American Journal of Biometrics & Biostatistics

1. American Journal of Biometrics & Biostatistics Mini Review Chakrabarty’s Tables of Random Numbers: A Comparison of Degree of Randomness with that of Some Existing Random Numbers Tables- Dhritikesh Chakrabarty* Department of Statistics, Handique Girls’ C ollege, Gauhati University *Address for Correspondence:Dhritikesh C hakrabarty, Department of Statistics, Handique Girls’ C ollege, Gauhati University, Guwahati-781001, Assam, India, Tel: +036-125-437-93/995-472-5639; E-mail: Submitted: 05 April 2018; Approved: 18 May 2018; Published: 19 May 2018 Cite this article: C hakrabarty D. C hakrabarty’s Tables of Random Numbers: a C omparison of Degree of Randomness with that of Some Existing Random Numbers Tables. American J Biom Biostat. 2018;2(1): 003-010. Copyright: © 2018 C hakrabarty D. This is an open access article distributed under the C reative C ommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2. American Journal of Biometrics & Biostatistics ABSTRACT A comparative study has been carried out on the degree of randomness of the two tables of random numbers due to Chakrabarty with the four tables of random numbers due to (1) Tippet, (2) Fisher & Yates, (3) Kendall & Smith and (4) Rand Corporation The degree of randomness has been examined on the basis of the signifi cance of difference between the observed frequencies and the corresponding expected frequencies of the 10 digits. Chi-square (χ2) test has been applied in examining the said signifi cance. This paper describes the examination of randomness of the six random numbers tables and a comparison of the degree of randomness of them. Keywords: Random numbers table; Tippet; Fisher & Yates; Kendall & Smith; Rand corporation; Chakrabarty; Degree of randomness; Chi-square test INTRODUCTION Th e scientifi c method of selecting a random sample, that has been found to be the vital or basic work in almost every branch of experimental sciences, consists of the use of random numbers table. Several tables of random numbers have already been constructed by the renowned scientists. Some of them are due to Fisher & Yates [1], Hald [2], Hill & Hill [3], Kendall & Smith [4], Mahalanobis [5], Manfred [6], Moses & Oakford [7], Quenouille [8], RAND Corporation [9], Rao, Mitra & Matthai [10], Rohlf & Sokal [11], Royo & Ferrer [12], Snedecor and Cochran [13], Tippett [14], etc. Recently, Chakrabarty [15,16] has constructed two tables of random numbers, one for random two-digit numbers [15] and the other for random three-digit numbers [16]. The procedure of the test includes the following steps 1. Compute the value of the chi-squared (χ2)test statistic which resembles a normalized sum of squares of the deviations between the observed frequencies and the corresponding theoretical frequencies. 2. Determine the degrees of freedom of that statistic, which is essentially the number of categories reduced by the number of parameters of the fi tted distribution. 3. Choose the level of signifi cance of the test. 4. Compare the observed (computed) value of χ2 to the corresponding theoretical value from the chi-squared distribution with degrees of freedom and the selected level of signifi cance. Fisher & Yates had selected the numbers from the 10th to 19th digits of A.S. Th ompson’s 20-fi gure logarithmic tables and recognized them as random numbers. In choosing from those digits, an element of randomness was introduced by using playing cards for the selection of half pages of the tables and of a column between 10th to 19th and fi nally for allotting these digits to the 50th place in a block. Th e method applied in selecting the numbers, by Fisher & Yates, creates a doubt on the randomness of the numbers generated. Th is leads to the necessity of examining the degree of randomness of the random numbers table constructed by Fisher and Yates. Similarly, there is also necessity of