ABSTRACT

In cricket, a batsman’s performance is often assessed by their average (arithmetic mean) runs scored over their career. However, the mean alone doesn’t provide a complete picture. The next step is to calculate the standard deviation, which describes how ‘spread out’ a batsman’s scores are. Better batsmen, with higher averages, tend to have larger standard deviations due to their higher scores. Consequently, ‘most consistent’ performers might appear to be those with lower average scores. This paper aims to test the significance of the consistency in the performances of the top 20 ICC T20 ranked batsmen as of 16 July 2024. To achieve this, secondary data was collected and analyzed.

INTRODUCTION

Cricket is a bat-and-ball game played between two teams of eleven players on a cricket field, with a 22-yard-long rectangular pitch at the center, featuring a wicket (a set of three wooden stumps) at each end. One team bats, aiming to score as many runs as possible, while the opposing team fields. Each phase of play is called an innings. After ten batsmen have been dismissed or a set number of overs have been completed, the innings ends, and the teams swap roles. The team with the most runs at the end of their innings wins the game. The International Cricket Council (ICC), the governing body of cricket, oversees its rules and regulations.

The ICC Player Rankings are a sophisticated moving average system, rating players on a scale from 0 to 1000 points. A player’s points increase if their performance improves compared to their past record and decrease if their performance declines. The value of each performance within a match is calculated using an algorithm that considers various match circumstances.

For a batsman, the rating factors include runs scored and the ratings of the opposing bowling attack—the higher the combined ratings of the attack, the more value is attributed to the batsman’s innings. Additional factors include the level of run-scoring in the match and performance in victories, with a higher bonus for victories against highly rated teams. In T20 (Twenty20 cricket), batsmen gain significant credit for rapid scoring and only a small amount of credit for remaining not out.

T20 cricket is a short form of cricket recognized by the ICC, initially introduced by the England and Wales Cricket Board (ECB) in 2003 for inter-county competitions. In a Twenty20 game, each team has a single innings limited to a maximum of 20 overs, with a break between the innings.

OBJECTIVES OF THE STUDY:

This study aims to assess the batting consistency of the top 20 ranked T20 cricketers in July 2024. Using appropriate techniques, the players are compared based on their batting average and consistency.

MATERIALS USED IN THE STUDY:

T20 run data for the top 20 ICC-ranked cricketers in July 2024 was obtained from websites ICC Men’s Ranking Batting and Player Stats (by adding the player name in Player Stats page).

DIFFERENT MEASURES USED IN THE PRESENT STUDY

The measures used in the present study are as follows:

Mean (or Average): The mean is the most popular and well-known measure of central tendency. It provides an overall indication of the typical value in a data set. For a data set with \(n\) values, represented as ( \(x_1, x_2,\ldots,x_n\)), the mean is calculated using the following formula:

\(Mean(\bar{x})=\frac1n\Sigma_{i=1}^{n}x_{i}\)

Standard Deviation: The standard deviation measures the amount of variation or dispersion in a data set. It indicates how spread out the values are from the mean. For the same data set, the standard deviation (\(\sigma\)) is calculated as follows:

\({\sigma=\sqrt{\frac1n\sum_{i=1}^n(x_{i}-\bar{x})^2}}\)

These measures provide a basis for analyzing the consistency and variability in the performances of the top 20 ICC T20 ranked batsmen.

The names, ranks and the points along with other data of top twenty Cricketer of T20 ICC ranking are shown in the following table:

Rank Name Country Innings.played Average.Runs Points Age Strike.Rate Total.Runs Not.Out Adjusted.Innings
1 Travis Head Australia 32 32.53 844 30 150.08 911 4 28
2 Suryakumar Yadav India 65 43.33 797 33 167.74 2340 11 54
3 Phil Salt England 29 35.40 797 27 165.11 885 4 25
4 Babar Azam Pakistan 116 41.03 755 29 129.09 4145 15 101
5 Mohammad Rizwan Pakistan 89 48.72 746 32 126.45 3313 21 68
6 Yashasvi Jaiswal India 19 37.82 743 22 162.78 643 2 17
7 Jos Butler England 114 35.87 716 33 146.37 3264 23 91
8 Ruturaj Gaikwad India 20 39.56 684 27 143.54 633 4 16
9 Brandon King West Indies 53 29.06 656 29 134.39 1395 5 48
10 Johnson Charles West Indies 56 23.25 655 35 131.38 1302 0 56
11 Aiden Markram South Africa 44 33.54 646 29 143.63 1241 7 37
12 Nicholas Pooran West Indies 87 26.61 639 28 135.86 2076 9 78
13 Rahmanullah Gurbaz Afghanistan 63 26.30 636 22 135.49 1657 0 63
14 Quinton De Kock South Africa 91 31.51 631 31 138.33 2584 9 82
15 Jonny Bairstow England 72 29.83 629 34 137.53 1671 16 56
16 Reeza Hendricks South Africa 67 29.87 629 34 129.12 1942 2 65
17 Glenn Phillips New Zealand 69 32.89 620 27 142.08 1875 12 57
18 Mitchell Marsh Australia 59 32.43 617 32 133.65 1557 11 48
19 Finn Allen New Zealand 47 24.27 612 25 158.69 1141 0 47
20 Rilee Rossouw South Africa 27 34.86 610 34 159.79 767 5 22

The runs scored by the top players along with other information is displayed in the following table:

’ symbol represents Not Out in an inning.

To prepare a modified table of a player’s runs by adding the runs from not-out innings to the preceding or succeeding innings:

Name Country Match.by.match.runs.scored…yij. Total.Runs Adjusted.Innings Average.Runs
1 Travis Head Australia 2,26,45,40,4,30,57,6,15,27,48,19,26,6,18,91,35,31,28,24,45,33,12,68,68,31,0,76 911 28 32.53
2 Suryakumar Yadav India 57,32,50,17,87,1,34,8,65,0,15,39,15,117,24,11,76,24,86,13,34,6,46,0,119,61,8,66,68,91,125,13,7,163,47,50,21,1,83,61,80,19,39,1,5,56,100,2,57,53,6,31,47,3 2340 54 43.33
3 Phil Salt England 57,0,3,8,10,30,8,8,91,20,10,38,25,0,40,134,119,38,13,45,37,12,98,36,5 885 25 35.40
4 Babar Azam Pakistan 89,56,27,43,38,86,45,48,1,35,91,18,114,119,45,50,7,40,79,38,90,23,65,13,3,86,50,6,66,56,21,82,51,0,5,44,14,50,122,24,2,41,52,85,22,11,51,77,51,70,66,39,7,1,19,0,7,79,66,10,9,14,0,30,5,31,118,36,9,91,22,100,55,15,0,4,4,6,25,53,32,9,102,19,57,66,58,19,13,14,37,5,69,57,0,75,32,36,44,13,65 4145 101 41.03
5 Mohammad Rizwan Pakistan 6,50,16,6,24,4,3,26,31,14,5,17,22,89,155,115,72,82,103,63,112,125,33,87,15,67,11,39,40,78,38,87,23,121,91,14,55,68,96,88,63,79,4,16,69,34,4,14,49,4,32,57,15,8,50,104,25,7,114,83,23,131,0,23,9,31,70 3313 68 48.72
6 Yashasvi Jaiswal India 85,5,24,18,100,0,21,53,6,37,21,0,60,68,4,36,105 643 16 37.82
7 Jos Butler England 13,3,7,7,38,15,12,13,132,17,27,0,20,22,8,0,67,3,32,2,34,6,26,10,11,33,34,54,30,21,72,68,89,15,0,10,31,30,5,46,2,2,61,69,14,34,13,15,2,57,44,84,89,28,83,9,120,59,45,89,127,29,0,4,18,22,29,14,68,82,18,0,73,108,28,67,4,53,40,39,5,51,55,11,84,39,42,24,25,100,23 3264 91 35.87
8 Ruturaj Gaikwad India 21,14,4,23,1,57,5,29,58,25,40,58,155,10,84,49 633 16 39.56
9 Brandon King West Indies 4,12,1,31,5,33,43,13,0,11,1,67,43,52,10,26,34,4,22,57,7,68,20,13,53,12,23,79,23,1,36,28,0,42,103,22,90,0,3,53,5,79,36,44,34,13,9,30 1395 48 29.06
10 Johnson Charles West Indies 36,21,24,37,24,36,16,84,12,8,10,0,57,26,10,1,0,16,7,4,0,34,0,10,32,22,52,1,79,43,7,10,5,3,29,45,24,28,118,0,3,2,12,27,42,24,4,1,7,69,0,44,0,43,38,15 1302 56 23.25
11 Aiden Markram South Africa 3,15,51,54,63,11,23,20,70,39,8,69,91,19,52,107,27,25,33,10,52,20,17,52,42,49,41,30,25,12,0,4,15,46,1,18,27 1241 37 33.54
12 Nicholas Pooran West Indies 5,4,16,57,37,29,58,1,11,20,19,17,38,27,14,1,7,0,8,23,9,26,16,20,49,16,31,13,63,12,40,46,4,18,26,91,24,70,22,21,61,62,61,108,18,14,22,24,3,15,1,2,2,5,7,13,16,2,41,41,67,20,1,47,13,5,82,39,10,18,18,1,27,22,17,98,63,1 2076 78 26.61
13 Rahmanullah Gurbaz Afghanistan 43,0,61,29,0,15,79,28,35,42,87,9,18,46,10,5,19,6,0,3,33,1,26,1,53,24,4,40,11,84,17,0,10,28,30,10,15,20,16,44,18,16,8,100,21,20,23,14,50,13,13,70,0,3,6,76,80,11,0,11,60,43,0 1657 63 26.30
14 Quinton De Kock South Africa 30,5,19,64,30,43,67,41,25,4,0,29,6,46,0,48,12,44,7,44,25,52,45,47,9,0,59,20,5,26,22,13,131,31,65,35,2,70,5,30,30,17,37,26,72,60,60,20,27,94,66,12,16,34,22,14,2,15,0,7,7,70,115,63,0,13,0,100,21,4,41,19,20,0,18,10,74,65,12,5,39 2584 82 31.51
15 Jonny Bairstow England 28,63,15,4,12,1,18,7,38,8,61,47,27,14,28,25,68,12,37,35,0,8,47,23,35,64,2,44,0,8,9,55,89,46,65,7,13,51,11,13,5,17,16,1,13,90,30,27,90,12,73,49,15,31,64 1671 56 29.83
16 Reeza Hendricks South Africa 0,18,49,12,42,41,3,7,0,70,26,7,19,19,74,28,5,8,65,66,6,28,14,16,13,54,42,2,17,42,17,2,69,38,74,39,11,4,2,4,23,57,53,70,74,42,21,68,83,56,3,42,49,8,87,34,6,4,3,0,43,11,19,29,4 1942 65 29.87
17 Glenn Phillips New Zealand 5,11,56,17,3,5,12,5,26,22,108,23,31,30,8,13,35,96,13,33,41,18,0,34,69,79,23,14,17,76,41,41,60,29,12,104,62,17,6,12,54,17,5,2,41,22,69,42,9,1,19,13,89,45,82,18,40 1875 57 32.89
18 Mitchell Marsh Australia 36,13,13,37,28,6,21,21,19,6,58,45,6,6,10,51,54,9,75,30,45,45,51,11,4,27,53,28,88,3,36,45,0,16,17,28,137,94,16,29,89,26,14,35,26,1,12,37 1557 48 32.43
19 Finn Allen New Zealand 0,17,71,15,12,41,1,35,14,101,6,8,13,13,16,62,32,12,42,1,16,32,4,0,3,35,11,3,21,3,83,16,1,2,38,34,74,137,8,22,32,6,13,0,26,9,0 1141 47 24.27
20 Rilee Rossouw South Africa 78,12,55,15,31,57,26,18,19,16,0,100,31,0,100,109,0,7,25,10,16,42 767 22 34.86

Methodology:

This study aimed to identify the most consistent batsman among the top 20 ICC T20-ranked cricketers. To achieve this, secondary data encompassing the complete T20 career scores of these players was collected from ICC Men’s Batting and Player Stats.

