-exp(-SUM/2)= .86139 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 2 to 9 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 900 944.3 2.078 2.078 r =5 21739 21743.9 .001 2.079 r =6 77361 77311.8 .031 2.111 p=1-exp(-SUM/2)= .65194 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 3 to 10 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 865 944.3 6.660 6.660 r =5 21826 21743.9 .310 6.970 r =6 77309 77311.8 .000 6.970 p=1-exp(-SUM/2)= .96934 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 4 to 11 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 953 944.3 .080 .080 r =5 21675 21743.9 .218 .298 r =6 77372 77311.8 .047 .345 p=1-exp(-SUM/2)= .15858 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 5 to 12 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 938 944.3 .042 .042 r =5 21828 21743.9 .325 .367 r =6 77234 77311.8 .078 .446 p=1-exp(-SUM/2)= .19973 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 6 to 13 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 886 944.3 3.600 3.600 r =5 21647 21743.9 .432 4.031 r =6 77467 77311.8 .312 4.343 p=1-exp(-SUM/2)= .88599 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 7 to 14 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 903 944.3 1.806 1.806 r =5 21628 21743.9 .618 2.424 r =6 77469 77311.8 .320 2.744 p=1-exp(-SUM/2)= .74638 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 8 to 15 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 944 944.3 .000 .000 r =5 21848 21743.9 .498 .498 r =6 77208 77311.8 .139 .638 p=1-exp(-SUM/2)= .27307 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 9 to 16 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 983 944.3 1.586 1.586 r =5 21901 21743.9 1.135 2.721 r =6 77116 77311.8 .496 3.217 p=1-exp(-SUM/2)= .79980 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 10 to 17 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 955 944.3 .121 .121 r =5 21676 21743.9 .212 .333 r =6 77369 77311.8 .042 .376 p=1-exp(-SUM/2)= .17120 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 11 to 18 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 979 944.3 1.275 1.275 r =5 21750 21743.9 .002 1.277 r =6 77271 77311.8 .022 1.298 p=1-exp(-SUM/2)= .47750 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 12 to 19 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 944 944.3 .000 .000 r =5 21502 21743.9 2.691 2.691 r =6 77554 77311.8 .759 3.450 p=1-exp(-SUM/2)= .82182 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 13 to 20 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 967 944.3 .546 .546 r =5 21439 21743.9 4.275 4.821 r =6 77594 77311.8 1.030 5.851 p=1-exp(-SUM/2)= .94636 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 14 to 21 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 933 944.3 .135 .135 r =5 21658 21743.9 .339 .475 r =6 77409 77311.8 .122 .597 p=1-exp(-SUM/2)= .25799 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 15 to 22 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 890 944.3 3.123 3.123 r =5 21885 21743.9 .916 4.038 r =6 77225 77311.8 .097 4.136 p=1-exp(-SUM/2)= .87354 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 16 to 23 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 958 944.3 .199 .199 r =5 21649 21743.9 .414 .613 r =6 77393 77311.8 .085 .698 p=1-exp(-SUM/2)= .29467 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 17 to 24 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 933 944.3 .135 .135 r =5 21810 21743.9 .201 .336 r =6 77257 77311.8 .039 .375 p=1-exp(-SUM/2)= .17099 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 18 to 25 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 872 944.3 5.536 5.536 r =5 21822 21743.9 .281 5.816 r =6 77306 77311.8 .000 5.817 p=1-exp(-SUM/2)= .94544 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 19 to 26 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 915 944.3 .909 .909 r =5 21827 21743.9 .318 1.227 r =6 77258 77311.8 .037 1.264 p=1-exp(-SUM/2)= .46854 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 20 to 27 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 951 944.3 .048 .048 r =5 21502 21743.9 2.691 2.739 r =6 77547 77311.8 .716 3.454 p=1-exp(-SUM/2)= .82220 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 21 to 28 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 993 944.3 2.511 2.511 r =5 21593 21743.9 1.047 3.559 r =6 77414 77311.8 .135 3.694 p=1-exp(-SUM/2)= .84227 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 22 to 29 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 959 944.3 .229 .229 r =5 21908 21743.9 1.238 1.467 r =6 77133 77311.8 .414 1.881 p=1-exp(-SUM/2)= .60952 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 23 to 30 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 958 944.3 .199 .199 r =5 21772 21743.9 .036 .235 r =6 77270 77311.8 .023 .258 p=1-exp(-SUM/2)= .12087 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 24 to 31 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 919 944.3 .678 .678 r =5 21705 21743.9 .070 .748 r =6 77376 77311.8 .053 .801 p=1-exp(-SUM/2)= .32995 Rank of a 6x8 binary matrix, rows formed from eight bits of the RNG xr232utest.dat b-rank test for bits 25 to 32 OBSERVED EXPECTED (O-E)^2/E SUM r<=4 980 944.3 1.350 1.350 r =5 21643 21743.9 .468 1.818 r =6 77377 77311.8 .055 1.873 p=1-exp(-SUM/2)= .60796 TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices These should be 25 uniform [0,1] random variables: .861385 .651942 .969342 .158579 .199735 .885988 .746376 .273071 .799800 .171204 .477501 .821823 .946364 .257995 .873538 .294673 .170987 .945436 .468535 .822197 .842272 .609524 .120869 .329952 .607956 brank test summary for xr232utest.dat The KS test for those 25 supposed UNI's yields KS p-value= .764672 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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: 141438 missing words, -1.10 sigmas from mean, p-value= .13540 tst no 2: 141951 missing words, .10 sigmas from mean, p-value= .53878 tst no 3: 141766 missing words, -.33 sigmas from mean, p-value= .36886 tst no 4: 141277 missing words, -1.48 sigmas from mean, p-value= .06978 tst no 5: 142431 missing words, 1.22 sigmas from mean, p-value= .88855 tst no 6: 141380 missing words, -1.24 sigmas from mean, p-value= .10809 tst no 7: 141701 missing words, -.49 sigmas from mean, p-value= .31322 tst no 8: 142102 missing words, .45 sigmas from mean, p-value= .67371 tst no 9: 141322 missing words, -1.37 sigmas from mean, p-value= .08499 tst no 10: 142455 missing words, 1.27 sigmas from mean, p-value= .89883 tst no 11: 141253 missing words, -1.53 sigmas from mean, p-value= .06258 tst no 12: 141379 missing words, -1.24 sigmas from mean, p-value= .10766 tst no 13: 141544 missing words, -.85 sigmas from mean, p-value= .19667 tst no 14: 141484 missing words, -.99 sigmas from mean, p-value= .16017 tst no 15: 141587 missing words, -.75 sigmas from mean, p-value= .22569 tst no 16: 142117 missing words, .49 sigmas from mean, p-value= .68624 tst no 17: 141553 missing words, -.83 sigmas from mean, p-value= .20255 tst no 18: 141943 missing words, .08 sigmas from mean, p-value= .53135 tst no 19: 142157 missing words, .58 sigmas from mean, p-value= .71859 tst no 20: 142759 missing words, 1.99 sigmas from mean, p-value= .97644 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p OPSO for xr232utest.dat using bits 23 to 32 141927 .061 .5243 OPSO for xr232utest.dat using bits 22 to 31 141605 -1.049 .1470 OPSO for xr232utest.dat using bits 21 to 30 142052 .492 .6886 OPSO for xr232utest.dat using bits 20 to 29 141679 -.794 .2135 OPSO for xr232utest.dat using bits 19 to 28 141645 -.911 .1810 OPSO for xr232utest.dat using bits 18 to 27 142489 1.999 .9772 OPSO for xr232utest.dat using bits 17 to 26 142040 .451 .6739 OPSO for xr232utest.dat using bits 16 to 25 141838 -.246 .4029 OPSO for xr232utest.dat using bits 15 to 24 141459 -1.553 .0602 