examining the degree of randomness of the tables of random numbers due to Tippet, Kendall & Smith and Rand Corporation respectively. In the meantime, some studies have been made on examining the degree of randomness of the four tables of random numbers due to Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation respectively [17-18]. In the current attempt, a comparative study has been carried out on the degree of randomness of the two tables of random numbers due to Chakrabarty with the four tables of random numbers due to [19] Tippet, [20] Fisher & Yates, [15] Kendall & Smith and [16] Rand Corporation. Th e degree of randomness has been examined on the basis of the signifi cance of diff erence between the observed frequencies and the corresponding expected frequencies of the 10 digits. Chi-square (χ2) test has been applied in examining the said signifi cance. Th is paper describes the examination of randomness of the six random numbers tables and a comparison of the degree of randomness of them. THE TEST STATISTIC USED Th e Chi-square statistic test innovated by Pearson [1,2,8,9,12,21- 27] which is a test statistic used for testing of the goodness of fi t, has suitably been applied here in testing the degree of randomness of table of random numbers. A test of goodness of fi t establishes whether or not an observed frequency distribution diff ers from a theoretical distribution. 5. Reject the null hypothesis that the diff erence between the observed frequency and the theoretical frequency is insignifi cant based on whether observed (computed) value of χ2 exceeds the corresponding theoretical value of χ2. The chi-squared test is based on the following assumptions Assumptions-1 (Randomness of data): Th e sampled data are assumed to be drawn by a random sampling from a fi xed distribution or population. Assumptions-2 (Adequate sample size): Th e size of the sample is assumed to be suffi ciently large. Assumptions-3 corresponding to each cell requires 5 or more. When this assumption is not met, Yates’s correction [27,14] is applied. (Adequate cell frequency): Frequency Assumptions-4 (Independence): Th e observations are assumed to be independent of each other. Chi-square statistic for the current study Th e random numbers tables under study are as follows: (1) Tippet’s Random Numbers Table [14]: Th is table consists of 10400 four-digit numbers giving in all 41600 single digits. (2) Fisher & Yates Random Numbers Table [1]: Th is table comprises 7500 random two-digit numbers giving in all 15000 single digits. (3) Kendall and Smith’s Random Numbers Table [4]: Th is table consists of 25000 random four- digit numbers giving in all 100000 single digits. SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -004

3. American Journal of Biometrics & Biostatistics Table-1: Observed value of χ2 Statistic of Random Numbers Tables. Value of χ2 of Table of Random Numbers due to Fisher & Yates & Smith 100 3.4 15.4 5.6 200 7.9 17.7 13.4 300 8.1923 18.386 7.265 400 11.25 13.1 9.75 500 14.8 6.28 10.16 600 15.6017 10.008 12.769 700 8.7705 6.55 11.687 800 5.8 6.75 11.226 900 10.5318 6.189 10.267 1000 6.42 5.48 7.84 1100 11.398 7.95 9.417 1200 13.1479 7.076 7.206 1300 10.9994 8.682 6.7687 1400 10.0284 7.995 5.643 1500 9.9269 7.644 7.026 1600 8.8126 7.961 5.174 1700 9.5345 8.2808 4.882 1800 9.398 8.702 4.167 1900 9.4842 7.714 3.147 2000 8.4 9.97 2.43 2100 6.9248 9.014 2.04 2200 4.5816 8.297 2.583 2300 4.3787 8.994 2.565 2400 4.0041 10.2405 4.225 2500 3.664 11.104 3.768 2600 5.3768 9.9448 3.7144 2700 4.2889 8.541 4.628 2800 4.6569 6.999 2.971 2900 5.7929 4.538 3.00 3000 5.6331 6.604 2.878 3100 6.0767 4.647 2.687 3200 6.4124 3.884 2.887 3300 6.7509 4.066 2.077 3400 6.8759 4.6707 2.695 3500 8.5087 4.9755 2.69 3600 6.6126 3.7949 3.071 3700 7.2001 3.358 2,944 3800 7.5998 3.1901 3.201 3900 7.3331 4.2056 4.498 4000 8.335 4.9175 4.4715 4100 8.4865 4.4239 4.312 4200 8.2709 3.8658 4.262 