A modified dataset was constructed by excluding ‘Not Out’ records. To calculate average runs, ‘Not Out’ scores were incorporated into the preceding or succeeding innings. A batsman will be categorized as ‘good’ if their average runs exceeded 30. To evaluate the significance of batting performance differences, either ANOVA or Kruskal-Wallis test is employed. Descriptive statistics (mean, standard deviation) and graphical analysis are also conducted to support the findings.

To choose ANOVA when:

Data is Normal: The data within each group follows a normal distribution.

Equal Variances: Groups have similar variances.

Continuous Data: The outcomes are measured on a continuous scale.

To choose Kruskal-Wallis test when:

Non-Normal Data: The data isn’t normally distributed.

Unequal Variances: Groups have different variances.

Ordinal Data: The outcomes are ordinal or ranks are more suitable.

ANOVA suits parametric data while Kruskal-Wallis is ideal for non-parametric data.

Basic Claim: There is no significant difference in the performance of the batsmen based on the averages being compared.

CALCULATION:

CALCULATION OF STANDARD DEVIATION:

The standard deviation is displayed in the following table:

Name Country Total.Runs Adjusted.Innings Average.Runs Standard.Deviation
Travis Head Australia 911 28 32.53 22.82774
Suryakumar Yadav India 2340 54 43.33 37.60418
Phil Salt England 885 25 35.40 36.74997
Babar Azam Pakistan 4145 101 41.03 31.99472
Mohammad Rizwan Pakistan 3313 68 48.72 39.12345
Yashasvi Jaiswal India 643 16 37.82 34.89849
Jos Butler England 3264 91 35.87 32.01792
Ruturaj Gaikwad India 633 16 39.56 37.74084
Brandon King West Indies 1395 48 29.06 25.70296
Johnson Charles West Indies 1302 56 23.25 24.14558
Aiden Markram South Africa 1241 37 33.54 24.79821
Nicholas Pooran West Indies 2076 78 26.61 24.48926
Rahmanullah Gurbaz Afghanistan 1657 63 26.30 25.33323
Quinton De Kock South Africa 2584 82 31.51 27.87289
Jonny Bairstow England 1671 56 29.83 24.72121
Reeza Hendricks South Africa 1942 65 29.87 25.26384
Glenn Phillips New Zealand 1875 57 32.89 27.91006
Mitchell Marsh Australia 1557 48 32.43 27.65254
Finn Allen New Zealand 1141 47 24.27 28.58857
Rilee Rossouw South Africa 767 22 34.86 33.16665

The players are then ranked according to their consistency.

The following objects are masked from data:

    Adjusted.Innings, Average.Runs, Country, Name, Total.Runs
Name Country Total.Runs Adjusted.Innings Average.Runs Standard.Deviation rank.on.consistency
Travis Head Australia 911 28 32.53 22.82774 1
Suryakumar Yadav India 2340 54 43.33 37.60418 18
Phil Salt England 885 25 35.40 36.74997 17
Babar Azam Pakistan 4145 101 41.03 31.99472 13
Mohammad Rizwan Pakistan 3313 68 48.72 39.12345 20
Yashasvi Jaiswal India 643 16 37.82 34.89849 16
Jos Butler England 3264 91 35.87 32.01792 14
Ruturaj Gaikwad India 633 16 39.56 37.74084 19
Brandon King West Indies 1395 48 29.06 25.70296 8
Johnson Charles West Indies 1302 56 23.25 24.14558 2
Aiden Markram South Africa 1241 37 33.54 24.79821 5
Nicholas Pooran West Indies 2076 78 26.61 24.48926 3
Rahmanullah Gurbaz Afghanistan 1657 63 26.30 25.33323 7
Quinton De Kock South Africa 2584 82 31.51 27.87289 10
Jonny Bairstow England 1671 56 29.83 24.72121 4
Reeza Hendricks South Africa 1942 65 29.87 25.26384 6
Glenn Phillips New Zealand 1875 57 32.89 27.91006 11
Mitchell Marsh Australia 1557 48 32.43 27.65254 9
Finn Allen New Zealand 1141 47 24.27 28.58857 12
Rilee Rossouw South Africa 767 22 34.86 33.16665 15

Average Runs and Standard deviations of the runs scored by the players:

Checking Normality:

The shape of the histogram is not bell shaped and highly skewed and also the Q-Q plot (quantile-quantile plot) line is not straight indicating that the underlying distribution of the variable is not normal distribution which suggests that ANOVA cannot be used to examine the equality of average runs of the batsmen.

