Journal of Biostatistics and Biometric Applications Volume 3 | Issue 1 Research Article
ISSN: 2348-9820 Open Access
Random Numbers Tables Due to Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation: Comparison of Degree of Randomness by Run Test Chakrabarty D* Department of Statistics, Handique Girls’ College, Gauhati University, Guwahati, Assam, India
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Corresponding author: Chakrabarty D, Department of Statistics, Handique Girls’ College, Gauhati University, Guwahati-781001, Assam, India, E-mail:
[email protected] *
Citation: Chakrabarty D (2018) Random Numbers Tables Due to Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation: Comparison of Degree of Randomness by Run Test. J Biostat Biometric App 3 (1): 103
Abstract The randomness of each of the four tables of random numbers namely (1) Tippet’s Random Numbers Table, (2) Fisher & Yates Random Numbers Table, (3) Kendall and Smith's Random Numbers Table and (4) Random Numbers Table due to Rand Corporation has been examined and a comparison of the merits of them has been studied with respect to the degree of randomness. Run test has been applied in examining the randomness of each of the four tables. This paper describes the testing of randomness of the four random numbers tables and a comparison of the degree of randomness of them. Keywords: Random numbers table; Tippet; Fisher & Yates; Kendall & Smith; Rand Corporation; Testing of randomness; Run test
Introduction The scientific 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 [1-28]. Some of them (in chronological order) are due to Tippett [28], Mahalanobis [9], Kendall & Smith [7,8], Fisher & Yates [4], Hald [5], Royo & Ferrer [16], RAND Corporation [13], Quenouille [12], Moses & Oakford [11], Rao, Mitra & Matthai [14], Snedecor and Cochran [27], Rohlf & Sokal [15], Manfred [19], Hill & Hill [6] etc. Among these tables, the four tables namely (1) Tippet’s Random Numbers Table that consists of 10,400 four-digit numbers, (2) Fisher & Yates Random Numbers Table that comprises 7500 two-digit numbers, (3) Kendall and Smith's Random Numbers Table & (4) Random Numbers Table by Rand Corporation are widely used in drawing of simple random sample (with or without replacement) from a population. Fisher & Yates obtained the random numbers from the 10th to 19th digits of A.S. Thompson's 20-figure logarithmic tables. 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 finally for allotting these digits to the 50th place in a block. In this case, the question arises whether the method applied in selecting the numbers has made the numbers random. This creates the necessity of determining the degree of randomness of the random numbers table constructed by Fisher and. Similarly, there is necessity of examining the degree of randomness of the other tables of random numbers. In the meantime, several 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 [17-28]. There is also necessity of study of comparison of the degree of randomness of the tables. For this reason, two separate studies [2,3] have already been done on the comparison of degree of randomness of the four tables. However, the studies have been done by the application of frequency test (based on chi-square statistic) [3] and deviation test (based on t statistic) [2]. The findings obtained in these two studies will be more trustable if the same findings are obtained in another study on the same objective with the same data but by the application of another suitable test statistic. Accordingly, an attempt has here been made on the same study by another suitable test statistic. The randomness of each of the four tables of random numbers namely (1) Tippet’s Random Numbers Table, (2) Fisher & Yates Random Numbers Table, (3) Kendall and Smith's Random Numbers Table and (4) Random Numbers Table due to Rand Corporation has been examined and a comparison of the merits of them has been studied with respect to the degree of randomness. Run test has been applied in examining the Annex Publishers | www.annexpublishers.com
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randomness of each of the four tables. This paper describes the testing of randomness of the four random numbers tables and a comparison of the degree of randomness of them.