4300 7.7256 4.2776 5.1503 4400 10.6277 4.0041 5.2983 4500 9.53 3.7807 5.858 4600 8.027 7.2959 5.321 4700 9.932 2.4961 4.077 4800 10.103 3.245 4.013 4900 11.369 2.898 5.245 5000 11.392 2.7952 5.484 5100 10.5675 3.589 5.783 5200 11.1768 5.777 6.501 5300 11.4629 3.6998 5.412 5400 10.5303 3.8445 5.036 5500 10.4275 3.6058 5.388 5600 10.5318 5.0036 5.462 5700 11.3367 5.7851 4.681 5800 15.814 6.3243 4.092 5900 10.135 7.0341 4.307 6000 10.805 6.688 5.362 6100 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10000 10100 10200 10300 10400 10500 10600 10700 10800 10900 11000 11100 11200 11300 11400 11500 11600 11700 11800 11900 12000 12100 12200 12300 12400 9.8492 9.5807 9.4377 10.669 9.3656 10.458 10.372 8.3655 8.6899 8.0287 7.8962 8.6185 8.9613 8.9986 8.333 9.1746 9.699 8.7332 9.2667 8.7695 7.3679 8.251 7.6912 7.567 6.9876 7.2584 7.7045 7.4427 7.896 7.8026 7.4145 8.5159 8.1087 9.3894 10.404 9.7165 9.9068 9.8328 8.9808 8.296 9.2524 8.2079 8.0387 8.1328 8.591 9.4579 9.9099 9.4313 8.2084 6.5808 6.6414 5.848 5.5923 5.0608 4.484 4.869 5.0943 4.536 4.531 4.3896 5.287 5.052 5.073 5.608 7.3731 7.4613 7.3573 7.279 6.636 6.2479 5.7576 3.5674 5.406 3.4273 5.084 5.203 3.047 5.1216 6.071 5.544 4.8017 6.1348 5.2715 5.504 4.8868 4.6364 4.1867 3.5018 3.072 3.2025 3.7743 4.5071 3.9144 4.248 4.2746 4.4044 5.6684 5.813 5.8473 5.9722 4.8202 5.7298 5.0662 5.432 4.992 4.8175 4.4821 3.8945 3.872 4.0281 4.677 4.594 4.203 3.9865 4.3925 4.651 4.5079 4.0389 3.8884 4.2668 5.293 6.614 5.7344 5.671 5.4714 6.1468 5.469 7.8212 6.423 6.395 5.806 6.675 6.774 6.649 7.77 8.027 6.892 6.206 7.088 6.621 6.239 6.565 6.193 6.524 7.528 7.336 7.863 7.771 6.998 6.919 6.461 6.956 8.74 9.522 9.523 10.568 9.971 9.841 10.378 10.162 9.965 9.531 9.404 9.039 9.421 9.067 9.419 9.391 9.03 9.682 10.189 9.105 8.75 8.944 8.066 7.659 8.427 8.571 8.451 9.253 8.567 7.441 7.708 8.216 8.937 8.396 7.337 6.824 7.427 7.308 7.215 6.288 5.587 7.04 6.821 5.381 6.681 5.907 5.647 5.77 6.889 7.066 5.961 5.696 4.827 5.289 4.659 5.51 5.732 5.661 6.885 6.774 6.72 7.784 7.071 6.095 6.519 7.955 8.123 8.805 7.345 6.473 6.729 7,748 8.078 7.944 7.761 9.219 9.074 9.615 9.217 8.442 9.132 8.45 8.669 9.223 8.702 8.935 8.911 9.49 9.615 8.815 8.933 9.187 9.346 9.599 9.912 10.023 9.838 9.102 9.321 9.573 10.79 9.612 9.704 8.875 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Number of Digits (1st) Kendall Rand Chakrabarty (two-digit) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Chakrabarty (three-digit) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Tippett Corporation 5.7 11.6 9.52 6.416 8.52 6.772 6.285 7.202 7.967 10.36 7.289 6.014 6.614 9.729 7.815 6.961 8.858 8.798 7.62 7.81 6.905 6.138 5.251 4.842 4.976 5.13 5.295 5.871 5.889 6.544 6.157 6.363 3.829 3.954 3.206 2.832 0.024 2.395 2.303 1.738 1.668 2.484 1.962 1.394 1.848 2.288 2.267 3.216 2.726 3.136 2.709 2.776 3.558 3.457 3.874 3.933 3.841 4.438 4.608 4.843 SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -005

4. American Journal of Biometrics & Biostatistics 12500 12600 12700 12800 12900 13000 13100 13200 13300 13400 13500 13600 13700 13800 13900 14000 14100 14200 14300 14400 14500 14600 14700 14800 14900 15000 15100 15200 15300 15400 15500 15600 15700 15800 15900 16000 16100 16200 16300 16400 16500 16600 16700 16800 16900 17000 17100 17200 17300 17400 17500 17600 17700 17800 17900 18000 18100 18200 18300 18400 18500 18600 18700 18800 6.32 6.7166 7.1556 7.4749 7.1319 6.3431 6.0259 5.5684 5.7254 6.3911 11.8595 4.9459 4.7411 3.5648 4.885 12.3586 8.2736 4.9539 4.4883 4.3462 4.1505 6.1549 4.4307 4.1109 3.449 3.449 3.1974 3.094 3.617 3.3507 3.3773 3.3915 2.994 3.3114 2.3523 3.182 3.2736 2.9751 2.3912 2.7744 2.5864 2.4243 2.2081 2.321 2,5904 1.8203 1.885 1.5281 1.5152 1.6973 1.6859 1.537 1.546 1.6856 1.546 2.3487 1.8919 2.282 2.313 2.549 2.4539 2.3757 2.4022 2.6152 2.157 8.2478 8.509 8.8626 8.3969 8.9979 9.469 8.8581 9.664 9.766 8.4756 9.497 6.5339 9.3647 8.817 8.5059 6.32 