OPSO for xr232utest.dat using bits 14 to 23 141990 .278 .6096 OPSO for xr232utest.dat using bits 13 to 22 142391 1.661 .9516 OPSO for xr232utest.dat using bits 12 to 21 142772 2.975 .9985 OPSO for xr232utest.dat using bits 11 to 20 141771 -.477 .3167 OPSO for xr232utest.dat using bits 10 to 19 141557 -1.215 .1122 OPSO for xr232utest.dat using bits 9 to 18 142261 1.213 .8874 OPSO for xr232utest.dat using bits 8 to 17 141865 -.153 .4393 OPSO for xr232utest.dat using bits 7 to 16 141624 -.984 .1626 OPSO for xr232utest.dat using bits 6 to 15 141611 -1.029 .1518 OPSO for xr232utest.dat using bits 5 to 14 141952 .147 .5585 OPSO for xr232utest.dat using bits 4 to 13 141465 -1.532 .0627 OPSO for xr232utest.dat using bits 3 to 12 141749 -.553 .2902 OPSO for xr232utest.dat using bits 2 to 11 141813 -.332 .3699 OPSO for xr232utest.dat using bits 1 to 10 142330 1.451 .9266 OQSO test for generator xr232utest.dat Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p OQSO for xr232utest.dat using bits 28 to 32 142551 2.175 .9852 OQSO for xr232utest.dat using bits 27 to 31 141859 -.171 .4323 OQSO for xr232utest.dat using bits 26 to 30 141967 .195 .5775 OQSO for xr232utest.dat using bits 25 to 29 142229 1.084 .8607 OQSO for xr232utest.dat using bits 24 to 28 141526 -1.299 .0969 OQSO for xr232utest.dat using bits 23 to 27 142159 .846 .8013 OQSO for xr232utest.dat using bits 22 to 26 142497 1.992 .9768 OQSO for xr232utest.dat using bits 21 to 25 142129 .745 .7718 OQSO for xr232utest.dat using bits 20 to 24 142148 .809 .7908 OQSO for xr232utest.dat using bits 19 to 23 142193 .962 .8319 OQSO for xr232utest.dat using bits 18 to 22 142312 1.365 .9139 OQSO for xr232utest.dat using bits 17 to 21 141975 .223 .5881 OQSO for xr232utest.dat using bits 16 to 20 141920 .036 .5144 OQSO for xr232utest.dat using bits 15 to 19 142101 .650 .7421 OQSO for xr232utest.dat using bits 14 to 18 142509 2.033 .9790 OQSO for xr232utest.dat using bits 13 to 17 142084 .592 .7231 OQSO for xr232utest.dat using bits 12 to 16 141928 .063 .5252 OQSO for xr232utest.dat using bits 11 to 15 141784 -.425 .3355 OQSO for xr232utest.dat using bits 10 to 14 142306 1.345 .9106 OQSO for xr232utest.dat using bits 9 to 13 141395 -1.743 .0406 OQSO for xr232utest.dat using bits 8 to 12 141706 -.689 .2453 OQSO for xr232utest.dat using bits 7 to 11 141772 -.466 .3208 OQSO for xr232utest.dat using bits 6 to 10 142108 .673 .7497 OQSO for xr232utest.dat using bits 5 to 9 141742 -.567 .2853 OQSO for xr232utest.dat using bits 4 to 8 142031 .412 .6600 OQSO for xr232utest.dat using bits 3 to 7 141880 -.099 .4604 OQSO for xr232utest.dat using bits 2 to 6 141892 -.059 .4766 OQSO for xr232utest.dat using bits 1 to 5 141807 -.347 .3643 DNA test for generator xr232utest.dat Output: No. missing words (mw), equiv normal variate (z), p-value (p) mw z p DNA for xr232utest.dat using bits 31 to 32 141802 -.317 .3758 DNA for xr232utest.dat using bits 30 to 31 142393 1.427 .9232 DNA for xr232utest.dat using bits 29 to 30 141436 -1.396 .0813 DNA for xr232utest.dat using bits 28 to 29 141808 -.299 .3825 DNA for xr232utest.dat using bits 27 to 28 142313 1.191 .8831 DNA for xr232utest.dat using bits 26 to 27 141785 -.367 .3569 DNA for xr232utest.dat using bits 25 to 26 142385 1.403 .9197 DNA for xr232utest.dat using bits 24 to 25 142504 1.754 .9603 DNA for xr232utest.dat using bits 23 to 24 141777 -.390 .3481 DNA for xr232utest.dat using bits 22 to 23 142056 .433 .6674 DNA for xr232utest.dat using bits 21 to 22 142188 .822 .7945 DNA for xr232utest.dat using bits 20 to 21 142236 .964 .8324 DNA for xr232utest.dat using bits 19 to 20 142271 1.067 .8570 DNA for xr232utest.dat using bits 18 to 19 141440 -1.384 .0831 DNA for xr232utest.dat using bits 17 to 18 141619 -.856 .1959 DNA for xr232utest.dat using bits 16 to 17 142164 .751 .7737 DNA for xr232utest.dat using bits 15 to 16 142698 2.326 .9900 DNA for xr232utest.dat using bits 14 to 15 141837 -.213 .4155 DNA for xr232utest.dat