Statistical Study about Ranking of Players in IPL

―Chase your dreams but make sure you don‘t find shortcuts‖ – Sachin Tendulkar. It is generally believed that cricket originated as children‘s games in the south-eastern countries of England. Cricket spread globally with the expansion of the British Empire, leading to the first international matches in second half of the 19th century. The games governing body of the international cricket council (ICC), which has over 100 members, twelve of which are full members who play test matches. The games rules are held in a code called the laws of cricketer which is owned and maintained by Marylebone cricket club (MCC) in London. The sport is followed primarily in Australia, Great Britain, Ireland, the Indian subcontinent, South Africa and West Indies. Women‘s cricket is organized separately and achieved the international standards. The most successful side playing international cricket is Australia having won several one-day international trophies, including five world cups more than any other country. In India, cricket is the biggest sport and in every street of India cricket is being played. India became the member of the Elite club in June 1932 joining Australia, England, South Africa, New Zealand and West Indies. India recorded its first test victory in 1952, beating England by an innings in Madras. In 1971, they won a test series in England for the first time ever. In 1983, India were surprise winners of the cricket world cup under the captaincy of Kapil Dev. In September 2007, India won the first ever Twenty 20 world cup held in South Africa and won the 50 over cricket world cup in 2011 under the captainship of Mahendra Singh Dhoni held in India. In 2008, BCCI announced the launch of a franchise based Twenty20 cricket tournament called Indian Premier League, one of the biggest sporting events of the world. The league's format was like that of the Premier league of England and the NBA in the United States. Each team play against each other twice in a home and away game in a round-robin format in the league phase. At the conclusion of the league stage, the top four teams will qualify for the playoffs. The winner of the second Qualifying match will move onto the final to play the winner of the first Qualifying match in the IPL final where the winner will be crowned as the Champion of the season. Quantifying individual player‘s contribution is an important task in all team sports. Ranking based on their performance and quantifying the quality of each player and to find the key players in each season of IPL from 2008 to 2013 using principal component analysis. Principal component analysis (PCA) uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

Data collection Secondary data of all IPL matches that took place from 2008 to 2013 is considered. Details of these

For ranking the batsmen, using the key variables are innings, runs, averages, strike rate, fours, sixes. There are several other variables that carry information about the contributions to a team by batsmen, and some of the variables may indeed be correlated. Here, we use the correlation matrix to accommodate discrepancies‘ in the magnitude of the measurement‘s units of the variables. Below is a brief description of our selection for the critical variables used to quantify the quality of the players. Runs: Total no of runs scored by a batsman in all innings. Batting average: Total no of runs scored divided by total number of innings. If the batsman is not out, the average increases. The number overrates the performance of a batsmen with several not out cases which is the weakness of this measure. Batting strike rate: The no of runs scored per 100 or the ratio of the number of runs scored and the number of balls faced by a player. Higher value of S.R indicate stronger performance as an aggressive batting style is always advantages in shorter version limited over cricket match like T20. Fifties: Scoring 50 runs in an innings is known as a half-century or Fifties. Sixes: Total no of sixes hit by the player in all innings. For ranking the bowlers, here is a brief description of the critical variables used to quantify the quality of the bowlers. Wickets: The number of wickets taken by a bowler in all innings, goal is to obtain many wickets. Bowling Strike rate: The average number of balls bowled per wicket taken. better bowlers have lower strike rate. Economy Rate: The average number of runs conceded per over. Better bowlers have lower economic rate. Bowling Average: It is the number of runs conceded by a bowler per wicket taken.

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to the complexity in high-dimensional data while retaining trends and patterns. It does this by transforming the data into fewer dimensions, which act as summaries of features. PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Briefly, if XI=[x1,x2,…..,xk] is a k-vector of random variable with variance covariance Matrix £ and corresponding eigenvalue- eigenvector pairs (ᵟ1,e1),(ᵟ2,e2),….,(ᵟk, ek) where ᵟ1 ≥ ᵟ2 ≥…..≥ᵟk, then the principal components p1,p2,….,pk are defined by, Furthermore, it can show that

Consequently, the proportion of total variability due to the ith principal component is given by

sample correlation matrix with SPSS 20.0. Next, we obtained all eigenvalues and associated eigenvectors for the correlation matrix and identified the largest eigenvalue, ∆1=, as well as its corresponding eigenvector e = [0.]‟. The latent value was the only eigenvalue exceeding 1.0 and SPSS 20.0 reports that the first principal component P1=e „1 X accounts for % of the total variability identified in equation (1). So, it is possible to concentrate on just the First Principal Component (FPC), as it accounts for a substantial portion of the total variability. Accordingly, we choose to rank the IPL batsmen from 2008 to 2013 based on their individual scores produces by the first principal component computation.