6.30 6.485 7.283 8.566 7.529 7.889 7.999 8.239 7.153 6,262 6.574 6.998 6.755 6.64 5.635 5.127 4.769 4.736 4.918 5.397 5.412 5.507 5.161 5.246 5.111 5.608 5.289 4.314 4.554 4.276 3.483 3.261 3.445 3.39 3.732 3.492 3.327 2.836 3.04 3.171 3.562 3.433 3.137 3.536 3.955 3.93 4.226 4.004 4.045 4.129 4.501 4.771 4.868 4.318 4.837 5.262 5.083 4.958 4.589 4.312 3.887 3.72 3.812 9.192 9.843 9.517 8.926 8.926 8.333 8.057 8.389 8.849 10.001 9.787 10.303 10.039 9.542 9.567 9.076 9.158 9.466 8.419 7.866 7.696 8.59 8.01 7.612 8.008 8.463 7.927 7.82 7.388 6.506 6.058 7.094 7.338 7.343 6.839 6.88 6.991 6.323 6.673 6.565 6.777 7.668 7.3682 7.319 7.578 7.637 7.773 8.28 7.513 6.96 6.765 6.758 7.219 7.76 7.859 7.851 8.082 8.06 8.19 7.977 7.446 7.512 6.953 7.09 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18900 19000 19100 19200 19300 19400 19500 19600 19700 19800 19900 20000 20100 20200 20300 20400 20500 20600 20700 20800 20900 21000 21100 21200 21300 21400 21500 21600 21700 21800 21900 22000 22100 22200 22300 22400 22500 22600 22700 22800 22900 23000 23100 23200 23300 23400 23500 23600 23700 23800 23900 24000 24100 24200 24300 24400 24500 24600 24700 24800 24900 25000 25100 25200 2.656 3.2563 2.377 2.714 2.468 2.802 3.138 2.989 2.864 3.0296 2.912 2.798 3.203 3.637 3.8863 3.65 3.789 3.817 3.759 4.119 4.388 4.692 3.752 4.1596 4.0916 3.848 3.926 3.724 5.021 3.382 3.224 3.3899 3.5105 3.1577 3.5166 3.5672 3.3201 3.0983 2.9388 3.153 2.9298 2.9128 2.574 2.5741 2.77129 2.3837 2.4918 2.2985 2.2616 2.4785 2.274 2.2924 2.326 2.2385 2.162 2.3401 2.7055 2.821 2.8491 3.5152 2.6747 3.3152 3.8342 3.9595 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 4.079 3.855 3.852 4.476 3.652 3.791 3.256 3.921 3.474 3.346 3.454 3.203 3.061 3.582 3.714 3.89 3.769 3.629 3.657 3.635 3.84 3.634 3.978 4.074 3.962 3.96 3.824 3.607 3.65 3.445 3.401 3.577 3.576 4.393 4.498 3.791 4.632 3.832 3.957 4.306 5.351 3.543 3.185 3.669 3.833 3.649 3.713 3.57 3.52 3.734 4.15 4.169 3.766 3.858 3.968 4.298 4.698 4.556 4.24 3.772 3.961 4.32 4.372 4.155 6.965 5.575 7.087 7.228 7.469 7.755 8.146 8.114 7.868 8.371 8.712 8.232 7.638 8.312 7.908 7.932 6.835 7.956 7.706 7.956 8.532 9.224 9.031 9.387 9.174 9.5325 8.824 8.916 8.03 8.3197 8.504 8.416 8.729 9.111 9.601 9.256 8.53 8.742 9.347 8.844 8.904 9.227 9.44 9.958 10.081 10.334 10.214 10.232 9.951 9.931 10.213 10.257 10.613 10.798 9.958 10.019 10.613 10.778 11.301 11.589 11.111 10.604 10.405 11.133 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8.566 9.5832 10.449 10.526 10.192 9.941 9.987 9.725 8.614 26.118 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -006

5. American Journal of Biometrics & Biostatistics 25300 25400 25500 25600 25700 25800 25900 26000 26100 26200 26300 26400 26500 26600 26700 26800 26900 27000 27100 27200 27300 27400 27500 27600 27700 27800 27900 28000 28100 28200 28300 28400 28500 28600 28700 28800 28900 29000 29100 29200 29300 29400 29500 29600 29700 29800 29900 30000 30100 30200 30300 30400 30500 30600 30700 30800 30900 31000 31100 31200 31300 31400 31500 31600 4.098 4.3545 4.1754 3.921 4.3292 4.5341 4.3937 4.4678 4.4131 4.6464 4.7937 4.7704 4.7163 4.7927 4.6195 4.5859 4.5969 4.9096 4.9112 4.4823 4.667 4.8887 4.727 4.7702 4.7248 4.8596 4.7382 4.7695 4.4433 4.5722 3.9653 3.9282 4.0976 4.4385 5.487 4.1992 4.258 4.6386 4.3189 4.1495 4.6732 4.6489 5.2756 4.9919 4.9808 5.2645 5.4562 6.0744 5.9296 6.1662 5.8754 6.4819 6.1508 6.3709 6.3003 6.5163 6.7915 6.4007 6.6209 6.5731 6.6937 6.8152 6.901 6.7705 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 3.943 3.903 3.77 3.475 3.281 3.35 3.138 3.16 3.164 3.23 2.966 2.833 2.582 3.012 3.044 2.878 2.69 2.466 2.265 2.534 2.4009 2.554 2.558 2.65 2.60 2.66 2.92 2.496 2.742 2.797 2.737 2.721 2.929 2.989 3.188 3.036 3.071 3.387 3.46 3.45 3.88 3.537 3.427 3.61 3.612 3.662 3.428 3.32 3.396 3.773 3.936 4.124 4.082 4.433 4.656 4.51 4.206 4.048 4.028 4.26 4.209 4.2 4.47 4.224 11.399 11.031 11.336 11.474 