Measuring batting consistency and comparing batting greats in test cricket: innovative applications of statistical tools

Abstract

This paper examines the intriguing problem of comparing great batsmen in test cricket across different eras. Traditional method of calculating a batsman’s batting average may be justified under the assumption that runs scored in various complete and incomplete innings by a batsman form a random sample from an exponential or a geometric distribution. This assumption, however, leads to undesirably having batting inconsistency or standard deviation uniquely determined by the batting mean. To correct this drawback, we propose use of the Weibull distribution model. First, the Weibull model is seen to provide a far superior fit to the test cricket data of our study. Second, the maximum likelihood estimate (MLE) of the batting standard deviation is found to provide a very sensible estimate of batting inconsistency. Third, the resulting MLE of the batting mean in case of Bradman turns out to be 109.42 instead of 99.94. Fourth, we define player longevity as a third criterion and introduce an index for quality-runs scored as a function of opposition strength and another measure for diversity of opponent teams encountered by a player. Fifth, the Mahalanobis distance is used for overall ranking of a select group of batting greats on the basis of various combinations of these five criteria, without assigning any subjective weights to them. Finally, multivariate statistical outlier detection technique affirms two players as truly outstanding—Bradman for his batting average and quality of runs scored, and Tendulkar for his longevity and opposition diversity he faced. The proposed techniques used here may easily be applied in sports management for ranking players available for procurement and in investment management for rating various financial assets.

Appendix

Table 3 List of batsmen considered for analysis

Table 4 Maximum likelihood estimates of batting mean and standard deviation under the Weibull(α, θ) model

Table 5 Ranks based on batting mean and batting consistency

Table 6 Ranks based on batting mean, batting consistency and longevity

Table 7 Opponent difficulty level for batsmen in home test matches

Table 8 Opponent difficulty level for batsmen in away test matches

Table 9 Batting average of players against respective opponents in home test matches

Table 10 Batting average of players against respective opponents in away test matches

Table 11 Relative performance index of players against various opponents in home test matches

Table 12 Relative performance index of players against various opponents in away test matches

Table 13 Composite performance indices and overall quality of runs scored

Table 14 Number of home test matches played by different countries over time

Table 15 Number of away test matches played by different countries over time

Table 16 Proportions of test innings played by batsmen in various periods

Table 17 Numbers of home test matches played by batsmen against respective opponents

Table 18 Numbers of away test matches played by batsmen against respective opponents

Table 19 Expected numbers of home matches to have been played by the players against respective opponents

Table 20 Expected numbers of away matches to have been played by the players against respective opponents

Table 21 Opposition diversity index

Table 22 Ranking through grouping and Mahalanobis distance based on five criteria

Table 23 Detecting outliers among the great batsmen considered