using bits 13 to 14 141541 -1.087 .1386 DNA for xr232utest.dat using bits 12 to 13 142163 .748 .7729 DNA for xr232utest.dat using bits 11 to 12 141847 -.184 .4271 DNA for xr232utest.dat using bits 10 to 11 141557 -1.039 .1493 DNA for xr232utest.dat using bits 9 to 10 142292 1.129 .8705 DNA for xr232utest.dat using bits 8 to 9 142067 .465 .6791 DNA for xr232utest.dat using bits 7 to 8 141198 -2.098 .0179 DNA for xr232utest.dat using bits 6 to 7 141965 .164 .5652 DNA for xr232utest.dat using bits 5 to 6 141564 -1.019 .1542 DNA for xr232utest.dat using bits 4 to 5 141947 .111 .5442 DNA for xr232utest.dat using bits 3 to 4 141694 -.635 .2627 DNA for xr232utest.dat using bits 2 to 3 142010 .297 .6168 DNA for xr232utest.dat using bits 1 to 2 141991 .241 .5952 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the COUNT-THE-1's TEST on a stream of bytes. :: :: Consider the file under test as a stream of bytes (four per :: :: 32 bit integer). Each byte can contain from 0 to 8 1's, :: :: with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let :: :: the stream of bytes provide a string of overlapping 5-letter :: :: words, each "letter" taking values A,B,C,D,E. The letters are :: :: determined by the number of 1's in a byte:: 0,1,or 2 yield A,:: :: 3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus :: :: we have a monkey at a typewriter hitting five keys with vari- :: :: ous probabilities (37,56,70,56,37 over 256). There are 5^5 :: :: possible 5-letter words, and from a string of 256,000 (over- :: :: lapping) 5-letter words, counts are made on the frequencies :: :: for each word. The quadratic form in the weak inverse of :: :: the covariance matrix of the cell counts provides a chisquare :: :: test:: Q5-Q4, the difference of the naive Pearson sums of :: :: (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Test results for xr232utest.dat Chi-square with 5^5-5^4=2500 d.of f. for sample size:2560000 chisquare equiv normal p-value Results fo COUNT-THE-1's in successive bytes: byte stream for xr232utest.dat 2564.81 .916 .820295 byte stream for xr232utest.dat 2551.57 .729 .767097 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 2415.77 -1.191 .116779 bits 2 to 9 2626.16 1.784 .962803 bits 3 to 10 2426.79 -1.035 .150238 bits 4 to 11 2513.20 .187 .574060 bits 5 to 12 2586.91 1.229 .890482 bits 6 to 13 2554.41 .770 .779211 bits 7 to 14 2509.58 .136 .553913 bits 8 to 15 2633.35 1.886 .970346 bits 9 to 16 2522.82 .323 .626567 bits 10 to 17 2428.48 -1.011 .155917 bits 11 to 18 2522.76 .322 .626233 bits 12 to 19 2443.60 -.798 .212529 bits 13 to 20 2619.81 1.694 .954902 bits 14 to 21 2497.76 -.032 .487373 bits 15 to 22 2545.08 .638 .738110 bits 16 to 23 2561.51 .870 .807815 bits 17 to 24 2592.04 1.302 .903484 bits 18 to 25 2544.43 .628 .735087 bits 19 to 26 2434.24 -.930 .176203 bits 20 to 27 2382.40 -1.663 .048148 bits 21 to 28 2560.52 .856 .803986 bits 22 to 29 2461.32 -.547 .292179 bits 23 to 30 2463.13 -.521 .301018 bits 24 to 31 2469.44 -.432 .332807 bits 25 to 32 2526.70 .378 .647144 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: 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 xr232utest.dat Of 12,000 tries, the average no. of successes should be 3523 with sigma=21.9 Successes: 3552 z-score: 1.324 p-value: .907282 Successes: 3476 z-score: -2.146 p-value: .015932 Successes: 3522 z-score: -.046 p-value: .481790 Successes: 3515 z-score: -.365 p-value: .357445 Successes: 3500 z-score: -1.050 p-value: .146807 Successes: 3574 z-score: 2.329 p-value: .990064 Successes: 3566 z-score: 1.963 p-value: .975204 Successes: 3565 z-score: 1.918 p-value: .972432 Successes: 3545 z-score: 1.005 p-value: .842447 Successes: 3486 z-score: -1.689 p-value: .045562 square size avg. no. parked sample sigma 100. 