10.904 10.981 10.536 11.166 11.307 9.703 10.454 9.120 8.994 9.041 9.417 10.228 10.544 10.378 10.433 10.653 10.594 9.15 9.365 9.472 9.817 9.685 9.44 9.274 9.67 9.464 9.512 9.413 9.738 10.333 10.829 10.354 9.647 9.335 9.494 9.82 9.217 9.676 9.363 9.431 9.841 9.973 10.743 9.997 11.192 11.337 11.387 11.942 12.226 12.248 12.03 11.722 11.464 12.093 12.253 11.696 11.715 11.779 11.965 11.854 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31700 31800 31900 32000 32100 32200 32300 32400 32500 32600 32700 32800 32900 33000 33100 33200 33300 33400 33500 33600 33700 33800 33900 34000 34100 34200 34300 34400 34500 34600 34700 34800 34900 35000 35100 35200 35300 35400 35500 35600 35700 35800 35900 36000 36100 36200 36300 36400 36500 36600 36700 36800 36900 37000 37100 37200 37300 37400 37500 37600 37700 37800 37900 38000 6.9166 6.307 6.7621 6.1931 6.0795 6.0855 6.0664 5.8082 6.07 6.257 6.2384 6.4827 6.0148 6.5653 6.6835 7.1698 7.7898 7.9719 7.7937 8.5374 8.9082 9.5029 9.2077 9.4735 9.0944 8.7398 8.6688 8.8238 8.7549 9.7251 9.9682 9.6288 10.2195 9.9896 10.6078 9.9749 9.8425 8.1661 10.2705 10.3578 10.2734 10.2379 10.0995 9.7082 36100 36200 36300 36400 36500 36600 36700 36800 36900 37000 37100 37200 37300 37400 37500 37600 37700 37800 37900 38000 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 4.634 5.018 5.05 4.709 4.987 4.573 4.93 5.061 4.87 4.766 4.927 5.44 5.315 4.881 5.213 5.588 5.998 4.898 5.328 5.384 5.127 5.056 5.382 4.88 5.014 4.962 5.245 5.517 5.094 5.408 5.875 6.093 6.172 6.114 6.065 5.793 5.662 5.487 6.06 6.084 5.928 5.734 5.832 5.516 5.362 5.592 5.763 6.084 5.674 5.722 5.505 5.522 5.726 5.853 6.11 5.94 6.06 6.401 6.129 5.90 6.15 5.884 6.371 6.317 11.576 11.642 10.801 10.67 10.111 9.891 9.596 9.681 9.719 9.27 9.129 9.113 9.608 9.636 9.82 9.586 9.906 10.065 9.973 10.017 9.799 10.061 10.367 9.546 9.628 9.559 9.909 9.438 9.444 9.233 10.254 12.518 9.902 9.423 9.44 10.023 10.224 10.068 10.57 9.935 10.063 9.848 10.104 10.257 10.095 9.434 9.136 9.053 8.968 8.919 8.945 8.943 8.72 8.437 8.035 8.167 7.672 7.923 7.614 7.731 7.524 7.375 7.827 7.531 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -007

6. American Journal of Biometrics & Biostatistics 38100 38200 38300 38400 38500 38600 38700 38800 38900 39000 39100 39200 39300 39400 39500 39600 39700 39800 39900 40000 40100 40200 40300 40400 40500 40600 40700 40800 40900 41000 41100 41200 41300 41400 41500 41600 41700 41800 41900 42000 42100 42200 42300 42400 42500 42600 42700 42800 42900 43000 43100 43200 43300 43400 43500 43600 43700 43800 43900 44000 44100 44200 44300 44400 38100 38200 38300 38400 38500 38600 38700 38800 38900 39000 39100 39200 39300 39400 39500 39600 39700 39800 39900 40000 8.6943 8.6632 8.921 9.0153 9.100 8.8044 9.3351 9.1598 9.5824 9.3375 9.5207 9.866 9.9955 9.9817 10.1614 10.3015 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 6.34 6.838 6.757 6.731 6.871 6.911 7.025 6.686 6.98 5.942 6.994 7.599 7.774 7.779 7.616 7.837 7.965 7.967 7.868 7.829 7.914 7.973 8.182 8.318 8.229 8.574 8.469 7.919 7.80 8.034 8.636 8.604 8.992 9.303 9.694 10.04 9.605 9.391 9.021 8.861 8.972 9.014 8.921 8.714 8.756 8.724 8.824 8.627 8.536 8.514 8.66 8.23 8.259 8.465 8.854 8.663 9.197 8.634 8.717 8.862 8.854 8.695 8.88 8.656 7.327 7.632 7.545 7.403 7.585 7.509 7.57 7.787 7.519 7.379 7.446 7.276 7.42 6.752 7.282 8.175 6.732 6.878 7.026 7.251 7.315 7.405 7.648 7.997 8.034 7.909 8.276 8.359 8.029 7.836 7.529 7.527 7.322 7.695 7.442 7.106 7.535 7.656 7.665 7.574 7.691 7.817 7.921 7.77 7.781 7.802 7.383 7.796 7.362 7.262 7.239 7.171 7.469 7.48 7.465 7.796 7.455 7.309 6.981 6.959 7.127 7.052 7.046 7.177 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 44500 44600 44700 44800 44900 45000 45100 45200 45300 45400 45500 45600 45700 45800 45900 46000 46100 46200 46300 46400 46500 46600 46700 46800 46900 47000 47100 47200 47300 47400 47500 47600 47700 47800 47900 48000 48100 48200 48300 48400 48500 48600 48700 