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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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