3530.100 33.472 KSTEST for the above 10: p= .946442 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE MINIMUM DISTANCE TEST :: :: It does this 100 times:: choose n=8000 random points in a :: :: square of side 10000. Find d, the minimum distance between :: :: the (n^2-n)/2 pairs of points. If the points are truly inde- :: :: pendent uniform, then d^2, the square of the minimum distance :: :: should be (very close to) exponentially distributed with mean :: :: .995 . Thus 1-exp(-d^2/.995) should be uniform on [0,1) and :: :: a KSTEST on the resulting 100 values serves as a test of uni- :: :: formity for random points in the square. Test numbers=0 mod 5 :: :: are printed but the KSTEST is based on the full set of 100 :: :: random choices of 8000 points in the 10000x10000 square. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: This is the MINIMUM DISTANCE test for random integers in the file xr232utest.dat Sample no. d^2 avg equiv uni 5 .2954 1.1600 .256888 10 .5895 .7988 .447043 15 .0728 .8092 .070522 20 .6374 .8280 .473021 25 .7586 .8089 .533449 30 1.2466 .8598 .714308 35 .5947 .8664 .449922 40 3.2424 .9794 .961562 45 .7559 1.0153 .532194 50 .0296 .9858 .029334 55 .2579 .9335 .228337 60 .0547 .9774 .053494 65 .7562 .9855 .532311 70 .9705 .9743 .622949 75 .2959 .9302 .257257 80 .7805 .9427 .543615 85 .7496 .9083 .529215 90 3.2456 .9242 .961686 95 .1177 .9122 .111581 100 1.4335 .9507 .763237 MINIMUM DISTANCE TEST for xr232utest.dat Result of KS test on 20 transformed mindist^2's: p-value= .153708 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: THE 3DSPHERES TEST :: :: Choose 4000 random points in a cube of edge 1000. At each :: :: point, center a sphere large enough to reach the next closest :: :: point. Then the volume of the smallest such sphere is (very :: :: close to) exponentially distributed with mean 120pi/3. Thus :: :: the radius cubed is exponential with mean 30. (The mean is :: :: obtained by extensive simulation). The 3DSPHERES test gener- :: :: ates 4000 such spheres 20 times. Each min radius cubed leads :: :: to a uniform variable by means of 1-exp(-r^3/30.), then a :: :: KSTEST is done on the 20 p-values. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The 3DSPHERES test for file xr232utest.dat sample no: 1 r^3= 2.755 p-value= .08774 sample no: 2 r^3= 11.907 p-value= .32759 sample no: 3 r^3= 7.062 p-value= .20975 sample no: 4 r^3= 8.434 p-value= .24508 sample no: 5 r^3= 19.784 p-value= .48288 sample no: 6 r^3= 14.727 p-value= .38792 sample no: 7 r^3= 7.071 p-value= .20999 sample no: 8 r^3= 20.417 p-value= .49366 sample no: 9 r^3= 5.766 p-value= .17485 sample no: 10 r^3= 28.923 p-value= .61867 sample no: 11 r^3= 8.065 p-value= .23572 sample no: 12 r^3= 23.795 p-value= .54759 sample no: 13 r^3= 2.627 p-value= .08385 sample no: 14 r^3= 17.576 p-value= .44337 sample no: 15 r^3= 38.318 p-value= .72120 sample no: 16 r^3= 27.848 p-value= .60476 sample no: 17 r^3= 41.159 p-value= .74640 sample no: 18 r^3= 41.454 p-value= .74888 sample no: 19 r^3= 64.838 p-value= .88482 sample no: 20 r^3= 16.370 p-value= .42055 A KS test is applied to those 20 p-values. --------------------------------------------------------- 3DSPHERES test for file xr232utest.dat p-value= .672302 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the SQEEZE test :: :: Random integers are floated to get uniforms on [0,1). Start- :: :: ing with k=2^31=2147483647, the test finds j, the number of :: :: iterations necessary to reduce k to 1, using the reduction :: :: k=ceiling(k*U), with U provided by floating integers from :: :: the file being tested. Such j's are found 100,000 times, :: :: then counts for the number of times j was <=6,7,...,47,>=48 :: :: are used to provide a chi-square test for cell frequencies. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: RESULTS OF SQUEEZE TEST FOR xr232utest.dat Table of standardized frequency counts ( (obs-exp)/sqrt(exp) )^2 for j taking values <=6,7,8,...,47,>=48: 1.3 -.3 .6 .3 -1.2 -.7 -.9 -1.0 .3 .1 -.6 2.1 .5 -1.1 -.7 .2 .7 -1.1 .8 .5 .4 -.5 -.7 .5 .6 -.5 .0 -.9 .0 -.3 -.3 1.3 -.4 -.6 1.0 -.5 .0 -.4 1.3 -1.3 .1 -1.0 .8 Chi-square with 42 degrees of freedom: 27.105 z-score= -1.625 p-value= .036355 ______________________________________________________________ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: The OVERLAPPING SUMS test :: :: Integers are floated to get a sequence U(1),U(2),... of uni- :: :: form [0,1) variables. Then overlapping sums, :: :: S(1)=U(1)+...