Quantifying the pressure on the teams batting or bowling in the second innings of limited overs cricket matches

Abstract

A team batting second in a limited overs match is under pressure to score the required number of runs for victory in the allotted number of overs without losing all its wickets. The bowling team has to prevent its opponents from reaching the target for victory. The objective of this study is to define methods that can be used to assess the pressure on the two teams’ batsmen and bowlers when playing the second innings of a Twenty20 or One-day match, and also to define methods to determine the best partnership and the turning point in such an innings. Methods are also defined to determine the batting and bowling performances of individual players in a specific match. These measures can be quite useful for selectors to assess how the players perform under pressure.

Keywords

Performance analysis one-day cricket psychological pressure sport analytics

Introduction

In a limited overs match, played between two teams A and B (say), team A bats first for a fixed number of overs. In these overs, team A tries to score as many runs it can against the fielding and bowling of team B. Team B tries to restrict the runs of team A. At the end of the innings of team A, a “target” is set for team B, which is one run more than the runs scored by team A. This target has to be achieved by team B within the specified number of overs without losing all its wickets. In a One-day match (50 overs in each innings), one over is equivalent to only 2% of the balls available for the batting team but in Twenty20 cricket one over is equivalent to 5% of the total number of balls the batting team has to face. Thus, a slow scoring rate at the start of the innings or losing too many early wickets creates pressure for the team batting second and it becomes hard to achieve the target. In an attempt to accelerate the rate at which runs are scored, the batsmen expose themselves to the possibility of getting out because aggressive batting can lead to the fall of wickets. Avoiding this risk may result in a slower scoring rate, which makes the job difficult for the team. A high scoring over, on the other hand, makes the team’s position more comfortable. The purpose of this study is to define criteria that can be used to measure the pressure experienced by both teams and to use these measures for various purposes. The pressure on the teams can change after every ball. A dot ball (a ball from which no run is scored, normally indicated by a dot on the scorecard) or the loss of a wicket increases the pressure on the batting team. A scoring shot like a six or a four, or a productive over of 20 runs, makes the task of the batting team easier and decreases their pressure level. This study attempts to quantify the pressure on both teams in the second innings of a Twenty20 or One-day match at any given point of the innings. The pressure on team B is inversely proportional to (i) the number of balls left and (ii) the number of wickets in hand, but directly proportional to (iii) the runs that remain to be scored.

According to Paccagnelia,¹ “Pressure usually refers to the feelings an athlete has about performing in a sporting situation. It is often experienced as a compelling or constraining influence on the mind, or an urgent demand that must be met. Pressure is a feeling that is created by ourselves, when we react to particular events or situations”. The pressure is exerted by the opposing team and can lead to anxiety. According to Quinn²: “Performance anxiety in sports is described as a decrease in athletic performance due to too much perceived stress”. Another description is: “A critical deterioration in the execution of habitual processes as a result of an elevation in anxiety levels under perceived pressure, leading to substandard performance”.³ Paccagnelia¹ added: “Pressure isn’t necessarily bad – it can enhance motivation, concentration and enjoyment. That feeling of stress that often accompanies a pressure situation can help keep you on your toes, ready to rise to a challenge”.

Review of literature

Cricket is a data-rich sport. A large number of papers have been published on performance measurement, decision making, and team selection. Resetting target scores in rain-truncated matches used the Duckworth/Lewis and other methods—cf. Duckworth and Lewis,⁴ Jayadevan,⁵ de Silva,⁶ Preston and Thomas,⁷ and Bhattacharya et al.⁸ Some other works that were concerned with the progress of scoring in the second innings of limited overs matches are Bailey and Clarke⁹ and de Silva et al.¹⁰ Bailey and Clarke⁹ attempted to predict the final score of the second innings of a One-day match while the game was in progress. They used the Duckworth–Lewis resource table at the end of each over of the second innings and tried to predict the final score of the match vis-à-vis the winner of the match. de Silva et al.¹⁰ again used the Duckworth–Lewis resource table to estimate the magnitude of victory in One-day matches. They suggested that this can be used to break ties between teams in the points table at the end of the league stage of any tournament.

Lemmer¹¹ defined a performance measure PR that describes the batting progress of the team batting second in a limited overs match. In the present notation PR = CRRR × RU where CRRR denotes the current required run rate and RU denotes the percentage of resources that had already been used according to the Duckworth–Lewis⁴ system. PR was based on the fact that the performance of the batting team not only depended on its required scoring rate, but also on its use of resources, i.e. wickets lost and balls already bowled. This measure was used to construct a method to measure “choking” in cricket. It was first introduced in a conference lecture.¹² See also Lemmer.¹³ Shah and Shah¹⁴ defined a “pressure index”, which can also be used to assess the team’s progress during its innings. The pressure index (PI) in Shah and Shah¹⁴ consists of various parts based on batting performance and wickets lost. Unfortunately, it lacks motivation and some symbols are undefined.

The term “index” has a wide variety of meanings. Borrowing from the Oxford English Dictionary, we describe a pressure index as “a number indicating the relative level of pressure compared with a standard”, where the standard refers to a base or starting value. In this study it is assumed that the base value will be 1 or 100, and that the index fluctuates around this value according to the pressure exerted by the opponents.