48800 48900 49000 49100 49200 49300 49400 49500 49600 49700 49800 49900 50000 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 8.769 9.059 9.463 9.522 9.91 10.16 10.272 10.38 10.18 10.378 10.54 10.66 9.959 9.52 10.004 9.91 9.95 9.161 9.416 9.42 9.368 9.598 9.94 9.372 9.555 9.754 9.147 9.308 9.844 9.473 9.279 10.207 10.692 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ 7.375 7.228 7.286 7.615 8.005 7.693 7.905 7.394 7.458 7.441 7.551 7.417 7.236 7.72 7.352 7.382 7.829 7.649 7.395 7.535 7.145 7.274 7.194 7.319 7.414 7.149 7.431 7.537 7.71 7.826 7.469 7.208 7.537 7.77 7.856 8.435 7.13 7.265 7.42 7.539 7.868 7.498 7.402 7.248 8.189 6.433 6.542 7.339 7.597 7.824 7.779 7.881 7.927 7.623 7.576 7.271 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (4) Random Numbers Table by Rand Corporation [9]: Th is table consists of one million single digits consisting of 200000 random numbers of fi ve-digits each. (5) Chakrabarty’s Table of Random Two-digit Numbers [15]: Th is table consists of 10000 random SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -008

7. American Journal of Biometrics & Biostatistics Th e observed values of Chi-square Statistic obtained have been shown in (Table-1). FINDINGS AND DISCUSSIONS Th e following observations/fi ndings have been obtained: Table-2: Name of the Random Numbers Table Rank with respect to the Degree of Randomness Due to Fisher & Yates 5 Due to Tippet 4 (1) In the case of Fisher & Yates Random Numbers Table, the highest observed value of chi-square with 9 degrees of freedom is 26.118. Th e theoretical value of chi-square with 9 degrees of freedom at 5% & 1% level of signifi cance are 16.919 & 21.666 respectively. Th us, the lack of randomness of Fisher & Yates Random Numbers Table can be regarded as signifi cant not only at 5% level of signifi cance but also at 1% level of signifi cance. However, the observed value of chi-square corresponds to its theoretical value at 0.055% level of signifi cance. Th us, the lack of randomness of Fisher & Yates Random Numbers Table can be regarded as insignifi cant up to 0.055% level of signifi cance and signifi cant at the level of signifi cance < 0.055% . Due to Kendall & Smith 3 Due to Rand Corporation 2 Due to Chakrabarty (both the tables) 1 two-digit numbers giving in all 20000 single digits. (6) Chakrabarty’s Table of Random Th ree-digit Numbers [16]: Th is table consists of 20000 random three-digit numbers giving in all 60000 single digits. Let us consider Fisher and Yates random number table. Th is table consists of a total of 15000 single digits comprising of 7500 two-digit numbers viz. (2) In the case of Tippet’s Random Numbers Table, the highest observed value of chi-square with 9 degrees of freedom is 15.814. Th e theoretical value of chi-square with 9 degrees of freedom at 5% level of signifi cance is 16.919. Th us, the lack of randomness of Tippet’s Random Numbers Table can be regarded as insignifi cant at 5% level of signifi cance. However, the observed value of chi-square corresponds to its theoretical value at 7.5%level of signifi cance. Th us, the lack of randomness of Tippet’s Random Numbers Table can be regarded as insignifi cant up to 7.5% level of signifi cance and signifi cant at the level of signifi cance < 7.5%. 00 , 01 , 02 , ........…… , 98 , 99 Th e test required in this study is to test whether the occurrences of the numbers appeared in the table is random. Th is is equivalent to a test to looked to make sure that equal numbers of 0s, 1s, 2s, 3s, ………..., 9s are present in the table. Let the number of occurrences of the ten digits in the table be N.. Let (3) In the case of Kendall & Smith’s Random Numbers Table, the highest observed value of chi-square with 9 degrees of freedom is 13.4 which is less than the corresponding theoretical value of chi-square at 5% level of signifi cance. Th us, the lack of randomness of Kendall & Smith’s Random Numbers