+U(100), S2=U(2)+...+U(101),... are formed. :: :: The S's are virtually normal with a certain covariance mat- :: :: rix. A linear transformation of the S's converts them to a :: :: sequence of independent standard normals, which are converted :: :: to uniform variables for a KSTEST. The p-values from ten :: :: KSTESTs are given still another KSTEST. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Test no. 1 p-value .162178 Test no. 2 p-value .214245 Test no. 3 p-value .297837 Test no. 4 p-value .861613 Test no. 5 p-value .016242 Test no. 6 p-value .608891 Test no. 7 p-value .963947 Test no. 8 p-value .599817 Test no. 9 p-value .988628 Test no. 10 p-value .761837 Results of the OSUM test for xr232utest.dat KSTEST on the above 10 p-values: .429538 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the RUNS test. It counts runs up, and runs down, :: :: in a sequence of uniform [0,1) variables, obtained by float- :: :: ing the 32-bit integers in the specified file. This example :: :: shows how runs are counted: .123,.357,.789,.425,.224,.416,.95:: :: contains an up-run of length 3, a down-run of length 2 and an :: :: up-run of (at least) 2, depending on the next values. The :: :: covariance matrices for the runs-up and runs-down are well :: :: known, leading to chisquare tests for quadratic forms in the :: :: weak inverses of the covariance matrices. Runs are counted :: :: for sequences of length 10,000. This is done ten times. Then :: :: repeated. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: The RUNS test for file xr232utest.dat Up and down runs in a sample of 10000 _________________________________________________ Run test for xr232utest.dat : runs up; ks test for 10 p's: .335159 runs down; ks test for 10 p's: .115278 Run test for xr232utest.dat : runs up; ks test for 10 p's: .399468 runs down; ks test for 10 p's: .987244 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: This is the CRAPS TEST. It plays 200,000 games of craps, finds:: :: the number of wins and the number of throws necessary to end :: :: each game. The number of wins should be (very close to) a :: :: normal with mean 200000p and variance 200000p(1-p), with :: :: p=244/495. Throws necessary to complete the game can vary :: :: from 1 to infinity, but counts for all>21 are lumped with 21. :: :: A chi-square test is made on the no.-of-throws cell counts. :: :: Each 32-bit integer from the test file provides the value for :: :: the throw of a die, by floating to [0,1), multiplying by 6 :: :: and taking 1 plus the integer part of the result. :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Results of craps test for xr232utest.dat No. of wins: Observed Expected 98705 98585.86 98705= No. of wins, z-score= .533 pvalue= .70294 Analysis of Throws-per-Game: Chisq= 12.52 for 20 degrees of freedom, p= .10291 Throws Observed Expected Chisq Sum 1 66827 66666.7 .386 .386 2 37760 37654.3 .297 .682 3 27020 26954.7 .158 .840 4 19168 19313.5 1.096 1.936 5 13734 13851.4 .995 2.931 6 9987 9943.5 .190 3.121 7 7191 7145.0 .296 3.417 8 5107 5139.1 .200 3.617 9 3610 3699.9 2.183 5.800 10 2671 2666.3 .008 5.808 11 1965 1923.3 .903 6.711 12 1373 1388.7 .178 6.889 13 952 1003.7 2.665 9.554 14 710 726.1 .359 9.913 15 513 525.8 .313 10.226 16 383 381.2 .009 10.235 17 268 276.5 .264 10.499 18 203 200.8 .023 10.522 19 163 146.0 1.983 12.505 20 106 106.2 .000 12.506 21 289 287.1 .012 12.518 SUMMARY FOR xr232utest.dat p-value for no. of wins: .702937 p-value for throws/game: .102910 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Results of DIEHARD battery of tests sent to file xr232utest.txt