The Test Statistic Based on Run Test The run test (Bradley, 1968) is a non-parametric statistical test that checks a randomness hypothesis for a two valued data sequence. More precisely it can be used to test the hypothesis that the elements of the sequence are mutually independent. A run test is based on the null hypothesis that each element in the sequence is independently drawn from the same distribution. Let us consider a sequence of runs from a table by writing A if the observation is above median and B if the observation is below the median. Let us consider the following hypothesis: Ho : The occurrences of numbers in a table are in random manner. H1 : The occurrences of the numbers in the table are not in random manner. Let U = Number of observed runs yielded by n successive numbers in a table Then, U follows a binomial distribution with expectation E(U) and variance V(U) given by n+2 , 2 n n−2 & V (U ) = ( ) 4 n −1 E (U ) =
Accordingly, Z defined by Z=
ZU − E (U ) V (U )
is a standard binomial variate. Under the null hypothesis Ho, the test statistic is thus given by Z. For large n, Z=
U − E (U ) N (0,1) V (U )
Thus one can accept or reject the null hypothesis Ho on comparing the values of |Z| with the corresponding theoretical value of |Z| namely 1.96 (at 5% level of significance) and 2.58 (at 1% level of significance). One is to accept Ho if and only if |Z| < 2.58 and to reject Ho if and only if |Z| > 2.58
Findings Obtained Tippets random number table consists of 10,400 four digits random numbers. Here only 5200 numbers have been used in testing of the randomness of the table. In order to test the randomness of the occurrence of the numbers one should note that if the occurrence is random then theoretical median of the observations will be 4999 + 5000 = 4999.5 2
Fisher & Yates random numbers table consists of 7500 two digit numbers Here all the 7500 numbers have been used in testing of the randomness of the table. In order to test the randomness of the occurrence of the numbers it should be noted that if the occurrence is random then the theoretical median of the observations will be
49 + 50 2 = 49.5 Annex Publishers | www.annexpublishers.com
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Journal of Biostatistics and Biometric Applications Kendall and Smith random numbers table consists of 25000 four digit numbers. However, a sample of 15000 numbers has been used in testing the randomness of the table. In order to test the randomness of the occurrence of the numbers one should note that if the occurrence is random then theoretical median of the observations will be 4999 + 5000 = 4999.5 2
Rand Corporation Random numbers table consists of 200,000 five digit numbers among which a sample of 20,000 numbers has been used in testing the randomness of the table. In order to test the randomness of the occurrence of the numbers one should note that if the occurrence is random then theoretical median of the observations will be 49999 + 50000 = 49999.5 2
The observed values of Z statistic obtained from the tables of random numbers due to (i) Tippet, (ii) Fisher & Yates, (iii) Kendall & Smith & (iv) Rand Corporation have been shown in Table 1.1-1.4 respectively. From Table 1.1 it is found, on comparing the observed values with the corresponding theoretical values of Z , that the lack of randomness in the three parts containing 19th, 21st, & 25th 200 trials respectively in Tippet’s Random Numbers Table can be regarded as significant at 5% level of significance but not at 1% level of significance while the lack of randomness in the other parts can be treated as insignificant. Random Numbers considered
Observed number of runs (U)
Value of IZI
Random Numbers considered
Observed number of runs (U)
Value of IZI
1st 200
100
0.14
14th 200
2nd 200
93
96
0.71
1.13
15th 200
112
1.56
3 200 4th 200
95
0.85
16 200
103
0.28
98
0.43
17th 200
111
1.42
5 200
102
0.14
18 200
109
1.13
6th 200
93
1.13
19th 200
115
1.99
7th 200
105
0.57
20th 200
107
0.85
8 200
113
1.70
21 200
87
1.99
9th 200
97
0.57
22nd 200
107
0.85
10 200
94
0.99
23 200