Top 50 bowlers in the IPL from 2008 to 2013 were included in this analysis, so that fifty total bowlers comprise this list. For each bowler we collected (5x 1) column vectors of the form X = (Innings, Wickets, Avg, Sr, Eco)‟, and using them computed the sample correlation matrix with SPSS 20.0. Next, we obtained all eigenvalues and associated eigenvectors for the correlation matrix and identified the largest eigenvalue, ∆1=, as well as its corresponding eigenvector e = []‟. The latent value was the only eigenvalue exceeding 1.0 and SPSS 20.0 reports that the first principal component P1=e „1 X accounts for % of the total variability identified in equation (1). So, it is possible to concentrate on just the First Principal Component (FPC), as it accounts for a substantial portion of the total variability. Accordingly, we choose to rank the IPL bowlers from 2008 to 2013 based on their individual scores produce by the first principal component computation.

For the analysis we took players of IPL from 2008 to 2013 and its output are given below

1. Gambhir and Jayasuriya have normal rankings 2 and 3, respectively, yet their FPC-rankings are just the opposite,3 and 2. 2. Gambhir scored 534 total runs with an average of 45.07 runs, and a strike rate of 140.9. He hit five-50s and 8 sixes. 3. Jayasuriya scored a total of 514 runs with an average 42.83 runs, and a strike rate of 166.38. He hit two 50s and 31 sixes. 4. Clearly, Jayasuriya has higher strike rate and more sixes, while Gambhir is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, Jayasuriya is ranked higher than Gambhir.

1. VY Mahesh is ranked number 8 in the Normal rankings but appears at number 5 in the list of top ten bowlers when using the FPC method. 2. VY Mahesh took 16 wickets with an average of 23.12. His economy rate is 8.7, and his strike rate is 15.8. 3. On the other hand, SK Warne, ranks 4th in the FPC-ranking method who ranks 2 in the normal rankings. He took 19 wickets with an

4. So, Warne is better than Mahesh with respect to the strike rate variable, which is the average number of balls per wicket. It is true that a low strike rate is a desirable attribute. 5. However, Warne has a lower average, which is better since it represents the number of runs conceded per wicket. Moreover, Warne has the lower economy rate, which is the average number of runs per over. 6. This justifies that Warne should be ranked higher than Mahesh.

1. SK Raina and Hodge have normal rankings 4 and 8 respectively, yet their FPC-rankings are 3 and 10. 2. Raina scored 434 total runs with an average of 31 runs, and a strike rate of 140.9. He hit two-50s and 21 sixes. 3. Hodge scored a total of 514 runs with an average 40.55 runs, and a strike rate of 117.74. He hit three 50s and 9 sixes. 4. Clearly, Raina has higher strike rate and more sixes, while Hodge is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, Raina is ranked higher than Hodge.

1. RP Singh is ranked number 1 in the Normal rankings but appears at number 5 in the list of top ten bowlers when using the FPC method. 2. RP Singh took 23 wickets with an average of 18.13. His economy rate is 6.98, and his strike rate is 15.5. 3. On the other hand, MM Patel, ranks 1st in the FPC-ranking method who ranks 7 in the normal rankings. He took 16 wickets with an average of 15.06. His economy rate is 6.91, and his strike rate is 13. 4. So, MM Patel is better than RP Singh with respect to the strike rate variable, which is the average number of balls per wicket. It is true that a low strike rate is a desirable attribute. 5. However, MM Patel has a lower average, which is better since it represents the number of runs conceded per wicket. Moreover, RP Sigh has the lower economy rate, which is the average number of runs per over. 6. This justifies that MM Patel should be ranked higher than RP Singh

NameInningsRunsAverageStrike rate50s6sFPCNormal rankingSK Raina1652047.27142.854222.063453JH Kallis1657247.66115.78691.780732RV Uthappa1437431.16171.553271.7201111M Vijay1545835.23156.842261.685915SR Tendulkar1561847.53132.61531.557571A Symonds1642930.64125.84181.407067SC Ganguly1449337.92117.664151.246314SS Tiwary1541929.92135.593181.203558RG Sharma1640428.85133.773141.059849YK Pathan1433327.75165.671241.043117