Table can be regarded as insignifi cant at 5% level of signifi cance. However, the observed value of chi-square with 9 degrees of freedom namely 13.4 corresponds to the theoretical value of chi-square with 9 degrees of freedom at 18.1% level of signifi cance. Th us, the lack of randomness of Kendall & Smith’s Random Numbers Table can be regarded as insignifi cant up to 18.1% level of signifi cance and signifi cant at the level of signifi cance < 18.1%. = Observed frequency of the digit i = Expected frequency of the digit i & (i = 0 , 1 , 2 , ........... , 9) among the N occurrences. Th en the χ2 statistic for testing the null hypothesis “the occurrences of the digits in the table is random” i.e. “each digit has the probability 0.1 to occur in any position” which is equivalent to testing of the null hypothesis that (4) In the case of Random Numbers Table due to Rand Corporation, the highest observed value of chi-square with 9 degrees of freedom is 12.518 which is less than the corresponding theoretical value of chi-square at 5% level of signifi cance. Th us, the lack of randomness of Random Numbers Table due to Rand Corporation can be regarded as insignifi cant at 5% level of signifi cance. However, the observed value of chi-square with 9 degrees of freedom namely 12.518 corresponds to the theoretical value of chi-square with 9 degrees of freedom at 24% level of signifi cance. Th us, the lack of randomness of Random Numbers Table due to Rand Corporation can be regarded as insignifi cant up to 24% level of signifi cance and signifi cant at the level of signifi cance < 24%. “the discrepancy between the observed frequencies and the corresponding expected frequencies of the digits is insignifi cant” is 2   O E   i i  χ2 = E i   which follows χ2 distribution with 9 degrees of freedom. Th is statistic can be applied to test the randomness of the whole table or of any part of the table. Th is statistic can similarly be applied in testing of the randomness of the other fi ve tables. (5) In the case of Random Numbers Table (of two-digit numbers) due to Chakrabarty, the observed values of chi-square have been found to be 0. Th us, there is no lack of randomness in this table and thus the table can be treated as properly random. It is mentioned here that the assumptions under which chi- square test is applicable hold good in the case of the occurrences of the numbers in each of the four tables. In the case of each of the six tables of random numbers, frequency test has been applied to (6) In the case of Random Numbers Table (of three-digit numbers) due to Chakrabarty, the observed values of chi-square have also been found to be 0. Th us, there is no lack of randomness in this table and thus this table also can be treated as properly random. 1st 100, 1st 200, 1st 300, ......................., 1st 15000 , ……………………. occurrences of the ten digits in the table. SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -009

8. American Journal of Biometrics & Biostatistics From the fi ndings obtained, one can conclude that the degree of the lack of randomness is highest (in other words, the degree of randomness is lowest) in the Fisher & Yates Random Numbers Table among the six tables of random numbers examined. On the other hand, the degree of the lack of randomness is lowest (in other words, the degree of randomness is highest) in the two Random Numbers Tables due to Chakrabarty among the six tables of random numbers examined. Th e six tables can be ranked with respect to the degree of randomness as follows: 13. Snedecor GW, Cochran WG. Statistical methods. Ames: Iowa state university press; 6th Edition. 1967. p593. https://goo.gl/Eet5NE 14. Tippett LHC. Random sampling numbers. England: Cambridge university press; 1927. p15. 15. Chakrabarty dhritikesh. One more table of random three-digit numbers. International journal of advanced research in science, engineering and technology. 