The purpose of the present study is to define and study statistics that can be used to measure the pressure that the batting team or the bowling team experiences at any stage of the match. The pressure on the batting team is obviously determined by a combination of two factors, namely its batting performance and its ability to retain wickets. Similarly, the pressure on the bowling team is determined by its ability to take wickets and to restrict runs scoring of its opponents.

Method

Performance criteria

We start by looking at criteria that can be used to measure pressure in a cricket match. It is customary to assess the scoring process by calculating the required run rate from time to time in order to see whether the scoring rate is satisfactory. Let T be the target runs for team B. At the start of the team’s innings, the initial required run rate is IRRR = 6.T/B with B the total number of balls available. During its innings, an eye should be kept on the current required run rate CRRR where CRRR = 6.Rr/Br with Rr the number of runs still required and Br the number of balls remaining. If this is compared with the initial required run rate, IRRR, it is easy to judge whether satisfactory progress is made. The ratio

CI = CRRR / IRRR

(1)

starts from the value 1 and it is desirable that it should decrease progressively until it reaches the value 0, but it can shoot up very high if the batting team fails. CI is a suitable measure for batting progress.

As far as bowling is concerned, two criteria can be considered. Firstly, the bowling team wants to take as many wickets as possible at a fast rate in order to weaken the batting team’s strength. As wickets fall, the team’s wicket strength deteriorates according to the number of wickets that had fallen. Instead of just counting the number of wickets down, it is important to take into account the ability of the batsmen whose wickets had been taken. When top order batsmen lose their wickets, the strength of the team is weakened more than when low order batsmen lose their wickets. In order to take this into account, the wicket weights of Lemmer¹⁵ are used—see Table 5. We denote by ∑ w_i the sum of the weights of the wickets that had fallen at any stage of the innings. When ∑ w_i increases it is an indication that the team’s wicket strength decreases. Taking into account that the sum of all the wicket weights is equal to 11, the ratio SW = ∑ w_i/11 reflects how far the wicket strength has deteriorated. Theoretically, the value of SW can vary between 0 (when no wicket is down) up to 0.983 (if batsman number 11, with wicket weight 0.19, is the only not out batsman). Unless both opening batsmen keep their wickets until the end of the match, the team’s wicket strength will necessarily deteriorate and its task becomes more difficult. As wickets fall, the value of SW will initially increase rapidly, but its growth will slow down when low order wickets fall. We use exp(SW) as factor in order to enlarge the effect of wickets falling and because its initial value is 1.

Secondly, the purpose of the bowling team is to prevent the batting team from reaching the target score before its resources are depleted. The Duckworth–Lewis full table for Twenty20 matches gives the percentage of resources left to the batting team at the end of each ball depending on the number of wickets lost—cf. Bhattacharya et al.⁸ Subtracting the value from 100 gives RU, which indicates the percentage of resources used. This depends on the number of overs team B has batted and the number of wickets it has lost. Thus RU can be used as an alternative to SW and the change in pressure due to resources used can be quantified by exp(RU/100).

Pressure indices

Shah and Shah¹⁴ defined a pressure index as

PI = CI \times 100 + [Wk . Wt / 180) \times T \times (Br / B) \times (Rr / T)

(2)

No motivation was given for their formula and the factor Wk.Wt was not defined. After obtaining the values of Wk.Wt from them, it was possible to calculate PI. The starting value of PI is always 100 and its value will normally fluctuate. We have calculated PI for various matches. It was found that in many cases the value of PI decreased when a wicket went down and the number of runs scored was below the required run rate, which is totally unrealistic. PI is not considered further in this study.

In the construction of pressure indices for batting and bowling, the criteria given in the previous section will be used in the best possible way. Firstly, the definitions of pressure indices for the batting team and their applications are discussed. Thereafter, a pressure index is defined for the bowling team.

Pressure indices for batting

Definitions

In a limited overs match, the batting team has to reach the target before depleting its resources by using the balls bowled to it and losing all its wickets. The challenge to the team thus consists of two parts, namely to score the runs and not to deplete its resources before reaching the target. In the construction of a pressure index, it is important to note that the pressure will go down if the batting progress is good AND if wickets/resources are kept until the target has been reached—not the one OR the other. Therefore, the measure should be the product of the two factors, as in the definition of PR of Lemmer.¹¹ Two methods are used to quantify the use of resources, namely the way wickets are lost and also by using the Duckworth–Lewis resources.

The first pressure index is defined as

{PI}_{1} = (\frac{CRRR}{IRRR}) \times exp (\sum w_{i} / 11) = CI \times exp (\sum w_{i} / 11) = CI \times exp (SW)

(3)

and the second definition is

{PI}_{2} = (\frac{CRRR}{IRRR}) \times \exp (RU / 100) = CI \times \exp (RU / 100)

(4)

PI₁ is sensitive for wickets falling, but not for balls used, whereas PI₂ depends on all the resources used.

