Wicket taking ability of bowlers

Abstract

The purpose of this study is to define a measure for the wicket taking ability of bowlers and to give perspective on the use of different bowling performance measures. The bowler who had taken the largest number of wickets in a match or a series of matches is normally called the best bowler. If more than one bowler had taken an equal number of wickets, the bowlers in such a group are ranked according to the number of runs conceded. This practice of giving the bowling award to the bowler who had taken the most wickets is challenged because it ignores the number of overs bowled. Various measures are considered and recommendations made about the assessment of the wicket taking ability and bowling performance of a bowler.

Keywords

Cricket bowling performance analysis sport analytics Twenty20 cricket

Introduction

For award winning purposes, bowlers are normally ranked according to the number of wickets taken. If two or more had taken the same number of wickets, the bowlers in this group are then ranked according to the number of runs conceded with the one who had conceded the smallest number of runs, in the top position. Equivalently, the bowler with the smallest average, A, in the group is ranked highest because A = R/W where W denotes the number of wickets taken and R the number of runs conceded. Alternatively, the ranking can be obtained by first ranking the bowlers according to increasing R and then according to decreasing W. We call this traditional procedure as the conditional ranking method. A formula is constructed for this conditional ranking method and it is modelled in such a way that the wicket taking performances of the bowlers can be measured on a continuous scale. Thereby it becomes possible to quantify the difference between the wicket taking performances of any two bowlers. It is shown that this measure complies with general statistical requirements. It is argued that the bowler who had taken the most wickets in a series is not necessarily the best bowler. By taking the number of overs bowled by a bowler into account, a formula is defined for the wicket taking ability of a bowler which is much more reasonable than the conditional ranking method.

Good wicket taking performance is best reflected by a large value of W augmented by a large value of 1/A. Consider

CRA = W + 1 / A

(1)

as a crude measure. If the bowlers are ranked according to CRA, the same result is obtained as by using the methods described above. But CRA as such is not a sensible measure of wicket taking performance because the values of 1/A are generally quite small, that is the value of CRA is only marginally larger than W. The purposes of this study are firstly to model the wicket taking performance of bowlers in a series by means of a suitable measure and then to define a measure for the wicket taking ability of bowlers. It will further be shown that, compared to other measures, the commonly used method where the bowler who had taken the largest number of wickets is called the best bowler, is not always the most reasonable method to determine the best bowler because it does not take other factors sufficiently into account.

The traditional bowling performance measures are the bowling average which is defined by A = R/W, the economy rate by E = R/O where O denotes the number of overs bowled, and the strike rate by S = B/W with B the number of balls bowled. Bairam et al.¹ called the strike rate ‘attacking bowling’ and the economy rate ‘defensive bowling’. According to Kimber² the average has traditionally been used to compare bowlers, but the economy rate and strike rate have more recently increased in popularity. Each of these measures is important in its own right, but some authors use combinations of these measures. Croucher³ defined the bowling index BI = A×S and used this to rank bowlers. Basevi and Binoy⁴ used CALC = A×E/6. A more comprehensive measure that has been designed to take A, E and S deliberately into account is the combined bowling rate

CBR = \frac{3 R}{(W + O + W \frac{R}{B})}

(2)

for use in limited overs matches.⁵ CBR is a weighted average between A, E and S in which the economy rate weights highly because in a limited overs match it is much more important to limit the scoring rate of batsmen than to take wickets. These and other methods do not address the topic of the present study, namely the wicket taking ability of bowlers.

Method

The quantity CRA defined in equation (1) does not assume values with a reasonable covering of the real line because the values of 1/A will obviously be very small. A large data set is used to determine a more appropriate model. The bowling figures of all bowlers who had taken at least six wickets in the World Cup Series of 2002/03, 2006/07, 2010/11 and 2014/15 have been taken from Cricinfo.⁶ For illustration purposes the bowlers of the smallest data set, that is that of the 2006/07 World Cup Series, is considered. For these bowlers, the values of 1/A vary between 0.020 and 0.073. For any given value of W the value of CRA is only slightly larger than W (between W + 0.020 and W + 0.073), that is it will never be close to W + 1. It is therefore better to consider

CRK = W + k / A

(3)

with k a positive real number in such a way that the values of CRK have a better spread over the real line than the values of CRA. The challenge now is to find an appropriate value for k in such a way that the values of CRK resemble values over the positive real line. If this can be executed in such a way that a continuous statistical distribution can be fitted satisfactorily to the CRK values, the model can be seen as a model for the wicket taking performance of bowlers in the series.

