## Cricket in pictures

We may say that cricket has nearly passed us by. In our youth we played and watched the game quite a bit. While in secondary school we were fairly interested in cricket statistics. The sources for statistics were not easily available those days. We would obtain them bit by bit from the occasional sports magazine our father might purchase, or from a cricket magazine at our local English library, or from sports quizzes on television, or they might stream in from acquaintances and certain relatives. But as we grew older this interest waned. When we were in college, our brother was passing through a similar phase of interest, which re-ignited ours and also now we sought to see it through the lens of statistical distributions and probability. However, those experiments were generally tedious and not the most exciting thing we had on our hands. As we aged into young adulthood and beyond, cricket kept increasingly passing out of our focus. Thus, we had reasonable familiarity with the heroics of Kapil Dev, Gavaskar, Shrikant, Tendulkar, Dravid, Sehwag and Laxman, less-so with Dhoni, and even less so with Kohli. Still, we say it has only nearly passed us by because, though we do not watch it anymore, we still have it streaming to us once in a way from friends and colleagues.

Due to such impingements we recently wondered about re-visiting certain statistical distributions in cricket. This urge finally precipitated due some discussion on Twitter regarding cricket statistics. We would like to acknowledge those discussions for inducing us to write this note that many might perceive as an unnecessary endeavor but a man should always pay attention to distributions and statistical properties for they can might help him in other walks of life. Here we take a look a some such for batting performances in Test cricket for we believe that it is the highest and the most satisfying form of the game. The sources of statistics for the below meanderings are the following: 1) A compilation of 73 top batsmen from the HowzStat Cricket Database; 2) A compilation of 301 batsmen scoring 2000 runs or more from Cricinfo; 3) A large data collection for all batsmen for all international matches, which was prepared by and kindly provided as a convenient csv file by Anupam Singh. The first two were leached from html and converted to csv files. The third file needed some post-processing for removing duplicates and null records. That said let us look at the data.

Basic features of batting in test matches

Figure 1

In the first panel of Figure 1 we see the frequency distribution of runs scored by the bat in an innings of a test match. It is a left truncated, right skewed distribution with a clear central tendency: a modal peak around 220 runs and a median of 239 runs. The second panel shows the frequency distribution of the number of fours in a given test innings. It again shows a similar distribution shape as the total number of runs scored by the bat in an innings. It has a modal peak at around 22-23 fours and a median value of 26 fours. That translates to a median value of 104 runs in a test innings being scored by fours. When we compare this to the first graph we can say that speaking in terms of central tendency about $44\%$ of the innings is scored by fours.

Sixes are much rarer in test cricket and their distribution is shown in panel 3 of Figure 1: the fraction of innings with $n=0, 1, 2, 3...$ sixes. This shows a non-linear decay law which might be approximated by an exponential function of the form $y=ae^{-bx}$. While this works in the range 1..14 sixes, it fails to captures the maximal 0 sixes fraction or the more extreme values. The Poisson distribution and power-law also do not approximate it well.

The strike rate of a batsman is defined as the ratio the number of runs scored to the number of balls faced expressed as as a percentage. In the fourth panel we have the frequency distribution of the mean strike rates for test batsmen computed for all scores greater than 10. It shows a strong central tendency close of $50\%$.

Figure 2

This sets the baseline for some further analysis. In Figure 2 we look at the frequency distribution of the scores for each innings. We did expect to see the median score fall with each innings. However, the non-linear nature of the fall and the changing shape of the distribution is notable. The first two innings have an almost triangular distribution, while the third innings is more of a skewed bell-shaped distribution. Obtaining functions that approximate these will be a problem of interest. The fourth innings has the lowest scores. This is due to two reasons: it is the innings of the final chase to win the match. Now, if the final total needed for a win is small then the forth innings scores would be low. Further, as the pitch deteriorates through the match fourth innings will favor the batsmen the least and the bowlers the most. This was even more in the past when the pitches were not protected overnight. However, this decline does suggests that barring special weather conditions, winning the toss and batting first gives the team which does so an advantage in the match.

Figure 3

In Figure 3 we look at frequency distribution of what fraction of big individual scores of a batsmen, i.e those of 50, 100, 150, 200 or greater are scored by fours and sixes. These are roughly normally distributed with a mean close to 0.5: thus, on an average about half of a big individual score in tests is attained by shots touching or clearing the fence.