1. Sachin Tendulkar and Raina have normal rankings 1 and 3 respectively, yet their FPC-rankings are 5 and 1. 2. Sachin scored 618 total runs with an average of 47.53 runs, and strike rate of 132.61. He hit five-50s and 3 sixes. 3. Raina scored a total of 520 runs with an average 47.27 runs, and a strike rate of 142.85. He hit four 50s and 22 sixes. 4. Clearly, Raina has higher strike rate and more sixes, while Sachin is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, Raina is ranked higher than Sachin

NameInningsWicketsAverageEconomiStrike arteFPCNormal rankingRJ Harris81416.647.5913.1-1.5548812DE Bollinger81217.256.6715.5-1.3491917KA Pollard121518.267.414.8-1.177596PP Ojha162120.427.2916.8-1.150361A Mishra141721.356.8418.7-0.813042Harbhajan Singh141722.177.0418.8-0.704633M Muralitharan121521.936.8519.2-0.684977SB Jakati111322.387.6517.5-0.509113SL Malinga131522.937.0219.6-0.482068R Ashwin121322.536.122.1-0.3961914

1. Ojha is ranked number 1 in the Normal rankings but appears at number 4 in the list of top ten bowlers when using the FPC method. 2. Ojha took 21 wickets with an average of 20.42. His economy rate is 7.29, and his strike rate is 16.28. 3. On the other hand, Pollard, ranks 3rd in the FPC-ranking method who ranks 6th in the normal rankings. He took 15 wickets with an average of 18.76. His economy rate is 7.4, and his strike rate is 16.8. 4. So, Pollard is better than ojha with respect to the economic rate variable, which is the number of runs given for total balls bowled. It is true that a low economic rate is a desirable attribute. 5. However, pollard has a lower average, which is better since it represents the number of runs conceded per wicket. Moreover, pollard has the lower economy rate, which is the average number of runs per over. 6. This justifies that Pollard should be ranked higher than Ojha.

NameInningsRunsAverageStrike rate50s6sCH Gayle1260867.55183.13344V Kohli1655746.41121.08416SR Tendulkar1655342.53113.3125SE Marsh1350442146.51420MEK Hussey1449241118.8446PC Valthaty1446335.61136.98220SK Raina1643831.28134.76417M Vijay1643427.12128.02320V Sehwag1142438.54176.66218JH Kallis1442435.33112.1646

CH Gayle1260867.55183.133443.825011SE Marsh1350442146.514201.314244V Sehwag1142438.54176.662181.074879V Kohli1655746.41121.084161.058482MS Dhoni1339243.55158.72231.0011114S Badrinath1339656.57126.51590.7699312PC Valthaty1446335.61136.982200.556316SK Raina1643831.28134.764170.417477MEK Hussey1449241118.84460.270275M Vijay1643427.12128.023200.159358

1. Virat kohli and marsh have normal rankings 2 and 4 respectively, yet their FPC-rankings are 4 and 2. 2. Kohli scored 557 total runs with an average of 46.41runs, and a strike rate of 121.08. He hit four-50s and 16 sixes. 3. Shaun marsh scored a total of 504 runs with an average 42 runs, and a strike rate of 146.51. He hit four 50s and 20 sixes. 4. Clearly, marsh has higher strike rate and more sixes, while Kohli is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, marsh is ranked higher than Kohli Top 10 Bowlers of 2011 in Normal Ranking NameInningsWicketAverageEconomiStrike rateSL Malinga162813.395.9513.5MM Patel152216.276.5814.8S Aravind132117.52813.1R Ashwin162019.46.1518.9A Mishra141918.846.7116.8DE Bollinger131719.35716.5R Sharma141617.065.4618.7Iqbal Abdulla151619.066.118.7PP Chawla1216218.1215.5RJ Harris131623.878.1217.6

NameInningsWicketsAverageEconomiStrike rateFPCNormal rankingSL Malinga162813.395.9513.5-2.509421MM Patel152216.276.5814.8-1.547442S Aravind132117.52813.1-1.084163R Ashwin162019.46.1518.9-1.020184R Sharma141617.065.4618.7-0.899937A Mishra141918.846.7116.8-0.895495Iqbal Abdulla151619.066.118.7-0.692438DE Bollinger131719.35716.5-0.602616YK Pathan141318.36.118-0.4989715Shakib Al Hasan71115.96.8613.9-0.2149821