2016; 3: 1667 - 1678. 16. Chakrabarty dhritikesh. One more table of random three-digit numbers. International journal of advanced research in science, engineering and technology. 2016; 3: 1851-1869. Th e fi ndings obtained are based on frequency test using chi- square statistic. However, there exist some other methods of testing of randomness. Th ere is necessity of obtaining more confi dence on the fi ndings by applying the other methods of testing of randomness. Th us one problem for researcher, at this stage, is to make attempt of studying the degree of randomness of these six tables of random numbers by the other testing methods. REFERENCES 17. Chakrabarty Dhritikesh & Sarmah Brajendra Kanta. Comparison of degree of randomness of the tables of random numbers due. Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation. Journal of reliability and statistical studies. 2017; 10: 27- 42. https://goo.gl/fuLvAf 18. Chakrabarty Dhritikesh. Random numbers tables due to tippet, fi sher & Yates, kendall & smith and Rand Corporation: comparison of degree of randomness by run test. Journal of biostatistics and biometric applications. 2018; 3: 1-7. https://goo.gl/QCtQKq 19. Bagdonavicius V, Nikulin MS. Chi-squared goodness of fi t test for right censored data. The international journal of applied mathematics & statistics. 2011; 24: 30-50. https://goo.gl/Wjp5HQ 1. Fisher R A, Yates F. Statistical tables for biological, agricultural and medical research. Longman Group Limited; England. 6th Edition. 1982. 37-38 & 134- 139. 20. Bradely James V. Distribution free statistical tests. First edition, Prentice Hall. 1968. 2. Hald A. Statistical tables and formulas. Wiley. 1952. https://goo.gl/JEhaAT 3. Hill I D, Hill P A. Tables of Random Times. 1977. 21. Chakrabarty Dhritikesh. Deviation Test: comparison of degree of randomness of the tables of random numbers due to tippet, fi sher & yates, kendall & smith and rand corporation. SM Journal of biometrics & biostatistics. 2017; 2: 1014. https://goo.gl/mJfvAK 4. Kendall MG, Smith BB. Randomness and random sampling numbers. Journal of the royal statistical society.1938; 101: 147-166. https://goo.gl/QJZYy4 5. Mahalanobis population. Sankya: The indian journal of statistic. 1934; 1: 289-328. https://goo.gl/f2CEPp PC. Tables of random samples from a normal 22. Chernoff H, Lehmann E L. The use of maximum likelihood estimates in χ2 tests for goodness of fi t. The annuls of mathematical statistics. 1954; 25: 579- 586. https://goo.gl/hk2mQB 6. Manfred Mohr. The little book of numbers at hasar, Artist Edition, Paris. 1971. 23. Corder GW, Foreman DI. Nonparametric statistics: a step-by-step approach. Wiley. 2014. https://goo.gl/1NZVnS 7. Moses EL, Oakford VR. Tables of random permutations. George Allen & Unwin. 1963. 24. Greenwood PE, Nikulin MS. A guide to chi-squared testing. Wiley. 1996. https://goo.gl/FngAmn 8. Quenouille MH. Tables of random observations from standard distributions. Biometrika. 1959; 46: 178-202. https://goo.gl/ESVSqR 25. Kendall MG, Smith BB. A table of random sampling numbers. England: University press; 1939. P 24. https://goo.gl/8Xin7n 9. Rand. Corporation a million random digits. Free Press, Glenoe. III. 1955. 10. Rao C R, Mitra S K, Matthai A. random numbers and permutations. Statistical publishing society. 1966. 26. Plackett RL. Karl pearson and the chi-squared test. International statistical review. 1983; 51: 59-72. https://goo.gl/N3oFzH 11. Rohlf F J, Sokal R R. Statistical tables. Freeman. 1969. https://goo.gl/Z7G2pu 27. F Yates. Contingency table involving small numbers and the chi-square test. Supplement to the journal of the royal statistical society. 1934; 1: 217- 235. https://goo.gl/mDhyMJ 12. Royo J, Ferrer S. Tables of random numbers obtained from numbers in the spanish national lottery. Trabajos de estadistica. 1954; 5: 247- 256. SCIRES Literature - Volume 2 Issue 1 - www.scireslit.com Page -010