A logical requirement about k is that the ranking according to CRK must not differ from that according to the conditional ranking method. This implies that if for any two bowlers i and j

W_{i} = W_{j} - 1, then W_{i} + k / A_{i} \leq W_{j} + k / A_{j}

This can be achieved if

W_{j} - 1 + k . max (1 / A_{i}) \leq W_{j} + k . min (1 / A_{j})

Solvek . max (1 / A_{i}) = 1 + k . min (1 / A_{i})

thatisk = \frac{1}{max (1 / Ai) - min (1 / Ai)}

thatisk = \frac{1}{1 / min (Ai) - 1 / max (Ai)}

(4)

For the 2006/07 data set min(A_i) = 13.73 and max(A_i) = 51 resulting in k = 18.79. The criterion is therefore

CRK = W + 18.79 / A

(5)

In order to fit a distribution to the data it must be taken into account that the study is based on bowlers who had taken six or more wickets. We therefore fit a distribution to the values of CRK − 6. Goodness of fit is tested by using the chi-squared and the Kolmogorov–Smirnov tests. The latter is specifically designed for data from continuous populations.^{7: 120} An exponential distribution with parameter λ = 0.1623 gives a good fit with p-value 0.421 according to the chi-squared goodness of fit test and a very small value of D = 0.099 according to the Kolmogorov–Smirnov test. From Table 1 of Lilliefors⁸ the 20% critical value for n = 39 is 0.138. See Lilliefors⁸ regarding the use of the Kolmogorov–Smirnov test in the case of an exponential distribution. The venture to construct CRK in order to obtain values from a continuous distribution was clearly highly successful. The exponential fit to CRK − 6 is much better than the fit to the W − 6 values which resulted in a p-value of 0.123. In Table 1 the bowlers are ranked according to CRK.

Table 1.

Ranking of the 2006/07 World Cup bowlers according to CRK.

CRK rank	Name	O	R	W	A	E	S	CBR	CRK	INC	CRM	CRM rank	S rank	CBR rank
1	GD McGrath	80.5	357	26	13.73	4.41	18.6	8.50	27.37	1.37	0.339	1	1	4
2	Muralitharan	84.4	351	23	15.26	4.14	22.0	8.52	24.23	1.23	0.286	4	4	5
3	SW Tait	84.3	467	23	20.30	5.52	22.0	10.89	23.93	0.93	0.283	5	3	22
4	GB Hogg	82.5	332	21	15.80	4.00	23.6	8.45	22.19	1.19	0.268	6	6	3
5	SL Malinga	58.2	284	18	15.77	4.86	19.4	9.37	19.19	1.19	0.329	2	2	10
6	NW Bracken	71.4	258	16	16.12	3.60	26.8	7.96	17.17	1.17	0.240	8	8	2
7	DL Vettori	97.4	447	16	27.93	4.57	36.6	10.65	16.67	0.67	0.171	23	22	20
8	A Flintoff	69	298	14	21.28	4.31	29.5	9.60	14.88	0.88	0.216	11	11	13
9	AJ Hall	76	335	14	23.92	4.40	32.5	10.02	14.79	0.79	0.195	15	15	15
10	C Langeveldt	66	361	14	25.78	5.46	28.2	11.67	14.73	0.73	0.223	10	9	31
11	DBL Powell	85	385	14	27.50	4.52	36.4	10.54	14.68	0.68	0.173	21	21	18
12	SE Bond	69.4	213	13	16.38	3.05	32.1	7.16	14.15	1.15	0.203	14	14	1
13	WPUJC Vaas	77.4	286	13	22.00	3.68	35.8	8.70	13.85	0.85	0.178	19	19	6
14	Abdur Razzak	73.4	344	13	26.46	4.66	34.0	10.66	13.71	0.71	0.186	16	16	21
15	DJ Bravo	64.5	361	13	27.76	5.56	29.9	12.05	13.68	0.68	0.211	13	13	35
16	A Nel	52.2	217	12	18.08	4.14	26.1	8.96	13.04	1.04	0.249	7	7	7
17	WB Rankin	59.2	324	12	27.00	5.46	29.6	11.82	12.70	0.70	0.214	12	12	32
18	JEC Franklin	65.2	331	11	30.09	5.06	35.6	11.60	11.62	0.62	0.178	20	18	30
19	W McCallan	58.4	233	10	23.30	3.97	35.2	9.28	10.81	0.81	0.184	17	17	9
20	JDP Oram	62.3	252	10	25.20	4.03	37.5	9.54	10.75	0.75	0.172	22	23	12
21	MF Maharoof	44	198	9	22.00	4.50	29.3	9.94	9.85	0.85	0.224	9	10	14
22	SB Styris	57	242	9	26.88	4.24	38.0	10.03	9.70	0.70	0.170	25	25	16
23	M Mortaza	71.2	323	9	35.88	4.52	47.5	11.12	9.52	0.52	0.134	28	28	25
24	Syed Rasel	62	239	8	29.87	3.85	46.5	9.54	8.63	0.63	0.139	26	26	11
25	DT Johnston	50.2	258	8	32.25	5.12	37.7	11.88	8.58	0.58	0.171	24	24	33
26	SI Mahmood	48	259	8	32.37	5.39	36.0	12.30	8.58	0.58	0.179	18	20	37
27	SM Pollock	84	296	8	37.00	3.52	63.0	9.18	8.51	0.51	0.101	38	36	8
28	M Rafique	72	302	8	37.75	4.19	54.0	10.59	8.50	0.50	0.118	30	30	19
29	JM Anderson	73	329	8	41.12	4.50	54.7	11.34	8.46	0.46	0.116	32	31	28
30	JS Patel	54.2	245	7	35.00	4.50	46.5	11.04	7.54	0.54	0.139	27	27	24
31	MS Panesar	64	283	7	40.42	4.42	54.8	11.15	7.46	0.46	0.117	31	32	26
32	D Lang-Smith	66	291	7	41.57	4.40	56.5	11.17	7.45	0.45	0.113	33	33	27
33	ST Jayasuriya	67.2	302	7	43.14	4.48	57.7	11.39	7.44	0.44	0.122	29	34	29
34	S Al Hasan	60.5	302	7	43.14	4.96	52.1	12.31	7.44	0.44	0.110	34	29	38
35	EChigumbura	22.4	116	6	19.33	5.11	22.6	10.30	6.97	0.97	0.308	3	5	17
36	C Collymore	63.2	268	6	44.66	4.23	63.3	10.93	6.42	0.42	0.101	36	38	23
37	PCollingwood	58	292	6	48.66	5.03	58.0	12.69	6.39	0.39	0.110	35	35	39
38	M Ntini	63	293	6	48.83	4.65	63.0	11.93	6.38	0.38	0.101	37	37	34
39	JE Taylor	64	306	6	51.00	4.78	64.0	12.28	6.37	0.37	0.100	39	39	36