1. Sharma is ranked number 7 in the Normal rankings but appears at number 5 in the list of top ten bowlers when using the FPC method. 3. On the other hand, Mishra ranks 6th in the FPC-ranking method who ranks 5th in the normal rankings. He took 19 wickets with an average of 18.84. His economy rate is 6.71, and his strike rate is 16.8. 4. So, Mishra is better than Sharma with respect to the economic rate variable, which is the number of runs given for total balls bowled. It is true that a low economic rate is a desirable attribute. 5. However, Mishra has a lower average, which is better since it represents the number of runs conceded per wicket. Moreover, Mishra has the lower economy rate, which is the average number of runs per over. 6. This justifies that Mishra should be ranked higher than Sharma

NameInningsRunsAverageStrike rate50s6sCH Gayle1473361.08160.74759G Gambhir1759036.87143.55617S Dhawan1556940.64129.61518AM Rahane1656040129.33310V Sehwag1649533161.23519CL White1347943.54149.68520R Dravid1646228.87112.1324SK Raina1844125.94135.69119RG Sharma1643330.92126.6318Mandeep Singh1643227126.3127

NameInningsRunsAveragesStrike rate50s6sFPCNormal rankingCH Gayle1473361.08160.747593.639131G Gambhir1759036.87143.556171.213282CL White1347943.54149.685201.149816V Sehwag1649533161.235191.083595S Dhawan1556940.64129.615180.936173KP Pietersen830561147.341200.6047526AB de Villiers1331939.87161.113150.4665225AM Rahane1656040129.333100.425414DJ Bravo1637146.37140.530200.1863615F du Plessis1239833.16130.923170.0757513

1. CL White and Dhawan have normal rankings 6 and 3 respectively, yet their FPC-rankings are 3 and 5. 2. White scored 479 total runs with an average of 43.54runs, and a strike rate of 149.68. He hit five-50s and 20 sixes. an average 40.64 runs, and a strike rate of 129.61. He hit five 50s and 19 sixes. 4. Clearly, white has higher strike rate and more sixes, while Dhawan is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, White is ranked higher than Dhawan.

1. Steyn is ranked number 5th in the Normal rankings but appears at number 3 in the list of top ten bowlers when using the FPC method. 2. Steyn took 18 wickets with an average of 15.83. His economy rate is 6.1, and his strike rate is 15.5. 3. On the other hand, Muralitharan, ranks 5th in the FPC-ranking method who ranks 10th in the normal rankings. He took 15 wickets with an average of 17.33. His economy rate is 6.5, and his strike rate is 16.6. 4. So, Steyn is better than Muralitharan with respect to the economic rate variable, which is the number of runs given for total balls bowled. It is true that a low economic rate is a desirable attribute. 5. However, Steyn has a lower average, which is better since it represents the number of runs has the lower economy rate, which is the average number of runs per over. 6. This justifies that Steyn should be ranked higher than Muralitharan.

NameInningsRunsAverageStrike rate50s6sMEK Hussey1773352.35129.5617CH Gayle1670859156.29451V Kohli1663445.28138.73622SK Raina1754842.15150.13418SR Watson1654338.78142.89222RG Sharma1953838.42131.54428KD Karthik1951028.33124.08214AM Rahane1848834.85106.55411R Dravid1747129.43110.8245MS Dhoni1646141.9162.89425

NameInningsRunsAverageStrike rate50s6sFPCNormal rankingCH Gayle1670859156.294512.396252MEK Hussey1773352.35129.56171.453481V Kohli1663445.28138.736221.255913DA Miller1241859.71164.563240.9544714MS Dhoni1646141.9162.894250.8665110SK Raina1754842.15150.134180.775834RG Sharma1953838.42131.544280.765236KA Pollard1842042149.463290.6581113SR Watson1654338.78142.892220.386885AJ Finch1445632.57135.714160.0284511

1. Chris Gayle and Hussey have normal rankings 2 and 1 respectively, yet their FPC-rankings are 1 and 2. 2. Gayle scored 708 total runs with an average of 59 runs, and a strike rate of 156.29. He hit four-50s and 51 sixes. 3. Mike Hussey scored a total of 733 runs with an average 52.35 runs, and a strike rate of 129.5. He hit six 50s and 17 sixes. 4. Clearly, Gayle has higher strike rate and more sixes, while Hussey is not superior in terms of some of the variables that we considered in the analysis. 5. Therefore, Gayle is ranked higher than Hussey.