Figure 4

We then analyzed the probability of a test inning containing a century. We found that the fraction of test innings containing $n=0, 1, 2, 3...$ innings can be well-described by a Poisson distribution. We empirically determined the $\lambda= 0.485$ for this Poisson distribution. The $p\approx 1$ for Poisson-predicted frequencies matching observed frequencies by the $\chi^2$ test. In figure 3 the hatched bars are the predicted values and the blue bars are the observed values.

Figure 5

In Figure 5 we analyzed the frequency distribution of individual scores of batsmen that are $\ge 100$. It shows a clear-cut decay law. This steep decay is best captured by a power law of the form $y=kx^a$, indicated by the blue line in Figure 5 (goodness of fit deviation fraction: 0.96). However, for scores $\ge 130$ an exponential decay fits the observed distribution at least as well as a power law (Red curve in Figure 5; goodness of fit deviation fraction: 0.94). Overall, it is fair to say that a power-law distribution approximately describes the distribution of scores $\ge 100$.

Apprehending the great batsmen
For this analysis we used a dataset of 301 batsmen who have scored over 2000 runs and have had an average career strike rate of $\ge 20$. This dataset records the number of innings played, number of times the player is unbeaten, total runs scored, highest score, career average, career strike rate, and 100s, 50s and 0s scored. We performed a principle component analysis using a subset of the numerical variables in this dataset for which good records are available (Highest score, number of 100s, number of 50s, career average, and strike rate) after scaling and centering. The first two components which together account for $\approx \tfrac{3}{4}$ of the variation, and their plot is show in Figure 6.

Figure 6

There is not much clumping but a set of “greatest” batsmen can be simply separated by choosing those with a first axis value $\le -2.3$ (red dots in Figure 6). At the extreme end of this this axis lie Tendulkar, Bradman and Lara (circled in green in Figure 6), who have quite unequivocally be mentioned as being among the greatest players. In this set two players are clearly separated from the rest (circled in dark red in Figure 6): Sehwag and Vivian Richards. These are two great batsmen marked by the rapid scoring rates and were a delight for the spectator. The one other Indian player who was notable for his scoring rate when I was young was Kapil Dev, who lies to the right side (circled in blue). While possessing a notable strike rate, as we can see, he was far too inconsistent to make it anywhere close to the great batsmen region. Finally, we may list this set of “greatest” batsmen:
1. V Sehwag (INDIA); 2. IVA Richards (WI); 3. BC Lara (WI); 4. ML Hayden (AUS); 5. GC Smith (SA); 6. RT Ponting (AUS); 7. DG Bradman (AUS); 8. MJ Clarke (AUS); 9. SPD Smith (AUS); 10. AB de Villiers (SA); 11. KC Sangakkara (SL); 12. SR Tendulkar (INDIA); 13. Inzamam-ul-Haq (PAK); 14. GS Sobers (WI); 15. Younis Khan (PAK); 16. GS Chappell (AUS); 17. DPMD Jayawardene (SL); 18. HM Amla (SA); 19. VVS Laxman (INDIA); 20. GA Gooch (ENG); 21. SR Waugh (AUS); 22. AN Cook (ENG); 23. Javed Miandad (PAK); 24. JH Kallis (SA); 25. SM Gavaskar (INDIA); 26. S Chanderpaul (WI); 27. R Dravid (INDIA); 28. KF Barrington (ENG); 29. MA Taylor (AUS); 30. AR Border (AUS); 31. WR Hammond (ENG); 32. L Hutton (ENG)

Figure 7

For this group of test batsmen we can visualize the distribution of their average and strike rate (panel 1 and 2 of Figure 7). We observed that both show a reasonable fit for a normal distribution: Average: Shapiro-Wilk $p=0.5866$; strike rate: Shapiro-Wilk $p=0.1437$. Thus, assuming a normal distribution of averages among top batsmen we can calculate the probability of a batsmen having an average like Bradman by chance alone to be a vanishingly small $p=1.99 \times 10^{-11}$. One may say that Bradman lived in a very different era when Australia’s main opponent was England and the other teams like India were not particularly strong. Yet, his record is unusually deviant and points to some special biological ability in him which might be likened to that possessed by a Ramanujan in mathematics. As evidence one might point to facts such as his surviving serious illness, his long lucid life and success in investing. Thus, he can be seen as the father of Australia itself.

Using the same distributions we can infer that the probability of a batsmen with Sehwag’s career strike-rate emerging by chance alone in this set of top batsmen is $p=0.00015$. When we combine it with his average we can infer that the chance of a Sehwag emerging by chance alone among these top players is a minuscule $p=2.3 \times 10^{-5}$.