Note that INC = CRK − W. The definition of CRM is given later. The first and last three columns give the ranks according to CRK, CRM, S and CBR.

Note that bowlers who had taken the same number of wickets are indeed ranked according to increasing A, which confirms that CRK ranks them according to the conditional ranking method. The values of the increment INC = CRK − W vary between 0.37 and 1.37, that is over an interval of unit length. Note that a CRK value of 27.37 for a bowler who had taken 26 wickets does not imply that he had actually taken 27 wickets, but rather that his overall performance in the series is best described by the value 27.37 which is determined by his number of wickets as well as his very good average, A = 13.73. The aim of spreading the values of W over a continuous interval by using CRK was successful.

From all these results, it is clear that CRK can indeed be seen as a useful measure of the wicket taking performance of bowlers in the series. The significance of the results given is that the measure CRK is not only suitable to rank the bowlers, but it actually quantifies the wicket taking performances into numbers which can be used to compare bowlers with a measure on a continuous scale like other bowling performance measures. In Table 1, the values of CRK in the case of the 2006/07 group are given in the tenth column. These values reflect the wicket taking performances on a continuous scale and are much more informative than just the values of W. The difference between the wicket taking performances of G. McGrath and M. Muralitharan is equal to 3.14. This indicates more than just stating that McGrath had taken three wickets more than Muralitharan.

In order to assist the reader, Table 2 can serve as a source of nomenclature of the various measures used in this study.

Table 2.

Summary of statistical measures.

Scaler variables
B	Number of balls bowled by a bowler
O	Number of overs bowled by a bowler
R	Number of runs conceded by a bowler
W	Number of wickets taken by a bowler
Ratios
A = R/W	Bowler’s average
E = R/O	Bowler’s economy rate
S = B/W	Bowler’s strike rate
Comprehensive measures
CRA = W + 1/A	Crude measure of wicket taking performance
BI = A×S	Bowling index
CALC = A×E/6	Measure of bowling performance
$CBR = \frac{3 R}{(W + O + W \frac{R}{B})}$	Combined bowling rate
CRK = W + k/A	Improved measure of wicket taking performance
CRM = CRK/O	Measure of wicket taking ability

Results

In the 2002/03 World Cup Series, the value of A varied between 12.25 and 49.5 which gave k = 16.28. According to the chi-squared test the exponential distribution with λ = 0.1821 fitted well with p = 0.335 (and D = 0.1024 for the Kolmogorov–Smirnov test). Similarly in the 2010/11 World Cup Series the value of A varied between 10.71 and 46, resulting in k = 13.96. The exponential distribution with λ = 0.2233 fitted well with p = 0.681 (and D = 0.0881). In the 2014/15 World Cup Series, the value of A varied between 10.18 and 68 which gave k = 11.97. The exponential distribution with λ = 0.201 fitted very well with p = 0.734 (and D = 0.0943).