98
0.43
11th 200
97
0.57
24th 200
110
1.28
12 200
103
0.28
25 200
115
1.99
13th 200
110
1.28
26th 200
114
1.84
rd
th
th
th
th
th
th
st
rd
th
Table 1.1: Observed value of Z statistic obtained from Tippet’s Random Numbers Table
From Table 1.2 it is found, on comparing the observed values with the corresponding theoretical values of Z , that the lack of randomness in Fisher & Yates Random Numbers table can be treated as insignificant. Random Numbers considered
Observed number of runs (U)
Value of IZI
Random Numbers considered
Observed number of runs (U)
Value of IZI
1 500
244
0.63
9 500
253
0.18
2 500
246
0.45
10 500
261
0.90
3rd 500
245
0.54
11th 500
233
1.61
4 500
254
0.27
12 500
254
0.27
5th 500
244
0.63
13th 500
244
0.63
6th 500
242
0.81
14th 500
255
0.36
7 500
250
0.09
15 500
246
0.45
254
0.27
st
nd
th
th
8 500 th
th
th
th
th
Table 1.2: Observed value of Z statistic obtained from Fisher & Yates Random Numbers Table
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From Table 1.3 it is found, on comparing the observed values with the corresponding theoretical values of Z , that the lack of randomness in the parts containing 1st , 2nd , 26th , 33rd , 36th , 45th , 46th , 48th , 65th , 68th , 70th 200 trials respectively in Kendall & Smith’s Random Numbers table can be regarded as significant at 5% level of significance but not at 1% level of significance while the lack of randomness in the other parts of the table can be treated as insignificant. Random Numbers considered
Observed number of runs (U)
Value of IZI
Random Numbers considered
Observed number of runs (U)
Value of IZI
1st 200
116
2.13
39th 200
105
0.56
2 200
88
1.84
40th 200
111
1.42
3 200
104
0.43
41 200
114
1.84
4th 200
109
1.13
42nd 200
90
1.56
5th 200
98
0.43
43rd 200
108
0.99
6 200
90
1.56
44 200
99
0.28
7th 200
96
0.71
45th 200
117
2.27
8 200
97
0.57
46 200
85
2.27
9th 200
98
0.43
47th 200
104
0.43
10 200
89
1.70
48 200
116
2.13
11th 200
101
0
49th 200
88
1.84
12 200
103
0.28
50 200
97
0.57
13th 200
95
0.85
51st 200
109
1.13
14 200
113
1.70
52 200
95
0.85
15th 200
91
1.42
53rd 200
100
0.14
16th 200
111
1.42
54th 200
89
1.70
17 200
111
1.42
55 200
117
2.27
18th 200
99
0.28
56th 200
110
1.28
19 200
92
1.28
57 200
103
0.28
20th 200
100
0.14
58th 200
96
0.71
nd rd
th
th
th
th
th
th
th
st
th
th
th
th
nd
th
th
21 200
105
0.57
59 200
105
0.57
22nd 200
105
0.57
60th 200
116
2.13
23 200
98
0.43
61 200
118
2.41
24th 200
110
1.28
62nd 200
117
2.27
25 200
102
0.14
63 200
99
0.28
26th 200
118
2.41
64th 200
101
0
27th 200
88
1.84
65th 200
119
2.55
28th 200
102
0.14
66th 200
85
2.27
29th 200
96
0.71
67th 200
111
1.42
30 200
109
1.13
68 200
87
1.98
31st 200
97
0.57
69th 200
90
1.56
32 200
101
0
70 200
109
1.13
33rd 200
115
1.99
71st 200
107
0.85
34 200
106
2.13
72
200
89
1.70
35th 200
90
1.56
73rd 200
102
0.14
36 200
116
2.13
74 200
115
1.99
37th 200
98
0.43
75th 200
101
0
38th 200
103
0.28
st
rd
th
th
nd
th
th
th
st
rd
th
th
nd
th
Table 1.3 Observed value of Z statistic obtained from Kendall & Smith’s Random Numbers Table
From Table 1.4 it is found, on comparing the observed values with the corresponding theoretical values of Z , that the lack of randomness in the four parts containing 35th , 52nd , 73rd , 94th 200 trials respectively in Rand Corporation Random Numbers Table can be regarded as significant at 5% level of significance but not at 1% level of significance while the lack of randomness in the other parts of the table can be treated as insignificant. Annex Publishers | www.annexpublishers.com
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Journal of Biostatistics and Biometric Applications Random Numbers considered
Observed number of runs (U)
Value of IZI
Random Numbers considered
Observed number of runs (U)
Value of IZI
1st 200
95
0.85
51st 200
100
0.14
2 200
97
0.57
52 200
118
2.41
3rd 200
100
0.14
53rd 200
100
0.14
4th 200
95
0.85
54th 200
110
1.28
5 200
102
0.14
55 200
92
1.28
6th 200
105
0.57
56th 200
105
0.57
7 200
110
1.28
57 200
96
0.71
8th 200
95
0.85
58th 200
100
0.14
nd
th
th
nd
th
th
9 200
102
0.14
59 200
91
1.42
10th 200
99
0.28
60th 200
