Data Analysis based on significant digits of runs made by one-day cricket batsmen

Abstract

          The pattern of first significant digit from collected numbers follows the Benford’s law. The first significant digit of runs made by several famous batsmen from various countries in one-day cricket is analyzed. Based upon this, the characteristics of the batsmen’s game plan are analyzed. For all the Batsmen, duck-outs are not considered.

1. Introduction

          The first significant digit of collected numbers doesn’t follow uniform distribution as we expected. According to Astronomer and Mathematician Siman Newcomb “the first pages of tables of logarithms wear out much faster than the last ones”.The law that follows for first significant digit is

                        P [first significant digit = d] = log₁₀(1+(1/d)) , d=1,2,…,9.

            Dr. Frank Benford (1938), a physicist, working for General Electric in 1930’s, worked independently on the significant digits of naturally collected numbers. He collected several data sets, such as, the area of rivers, American league baseball statistics, numbers appearing in Reader’s Digest, death rates and atomic weights of elements and invented that Benford’s law fits well to these data sets.

            Interestingly, M. Nigrini (1996) applied Benford’s law to the tax returns data and found the fraudulent data. Ley (1996) extended Benford’s law applications to stack market indexes. Theodore P.Hill (1995) provided a statistical derivation of the Benford’s law.

            We collected one-day cricket runs made by famous one-day batsmen in world cricket and applied Benford’s law to the first significant digit excluding duck-outs. The statistical significance of uniform distribution to the significant digit in the units place is also tested.

            Section2 explains the Benford’s law in detail. Section 3 consists of description of one-day cricket. Section 4 gives the applications of significant digits on runs made by several famous one-day batsmen and section 5 makes the conclusions.

            The data sets are available from http://www.howstat.com.au/.

2. Benford’s law

The first significant digit is distributed in the set {1,2,…,9} as P[first significant digit = d] = log₁₀((d+1)/d) , d= 1,2,…,9 . i.e., in any collected numbers, the digit 1 occurs 30.1 % [ log₁₀2 = 0.301 ] times, but the digit 9 occurs 4.6% [log₂(10/9) = 0.046] times.

            The first significant digit of 40 European countries that is in square kilometers (P.M. Lee, 1989) follows Benford’s law. The following table gives the details.

Digit	1           2            3            4           5          6         7         8          9
True data	25         17.5       15          15         7.5       5         5         2.5      7.5
Benford’s law	30.103   17.609  12.494    9.691   7.978   6.695  5.799  5.115  4.576

Table 2.1

            The quantities that are measured may vary. Instead of square kilometers, square miles may be considered. Because of the scale invariant property, Benford’s law is still applicable to the changed data. If the considered data set is converted from one base (suppose base is 10) to another base (base 100), Benford’s law is applicable i.e., Benford’s law satisfies base invariant properties. About the detailed discussion of scale and base invariant properties, reader can have a look in to the publication of Theodore P. Hill (1995)

3. One-day Cricket

            Cricket was born in England. Because of Great Britain’s rule in many countries in 18^thand 19^th centuries, cricket is also spread in to colonial states. People are used to play cricket for 5 days, which is called Test cricket. To increase the enthusiasm in the viewers, a limited sort of game playing for a day is introduced. It is called one-day cricket. Cricket is very much famous in Australia, Newzeland, England, Indian sub continent, South Africa and West Indian Islands.

To learn more about this game, the URL http://encyclopedia.thefreedictionary.com/One-day%20cricketmay be helpful.

4. Application

            Runs made by several batsmen from various countries are collected. The test statistics of Chi-square goodness of fit test that follows Benford’s law and its p-value for first significant digit are given in table 4.1.

Player	Chi-square test for Benford’s law on first significant digit
	Test statistic      P-value
Michael G Bevan	14.90506693	0.061017873
Andrew Flower	12.41756102	0.133523039
Jacques H Kallis	7.746278365	0.458638789
Desmond L Haynes	6.475067741	0.594174459
Rahul S Dravid	6.446699507	0.597325635
Stephen R Waugh	6.223625986	0.622198182
Adam C Gilchrist	5.901606244	0.658252569
Ricky T Ponting	5.858942527	0.663028872
Inzamam-Ul-Haq	5.779427034	0.671923685
Saurav C Ganguly	4.257584281	0.833167603
Pinnaduwage A De Silva	4.209530898	0.837741214
Mohammad Azharuddin	3.534059895	0.896530246
Mark E Waugh	2.993049155	0.934792991
Brian S Lara	2.885163313	0.941356018
Sachin R Tendulkar	2.285404993	0.97098821
Allan R Border	1.823578373	0.985950882
Herschelle H Gibbs	1.477624181	0.993073213
Sanath T Jayasurya	1.318275438	0.995330207

                                                     Table 4.1

By observing the table 4.1, it is confirmed that all the batsmen from the list are playing according to Benford’s law, where the test statistic is considered for 95% confidence interval.

For Michael G Bevan, values that are higher than 14.91 would be expected to occur about 6.1 % of the time, where as for Sanath T Jayasurya , values that are higher than the test statistic 1.318 would be expected to occur about 99.5% of the time. The test statistics is low and high for these two players respectively. The true data for these two players are in Table 4.2.

Digit	1	2	3	4	5	6	7	8	9
Bevan’s true data	23.037	13.089	19.895	13.613	11.518	5.759	7.853	4.712	0.524
Jayasurya's true data	32.5	15.833	10.833	10.833	7.5	7.5	3.75	5.833	5.417
Benford’s law data	30.103	17.609	12.494	9.691	7.978	6.695	5.799	5.115	4.576

                                                                  Table 4.2

The significant digit of one-day runs of Andrew Flower and Michael G Bevan tried to avoid Benford’s law, but are unable i.e., Both of these players played well in most of the games and they tried to decrease the effect of 1’s in the first significant digit of their runs. This reveals that these two players tried to be around their average in every match. The P-values for these two players are 13.4% and 6.1% respectively that are least compared to others. In the case of Sanath T. Jayasurya p-value is (highest) 99.5%. In the case of Sachin R Tendulkar, the p-value is 97%. Since, he made more than 30 centuries, which has 1 in its significant digit may be showing some effect.

5. References:

[1] Benford, F. (1938), “The Law of Anamolous Numbers,”Proceedings of the American Philosophical Society, 78, 551-572.

[2] Theodorw P.Hill. (1995), ”A Statistical Derivation of the Significant-Digit Law,” Statistical Science, 86, 4, 354-363.

[3 ] Ley, E. (1996), “On the Peculiar Distribution of the U.S.Stock Indices Digits,” The American Statistician, 50, 311-313.

[4] Nigrini, M. (1996), “ A Taxpayer Compliance Application of Benford’s law,” Journal of the American Taxation Association, 18, 72-91.