The quantity CRK of the conditional ranking method is a measure of wicket taking performance, but not of wicket taking ability, because it does not take the number of overs bowled into account. How can one say that a bowler who had taken 21 wickets in 82.5 overs had performed better than one who had taken 18 wickets in 58.2 overs? This is obviously nonsense, but that is what the generally used conditional ranking method says. The method based on CRK can be developed into a wicket taking ability measure by taking the number of overs into account. We therefore define

CRM = CRK / O

(6)

This is a more logical measure of wicket taking ability than CRK and it is closely related to the inverse of the strike rate as can be seen in

CRM = (6 / 6 S S) (1 + k / R)

(7)

with k = 18.79 in the 2006/07 World Cup Series. Obviously CRM entails more than the strike rate. It contains a factor that takes the average into account, making it more reasonable than S. CRM is a measure of the wicket taking ability of bowlers that can be used to measure the difference between the wicket taking abilities of any two bowlers.

The correlation between CRK and CRM is 0.794 which confirms that they do not measure exactly the same characteristic. According to CRK, D. Vettori ranks seventh, but according to CRM he only ranks 23rd due to his high average. From equation (7) it is clear that CRM is closely related to the strike rate S with correlation coefficient = −0.997, but it is important to note that these two measures do not give identical rankings of bowlers. In the 2014/2015 World Cup Series K. Abbott ranked sixth according to S but fourth according to CRM due to his good average of 14.44. In the 2010/11 World Cup Series T. Dilshan ranked tenth according to S but fifth according to CRM. If the focus is on successful bowling, CRM is a better criterion to use than the conditional ranking method based on CRK and even the strike rate S.

CRM is a measure of wicket taking ability, but not of bowling performance in general. Bowlers’ overall performances should rather be compared by using measures like CALC and CBR. The combined bowling rate, CBR, is a well-established measure of bowling performance and the ranks in the last column give the ranking of the bowlers according to CBR. This differs quite markedly from the rankings according to CRK, CRM and S. Based on his overall performance, S.E. Bond was the best bowler according to CBR but he only ranks 12th according to CRK and 14th according to CRM and S. The correlation between CBR and CRK is −0.550 and with CRM it is −0.509. Wicket taking ability can be judged by using CRM, but bowling performance in general by using CBR.

Table 3 contains a summary of the advantages, disadvantages and purposes of the comprehensive measures contained in this study.

Table 3.

Comments on comprehensive measures.

Measure	Advantages of measure	Disadvantages of measure	Best use of measure
CRA	Better than simply using W to measure wicket taking performances	A crude measure of wicket taking performance	Comparison of bowlers’ wicket taking performances
BI	Measure of bowling performance	Does not take economy rate into account	Comparison of bowlers’ bowling performances
CALC	Measure of bowling performance better than BI	Theoretically not well motivated	Comparison of bowlers’ bowling performances
CBR	Best measure of bowling performance	Slightly more complicated than BI and CALC	Comparison of bowlers’ bowling performances
CRK	Good measure of wicket taking performance	Does not take number of overs bowled into account	Comparison of bowlers’ wicket taking performances
CRM	Measure of wicket taking ability		Comparison of bowlers’ wicket taking abilities

Discussion

An important observation is that the value of k depends only on the two most extreme values of A of the bowlers’ performances in the specific series. It is therefore appropriate to talk of the wicket taking performances of the bowlers in the series. Many factors contribute to this, for example the number of overs bowled, the number of matches in the series, the playing conditions, the pitch conditions, the general quality of the batsmen and fielders, etc. The wicket taking performance of a bowler is always limited to the series in which he played and strictly speaking the measure CRK can only be used to compare bowlers’ performances in the specific series. Thus, values of CRK in different series are not really comparable. CRM, on the other hand, can more readily be used to compare the wicket taking abilities of bowlers in different series because the number of overs bowled is taken into account. The fact that the value of k differs from series to series is not crucial. It was found that the goodness of fit was not influenced markedly by using values of k moderately smaller or larger than the value determined by equation (4). To facilitate the more general use of CRM over different series, it is desirable to decide on a specific value of k. The average value of k in the four World Cup Series is k = 15.25. Thus, in order to compare the CRM values from different series, use k = 15.25, that is

CRM = (W + 15.25 / A) / O

(8)

If this formula is used to compare the wicket taking abilities of McGrath and Muralitharan the difference in their CRM values is 0.0520 compared to a value 0.0524 obtained in the specific series where k = 18.79. Thus CRM is a useful continuous measure of the wicket taking abilities of bowlers.