95
0.85
11 200
88
1.84
61 200
96
0.71
12th 200
107
0.85
62nd 200
100
0.14
13 200
96
0.71
63 200
92
1.28
14th 200
97
0.57
64th 200
96
0.71
15th 200
106
0.71
65th 200
105
0.58
16 200
98
0.43
66 200
94
0.99
17th 200
97
0.57
67th 200
103
0.28
18 200
98
0.43
68 200
104
0.43
19th 200
106
0.71
69th 200
97
0.57
20 200
108
0.99
70 200
95
0.85
21st 200
89
1.70
71st 200
105
0.57
22 200
100
0.14
72 200
110
1.28
23rd 200
100
0.14
73rd 200
119
2.55
24 200
100
0.14
74 200
107
0.85
25th 200
108
0.99
75th 200
102
0.14
26th 200
100
0.14
76th 200
93
1.13
27 200
92
1.28
77 200
103
0.28
28th 200
89
1.70
78th 200
114
1.84
29 200
114
1.84
79 200
100
0.14
30th 200
110
1.28
80th 200
89
1.70
th
th
th
th
th
th
nd
th
th
th
th
st
rd
th
th
th
nd
th
th
th
31 200
104
0.43
81 200
108
0.99
32nd 200
100
0.14
82nd 200
102
0.14
33 200
108
0.99
83 200
104
0.43
34th 200
102
0.14
84th 200
93
1.13
35 200
119
2.55
85 200
100
0.14
36th 200
103
0.28
86th 200
110
1.28
37th 200
95
0.85
87th 200
94
0.99
38 200
107
0.85
88 200
95
0.85
39th 200
103
0.28
89th 200
105
0.59
40 200
93
1.13
90 200
96
0.71
41st 200
102
0.14
91st 200
88
1.84
42 200
104
0.43
92
200
89
1.70
43rd 200
112
1.56
93rd 200
105
0.57
44 200
90
1.56
94 200
118
2.41
45th 200
94
0.99
95th 200
103
0.28
46 200
102
0.14
96 200
114
1.28
47th 200
97
0.57
97th 200
94
0.99
48th 200
90
1.56
98th 200
102
0.14
st
rd
th
th
th
nd
th
th
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Random Numbers considered
Observed number of runs (U)
Value of IZI
Random Numbers considered
Observed number of runs (U)
Value of IZI
49th 200
99
0.28
99th 200
92
1.28
50 200
92
1.28
100 200
91
1.42
th
th
Table 1.4: Observed value of Z statistic obtained from Random Numbers Table due to Rand Corporation
Conclusion The findings, obtained in this study, imply that the degree of the lack of randomness is highest (in other words, the degree of randomness is lowest) in the Kendall & Smith’s Random Numbers Table among the four tables of random numbers examined. The four tables can be ranked with respect to the degree of randomness as follows (Table 2.1): Name of the Random Numbers Table
Number of parts deviated from randomness
Rank with respect to the degree of lack of randomness
Rank with respect to the degree of presence of randomness
Due to Tippet
3
3
2
Due to Fisher & Yates
0
4
1
Due to Kendall & Smith
11
1
4
Due to Rand Corporation
4
2
3
Table 2.1: Ranks of the four tables of random numbers as per run test obtained in the current study
Chakrabarty [2] have already made the same study. However, he has applied deviation test (based on t statistic) instead of run test (based on Z statistic) applied here. The findings obtained in that study have been shown in Table 2.2. Name of the Random Numbers Table
Rank with respect to the degree of lack of randomness
Rank with respect to the degree of presence of randomness
Due to Tippet
1
4
Due to Fisher & Yates
3
2
Due to Kendall & Smith
4
1
Due to Rand Corporation
2
3
Table 2.2: Ranks of the four tables of random numbers as per frequency test obtained in the study done by Chakrabarty [2]
In another study, Chakrabarty & Sarmah [3] have already made the same study. However, they have applied frequency test (based on chi-square statistic) instead of run test (based on Z statistic) applied here. The findings obtained in that study have been shown in Table 2.3. Name of the Random Numbers Table
Rank with respect to the degree of lack of randomness
Rank with respect to the degree of presence of randomness
Due to Tippet
2
3
Due to Fisher & Yates
1
4
Due to Kendall & Smith
3
2
Due to Rand Corporation 4 1 Table 2.3: Ranks of the four tables of random numbers as per frequency test obtained in the study done by Chakrabarty & Sarmah [3]
It is observed that the findings obtained in the three studies are not exactly identical. This leads to the necessity of searching for the reason(s) of the variation in the findings in the two studies. Moreover, one problem for researcher at this stage is to make attempt of constructing of random numbers table with more degree of randomness than that of the existing ones.