DJ Bravo183215.537.9511.7JP Faulkner162815.256.7513.5Harbhajan Singh1924196.5117.5MG Johnson172419.127.1716R Vinay Kumar162321.438.1915.6SP Narine162215.95.4617.4A Mishra172118.766.3517.7MM Sharma152016.36.4315.2SL Malinga172023.47.1619.6DW Steyn171920.215.6621.4

NameInningsWicketsAvaregeEconomiStrike rateFPCNormal rankingDJ Bravo183215.537.9511.7-1.980031JP Faulkner162815.256.7513.5-1.611522Harbhajan Singh1924196.5117.5-1.080423SP Narine162215.95.4617.4-1.071326MG Johnson172419.127.1716-0.890034MM Sharma152016.36.4315.2-0.827228A Mishra172118.766.3517.7-0.719417R Vinay Kumar162321.438.1915.6-0.421135DW Steyn171920.215.6621.4-0.2976310SL Malinga172023.47.1619.6-0.032069

1. Narine is ranked number 6th in the Normal rankings but appears at number 4th in the list of top ten bowlers when using the FPC method. 2. Narine took 16 wickets with an average of 15.9. His economy rate is 5.46, and his strike rate is 17.4. 3. On the other hand, Johnson, ranks 5th in the FPC-ranking method who ranks 4th in the normal rankings. He took 17 wickets with an average of 19.22. His economy rate is 7.17, and his strike rate is 16. 4. So, Narine is better than Johnson with respect to the economic rate variable, which is the number of runs given for total balls bowled. It is true that a low economic rate is a desirable attribute. 5. However, Narine has a lower average, which is better since it represents the number of runs conceded per wicket. Moreover, Johnson has the lower economy rate, which is the average number of runs per over. 6. This justifies that Narine should be ranked higher than Johnson

CONCLUSION

To summarize, a simple yet straightforward method for analyzing the overall performance of IPL players from 2008 to 2017 is the proposed method based on principal component analysis. It is transparent and can be directly applied to the type of correlated data gives different ranking based on considering overall performance. The ability of the first principal component method to consistently capture a significant proportion of variability in the cricket athletic performance is the key strength of the proposed method, which offers a transparent alternative or serves as an additional measure.

Allsopp, P.E., & Clarke, S.R. (2004). Rating teams and analyzing outcomes in one-day and test cricket. Journal of the Royal Statistical Society: Series A, 4, pp. 657–667 Duckworth, F. C., & Lewis, A.J. (2004). A successful operational research intervention in one-day cricket. Journal of the Operational Research Society, 55, pp. 749–759 Fernando, M., Manage, A.B.W., & Scariano, S. (2013). Is the home-field advantage in limited overs one-day international cricket only for day matches? South African Statistics Journal, 47, pp. 1–13 Lemmer, H.H. (2013). Team selection after a short cricket series. European Journal of Sport Science, 13, pp. 200–206 McHale, I.G., & Asif, M. (2013).A modified Duckworth-Lewis method for adjusting targets in interrupted limited overs cricket. European Journal of Operational Research, 225, pp. 353–362. Norton, P. & Phatarfod, R. (2008). Optimal strategies none-day cricket. Asia-Pacific Journal of Operational Research, 25, pp. 495–511. Swartz, T.B., Gill, P.S., Beaudoin, D., & deSilva, B.M. (2006). Optimal batting orders in one-day cricket. Computers and Operations Research, 33, pp. 1939–1950 Valero, J., & Swartz, T.B. (2012). An investigation of synergy between batsmen in opening partnerships. SriLankan Journal of Applied Statistics, 13, pp. 87–98 Van Staden, P.J. (2009). Comparison of cricketers‘ bowling and batting performances using graphical displays. Current Science, 96, pp. 764–766 Wright, M. (2014). Or analysis of sporting rules–a survey. European Journal of Operational Research, 232, pp. 1–8 Swartza (2015). A study of the power-play in one-day cricket. European Journal of Operational Research, DOI:10.1016/j.ejor.2015.02.004