From the results obtained the following procedure is recommended. To compare bowlers’ wicket taking performances in a specific series, calculate the value of k from equation (4) and use CRK. To compare their wicket taking abilities in the series, use CRM with the same k. The use of CRK should preferably be limited to comparisons within a series. CRM, on the other hand, can be seen as a measure of the wicket taking ability of a bowler in general provided that k = 15.25 be used in the case of One-Day International matches.

To obtain a good indication of a bowler’s wicket taking ability, the bowling figures of the four World Cup Series are combined into a data set of 184. By using the generally recommended value of k = 15.25 the values of CRM are calculated. The question now arises: What is a good CRM value? To assess a bowler’s wicket taking ability based on his CRM value we use our data set and construct 10 classes according to which bowlers are classified. The bootstrap technique^{9: 79} is used to estimate the deciles for the data set and these are used as class boundaries. Those 10% with the largest (best) CRM values are classified into class one, the next 10% into class two, etc. See Table 4.

Table 4.

Classification scheme of 10 classes for CRM values of ODIs and twenty20 matches.

Class number	Interval for ODIs	Interval for T20 matches
1	0.294+	0.656+
2	0.267+ – 0.294	0.561+ – 0.656
3	0.244+ – 0.267	0.508+ – 0.561
4	0.216+ – 0.244	0.478+ – 0.508
5	0.198+ – 0.216	0.430+ – 0.478
6	0.181+ – 0.198	0.393+ – 0.430
7	0.171+ – 0.181	0.350+ – 0.393
8	0.157+ – 0.171	0.300+ – 0.350
9	0.131+ – 0.157	0.257+ – 0.300
10	0.000 – 0.131	0.000 – 0.257

G. McGrath, L. Malinga and E. Chigumbura are in the top class with M. Muralitharan, S. Tait and G. Hogg in the second class. Chigumbura bowled only 22.4 overs and ranked third according to his wicket taking ability, but only 17th according to CBR for his bowling performance in general. This is due to his high economy rate.

It is fairly obvious that the same method can also be used in the case of Twenty20 matches. The bowling figures of the 2009, 2010, 2012/13 and 2013/14 Twenty20 World Cup Series¹⁰ were used for a study similar to the foregoing. In these series bowlers could only bowl four overs per match, therefore bowlers who had taken three or more wickets are included. The values of k in the different series are 10.87, 8.41, 9.33 and 12.65 with average 10.32. These fairly small values of k are due to the fact that batsmen tend to bat aggressively almost from the start of their innings and also in the last few overs, thereby making it easy for some clever bowlers to take their wickets. This results in small values of the bowlers’ averages. The formula for the wicket taking ability in Twenty20 matches thus becomes

CRM = (W + 10.32 / A) / O

(9)

The 10% classes for CRM for use in Twenty20 matches are also given in Table 4.

The initial challenge of this study was to define a formula that can be used to rank bowlers’ wicket taking performances in a series of matches according to the conditional ranking method. The formula CRA in equation (1) served the purpose, but in order to get measurements with a better spread over the real line, CRK in equation (3) was constructed under the restriction that the ranking should remain unaltered. The exponential distribution gives a very good fit to the CRK − 6 values. To measure wicket taking ability (rather than performance) it was necessary to take the number of overs bowled into account. The criterion CRM is a sensible measure of wicket taking ability and is not restricted to the specific series.

Conclusion

The venture to construct a measure, CRK, for the conditional ranking method that retains the order of the bowlers but measures the wicket taking performances on a continuous scale that is suitable to quantify the difference in performance between any two bowlers in a series, was successful. This work was instrumental in laying the foundation for the wicket taking ability measure.

A very useful measure, CRM, for the wicket taking ability of bowlers, was defined. This measure is better than the ordinary strike rate of bowlers and it can be used to measure the difference between the wicket taking abilities of any two bowlers.

Cricket authorities should not use their conditional ranking method (most wickets taken) to determine the best wicket taking bowler. They should rather use the measure CRM. The second best is the well-known strike rate S.

Footnotes

Acknowledgement

The author is grateful for the valuable comments of the referees that led to a substantial improvement of the article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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