References
1. Bradely JV (1968) Distribution Free Statistical Tests Ist (Edn.) Prentice Hall, VA, USA. 2. Dhritikesh C (2017) Deviation Test: Comparison of Degree of Randomness of the Tables of Random Numbers due to Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation. SM J Biometr Biostatis2: 1014. 3. Dhritikesh C, Kanta SB (2017) Comparison of Degree of Randomness of the Tables of Random Numbers due to Tippet, Fisher & Yates, Kendall & Smith and Rand Corporation. J Reliability Statis Stu 10: 27-42. 4. Fisher RA, Yates F (1938) Statistical Tables for Biological, Agricultural and Medical Research. 6th (Edn.) Longman Group Limited, England. 5. Hald A (1952) Table of random numbers, In: Statistical Tables and Formulas, AGRIS since.
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6. Hill ID, Hill PA (1977) Tables of Random Times. UK. 7. Kendall MG, Smith BB (1938) Randomness and Random Sampling Numbers. J Roy Stat Soc 101: 147-66. 8. Kendall MG, Smith BB (1939) A Table of Random Sampling Numbers. Tracts for Computers, Cambridge University Press, Cambridge, England. 9. Mahalanobis PC (1934) Tables of random samples from a normal population. Sankya 1: 289-328. 10. Manfred Mohr (1971) Le Petit Livre de Nombres au Hasar, Paris. 11. Moses EL, Oakford VR (1978) Tables of Random Permutations. Stanford Univ Pr, USA. 12. Quenouille MH (1959) Tables of Random Observations from Standard Distributions. Biometrika 46: 178-202. 13. Davis J (1990) A Million Random Digits. New England Rev 28: 161-3. 14. Rao CR , Mitra SK, Matthai A (1966) Random Numbers and Permutations. Statistical Publishing Society, Calcutta. 15. Rohlf FJ, Sokal RR (1969) Ten Thousand Random Digits. In: Rohlf & Sokal: Statistical Tables. 16. Royo J, Ferrer S (1954) Tables of Random Numbers Obtained from Numbers in the Spanish National Lottery. Trabajos de Estadistica 5: 247-56. 17. Sarmah BK, Chakrabarty D (2015) Testing of Randomness of the Number Generated by Fisher and Yates. Int J Eng Sci Res Technol 3: 632- 6. 18. Kanta SB, Dhritikesh C (2014) Examination of Proper Randomness of the Number Generated by L.H.C. Tippett. Int J Eng Sci Res Technol 3: 631-8. 19. Kanta SB, Dhritikesh C (2015) Testing of Proper Randomness of the Table of Number Generated by M.G. Kendall and B. Babington Smith (1939). Int J Eng Sci Res Tech 4: 260-82. 20. Kanta SB, Dhritikesh C, Nityananda B (2015) Testing of Proper Randomness of the Table of Number Generated by Rand Corporation (1955). Int J Eng Sci Mgmt 5: 97-119. 21. Sarmah BK, Chakrabarty D (2015) Testing of Randomness of the Numbers Generated by Fisher and Yates. AryaBhatta J Math Info 7: 87-90. 22. Kanta SB, Dhritikesh C (2015) Examination of Proper Randomness of the Numbers Generated by L.H.C. Tippett (1927). IOSR J Math 11: 35-7. 23. Kanta SB, Dhritikesh C (2015) Examination of Proper Randomness of the Numbers Generated by Rand Corporation (1955) Random Number Table: t-Test. IJIRSET 4: 9536-40. 24. Kanta SB, Dhritikesh C (2015) Testing of Proper Randomness of the Numbers Generated by Kendall and Babington Smith: t-Test. AryaBhatta J Math Info 7: 365-8. 25. Kanta SB, Dhritikesh C (2015) Examination of Proper Randomness of Numbers of M. G. Kendall and B. Babington Smith’s Random Numbers Table: Run Test. IJMRME I: 223-5. 26. Kanta SB, Dhritikesh C (2015) Testing of Randomness of the Numbers Generated by Fisher and Yates:: Run Test. Int J Innov Res Sci Eng Technol 4: 11956-58. 27. Snedecor GW, Cochran WG (1967) Statistical Methods. 6th (Edn.) Iowa State University Press, Ames, Iowa, USA. 28. Tippett L HC (1927) Random Sampling Numbers. Tracts for Computers no. 15, Cambridge University Press, Cambridge, England.
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