-

Checking heteroscedasticity:

Conducting Levene’s Test:

Levene's Test for Homogeneity of Variance (center = median)
        Df F value Pr(>F)
group   19  1.2206 0.2319
      1036

The boxplot shows marginal presence of unequal variances among the different batsmen and levene’s test has a p-value of 0.2319 (>0.05) resulting in acceptance of null hypothesis. Therefore, it can be concluded that heteroscedasticity is absent.

-

KRUSKAL-WALLIS TEST:

The Kruskal-Wallis test is a non-parametric statistical method used to compare three or more independent groups to determine if there are statistically significant differences in their medians. It serves as an alternative to the one-way ANOVA when the assumptions of normality and homogeneity of variance are not met.

Hypotheses:

Null Hypothesis (\(H_0\)): The medians of the groups are equal.

Alternative Hypothesis (\(H_1\)): At least one group median is different from the others.

Procedure:

Ranking all data: Combine all observations from the groups and rank them from lowest to highest.

Calculating the test statistic: Using the ranks to compute the Kruskal-Wallis H statistic, which reflects the differences between the average ranks of the groups.

Determining significance: Comparing the H statistic to a chi-square distribution to obtain a p-value. A p-value less than the significance level (\(\alpha\)) indicates that the null hypothesis can be rejected.


    Kruskal-Wallis rank sum test

data:  Runs by Name
Kruskal-Wallis chi-squared = 20.389, df = 19, p-value = 0.3715

Since the p-value is greater than the level of significance 0.05, the null hypothesis cannot be rejected at \(5\)% level of significance .

-

MODEL FITTING AND CLUSTERING:

The assumed regression function is: \(y_t=\alpha+\beta sin(\gamma t)\) for all \(t\) is the number of innings played by each player.

Applying Gauss Newton Theorem to find the estimates of the parameters of the function \(\hat\alpha\), \(\hat\beta\) and \(\hat\gamma\).

Fitting of the regression line with the estimated parameters:

               [,1]
dfdalpha  29.149692
dfdbeta  -16.498140
dfdgamma   7.749984

               [,1]
dfdalpha  32.542135
dfdbeta  -16.299451
dfdgamma   2.118673

             [,1]
dfdalpha 23.11928
dfdbeta  12.94644
dfdgamma  2.24834

                [,1]
dfdalpha  31.6583592
dfdbeta  -10.6977077
dfdgamma   0.9339909

              [,1]
dfdalpha 28.622201
dfdbeta  -8.598239
dfdgamma  1.202331

              [,1]
dfdalpha 27.638843
dfdbeta  13.101247
dfdgamma  2.829934

              [,1]
dfdalpha 24.056974
dfdbeta   7.313610
dfdgamma  3.037468

              [,1]
dfdalpha  22.91284
dfdbeta  -13.57615
dfdgamma  22.97408

              [,1]
dfdalpha 22.825754
dfdbeta  -7.784337
dfdgamma  2.783021

              [,1]
dfdalpha 23.706097
dfdbeta  -9.926362
dfdgamma  2.998074

              [,1]
dfdalpha 26.229157
dfdbeta   8.170593
dfdgamma  2.814021

              [,1]
dfdalpha 22.271984
dfdbeta   4.029426
dfdgamma  3.009327

               [,1]
dfdalpha  26.496474
dfdbeta  -12.147676
dfdgamma   1.642807

              [,1]
dfdalpha 26.587771
dfdbeta   6.557524
dfdgamma  2.871559

              [,1]
dfdalpha 20.550137
dfdbeta   6.940338
dfdgamma  2.005519

              [,1]
dfdalpha 28.656648
dfdbeta  -9.026537
dfdgamma  1.887713

              [,1]
dfdalpha 25.053708
dfdbeta   9.297533
dfdgamma  2.972163

              [,1]
dfdalpha 24.021432
dfdbeta   8.066466
dfdgamma  4.749136

             [,1]
dfdalpha 24.72247
dfdbeta  -8.43184
dfdgamma  2.96804

              [,1]
dfdalpha 21.490301
dfdbeta  12.601597
dfdgamma  1.678942

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[20,] 85.13519 78.95568 66.24953 82.28001 71.61704 84.71127 78.76547 68.63672
          [,9]    [,10]    [,11]    [,12]    [,13]    [,14]    [,15]    [,16]
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 [5,] 63.37192 84.24369 73.95269 81.37567 68.70226 69.40461 91.38381 78.39643
 [6,] 81.08637 64.57554 77.33046 75.54469 80.30567 79.51101 67.51296 99.50377
 [7,] 80.44253 73.36212 70.17122 77.18160 72.22880 85.70881 84.80566 65.61250
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 [9,]  0.00000 86.61986 80.47981 78.72738 71.17584 73.88505 77.09086 84.01190
[10,] 86.61986  0.00000 85.05292 84.68176 68.78227 72.58099 80.29944 79.20227
[11,] 80.47981 85.05292  0.00000 85.05880 78.69562 70.82372 72.73239 83.45658
[12,] 78.72738 84.68176 85.05880  0.00000 86.98276 87.53856 75.95393 67.46851
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         [,17]    [,18]    [,19]    [,20]
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 [6,] 67.52777 71.23202 74.07429 84.71127
 [7,] 89.17399 68.84766 74.86655 78.76547
 [8,] 62.85698 94.95789 65.03845 68.63672
 [9,] 69.73521 66.09841 87.51000 70.94364
[10,] 81.15417 67.54258 61.06554 95.57196
[11,] 63.65532 81.59657 67.50556 54.11100
[12,] 85.55115 68.27152 71.62402 71.77047
[13,] 64.14827 75.11990 62.70566 81.22192
[14,] 67.23095 68.52737 84.66995 72.48448
[15,] 81.46165 78.26877 69.13031 80.52329
[16,] 93.46122 86.08717 68.52737 69.37579
[17,]  0.00000 86.55634 79.21490 65.63536
[18,] 86.55634  0.00000 82.84323 87.01724
[19,] 79.21490 82.84323  0.00000 82.26178
[20,] 65.63536 87.01724 82.26178  0.00000

CONCLUSION

Based on the analysis, Travis Head (Australia) exhibits the lowest standard deviation (\(\sigma = 22.82774\)) among the top 20 ICC T20 cricketers in 16, July 2024. This indicates superior consistency compared to his peers. Given his lower variability in scores and an average of 32.53 runs, Head is likely to achieve scores closer to his average more frequently.

Johnson Charles (West Indies) and Nicholas Pooran (West Indies) follow Travis Head as the next most consistent players, based on their respective standard deviations and averages. Indian batsmen Suryakumar Yadav, Yashasvi Jaiswal and Ruturaj Gaikwad occupy the eighteenth, sixteenth and nineteenth positions, respectively, in terms of consistency among the top 20 ICC T20 cricketers in July, 2024.

Batsmen performances are homogeneous.

The dendrogram you have provided is a hierarchical clustering of the top 20 ICC T20I batsmen, where each number represents a specific batsman.

• Height: The vertical axis represents the “distance” or “similarity” between clusters. The higher the link on the dendrogram, the more dissimilar the clusters.

• Players are grouped based on their similarity in performance related to their ranking.

• Players in close proximity to each other (such as 11 and 20, 5 and 9 or 1 and 13 and 3 and 7) have similar performance metrics or playing styles. These groups could potentially represent players with similar strengths, weaknesses, or roles (e.g., similar strike rates or run-scoring patterns).

• Players 11 and 20 and 8 and 17 form clusters separately and merge at a higher distance, indicating that they are relatively dissimilar from each other but belong to a larger cluster.

• The batsmen can be divided into two broad groups, where each group might represent a different style or consistency level. For instance, the first group might include batsmen who are more aggressive or consistent, while the second group might contain batsmen with different strengths. The first major group includes players 7, 16, 3, 12, 18, 5, 9, 1 and 13. The second major group includes players 11, 20, 8, 17, 14, 4, 19, 15, 10, 2 and 6.

• Cutting the dendrogram around a height of 70 suggests 4-5 meaningful clusters.

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