From this set of 301 batsmen we can get a smaller set of 73 top batsmen based on the fact that they convert 50s to 100s at a rate of $40\%$ or higher. For this set we find that there is a strong linear correlation between the 100s they score and the number of innings they have played ( $r^2=0.84$; Figure 7, panel 3). As ever, Bradman stands apart from the rest. From this we can calculated that this creamy layer of batsmen score a test century once every 7.8 innings (median value; $\mu=8.27$) or approximately once in every four matches (Figure 7, panel 4).

Figure 8

Finally, for this set of 73 batsmen we can look the distributions of the fraction of the innings in which they have remained not out and their highest test scores (Figure 8). These two metrics bring out two of the greatest men in my cricket-watching days: 1) Kallis, perhaps the second greatest all-rounder to date (the first being Sobers whom we have never watched). He has remained unbeaten unusually high number of times. 2) Lara, the only man who score a 400 in test cricket. He was the last of the great black emperors of the Caribbean: what more needs to be said of him?

Some notable Indian test batsmen
We next took a closer look at some of the notable Indian batsmen whose innings we have watched in our career as a spectator of the game: 1) Sehwag, 2) Tendulkar, 3) Dravid, 4) Laxman, 5) Kohli, 6) Ganguly.

Figure 9

Figure 6 shows the probability density distribution of the scores of these 6 batsmen along with their career average. Kohli is still playing so his result will change in the future. The propensity for low scores is higher in Laxman and Ganguly, whose effect is seen in the form of their lower averages. The most notable features are: 1) Sehwag’s far-out right tail with secondary elevations in that region: he was clearly that man who could reach the big scores which balanced out his low scores. $7.7\%$ of his innings are adorned by scores of 150 or more. In contrast, the right tails of Tendulkar, Kohli, and Ganguly terminate more quickly, and Laxman and Dravid are in between. 2) However, both Tendulkar and Kohli have fat right tails keeping with their tendency to score numerous hundreds. Kohli in particular, displays a secondary peak around 100 which is consistent with the fact that he has 10 scores between 100 and 130. 3) Laxman’s peculiarity is a shoulder between 45 and 75. This comes from the fact in 49 of his 225 innings ( $\approx 22\%$) he has scored runs in this range.

Figure 10

We next look at the strike rates of these same 6 players for all innings where they have not scored a 0. As noted above, the ferocity of Sehwag’s batting stands out in this metric with a mean of 84.5 (Figure 7). On the other end, Dravid’s role as the slow-moving defensive formation in the battle array is displayed by his mean of 40.1, which is way below the average for tests (Figure 1). Most of these players have an approximately normally distributed strike rate. However, Kohli’s profile hints a bimodality which suggests that he has played some defensive innings like Dravid and also more attacking ones. But his central peak indicates that he has one characteristic strike rate (Figure 7).

Figure 11

We devised another way of visualizing the same: a scatter plot of runs scored in an innings versus balls faced. On this scatter plot, using all innings where the player faced 50 or more balls, we plot the minimal and maximal angles corresponding the least and highest strike rate for the innings meeting this criterion. The difference between these two is the characteristic strike rate angle (SRA) of the batsman. We also plot the angle corresponding to the median strike rate for innings meeting the above criterion of balls faced. Sehwag stands out right away: He has the maximum median angle and the narrowest SRA (Figure 8). This means that he was consistently the fastest of these great Indian batsmen in tests. In contrast, Dravid has the lowest median angle and the widest SRA. The former value indicates that he was the slowest of these great batsmen but the wide angle indicates that he was capable of fast innings on occasion. Kohli and Tendulkar have similar median angles that are larger than the remaining batsmen other than Sehwag. However, Kohli has a much narrower SRA closer to Sehwag. This suggests that, while Tendulkar and Kohli score(d) at the same overall rate Kohli is more consistent in scoring at that rate. Tendulkar, however, tended to score at very different rates in different innings. On the whole these features affirm Sehwag’s uniqueness in the constellation of great batsmen.

The laws of distributions hold their strong sway but once in a way a man of superhuman capacity might emerge. However, they used to say: “Cricket is a funny game.” It indeed is. Some might have the potential for greatness but, like the Khans passing away into the grasses of the steppe without history recording ever recording their name, they might stumble as an IR Bell or a DI Gower on the brink of greatness. On the other hand, others like a Gooch or SR Waugh, while lacking genius, might still make it to the club of greatness by their bulldog-like stickiness.

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