At the start of the second team’s innings both pressure indices are equal to 1 and will increase or decrease or fluctuate as the match progresses. At the end of the match, if team B wins, CI becomes 0 and therefore also PI₁ and PI₂.

Functioning of the pressure indices

To understand how the indices work they are applied in a match situation. A Twenty20 international match of the recent past with lots of ups and downs in the second innings is an ideal one to explain the working of the indices. The second Twenty20 match of Pakistan versus Sri Lanka, played on 1 August 2015, at the Premadasa Stadium, Colombo, is considered—cf. Cricinfo.¹⁶ Sri Lanka batted first and scored 172/7 in 20 overs. The target for Pakistan was 173. The target was attained by them for the loss of 9 wickets with 4 balls to spare. The initial required run rate (IRRR) for Pakistan was 8.65 per over, which was quite high. The ball-by-ball values of the pressure indices, i.e. both PI₁ and PI₂, are given in Table 6. The graphical representation of the same can be seen in Figure 1. The pressure increased almost consistently until a six was hit from the 98th ball. In Table 6, it can be seen that the pressure was largest in the 97th ball and that the pressure decreased rapidly thereafter due to good scoring.

Figure 1.

The pressure index graph for the Pakistan innings of the 449th International Twenty20 match.

The graphs in Figure 1 show that there is not much difference in the pressure values obtained through PI₁ and PI₂, and hence the interpretation is almost the same. After the start of the Pakistan innings, the pressure kept on increasing and was boosted up with the fall of wickets at regular intervals. The graph shows that though the sixth wicket partnership lasted longest, yet the pressure on Pakistan increased. The partnership was between Mohammad Rizwan and Shahid Afridi who scored 61 runs in 37 balls at a rate of 9.89 runs per over. But that rate was not sufficient, given the situation of the match, which is visualized in the graph, with a slight increase in the pressure at the end of the partnership. Then the eighth wicket partnership between Anwar Ali and Imad Wasim started. The partnership lasted for 27 balls and 58 runs were scored at a rate of almost 13 runs per over. In the partnership the pressure initially increased to the highest value in the match, but in the 17th over they scored 21 runs. This brought the pressure abruptly down. See also Table 2 for partnership statistics.

Table 1.

Pressure index values for different events in the second innings.

Sit. no.	Event	Situation	PI ₁	PI ₂	Increase in PI₁	Increase in PI₂
1	Low scoring 4th over	Opening value	1.3498	1.2636	0.066	0.1095
2	Low scoring 4th over	Closing Value	1.4158	1.3731	0.066	0.1095
3	Low scoring 9th over	Opening value	2.3229	2.0096	0.1326	0.1756
4	Low scoring 9th over	Closing Value	2.4555	2.1852	0.1326	0.1756
5	High scoring 12th over	Opening value	2.2959	2.2516	0.1119	0.2034
6	High scoring 12th over	Closing Value	2.4078	2.4550	0.1119	0.2034
7	High scoring 17th over	Opening value	3.4843	3.4466	−0.6794	−0.5722
8	High scoring 17th over	Closing Value	2.8049	2.8744	−0.6794	−0.5722
9	Ahmed Shehzad out	Before wicket	1.0502	1.1431	0.1483	0.0499
10	Ahmed Shehzad out	After wicket	1.1985	1.1912	0.1483	0.0499
11	Mukhtar Ahmed out	Before wicket	1.1839	1.1913	0.1614	0.0614
12	Mukhtar Ahmed out	After wicket	1.3453	1.2527	0.1614	0.0614
13	Md. Hafeez out	Before wicket	1.4041	1.4333	0.2089	0.0766
14	Md. Hafeez out	After wicket	1.6130	1.5099	0.2089	0.0766
15	Umar Akmal out	Before wicket	1.6790	1.6368	0.2341	0.0888
16	Umar Akmal out	After wicket	1.9131	1.7256	0.2341	0.0888
17	Shoaib Malik out	Before wicket	1.9628	1.7893	0.306	0.1391
18	Shoaib Malik out	After wicket	2.2688	1.9284	0.306	0.1391
19	Md Rizwan out	Before wicket	2.3661	2.4425	0.2984	0.1217
20	Md Rizwan out	After wicket	2.6645	2.5642	0.2984	0.1217
21	Shahid Afridi out	Before wicket	2.6460	2.5872	0.3292	0.2040
22	Shahid Afridi out	After wicket	2.9752	2.7912	0.3292	0.2040
23	Anwar Ali out	Before wicket	1.4025	1.5370	0.2622	0.2199
24	Anwar Ali out	After wicket	1.6647	1.7569	0.2622	0.2199
25	Sohail Tanvir out	Before wicket	1.6647	1.7693	0.0607	0.0347
26	Sohail Tanvir out	After wicket	1.7248	1.8040	0.0607	0.0347

Table 2.

Pressure index for batting partnerships.

Sit. no.	Partnership	Increase in PI₃	Partnership statistics
1	M Ahmed & A Shehzad	0.168	11 runs in 14 balls
2	M Ahmed & M Hafeez	0.131	5 runs in 3 balls
3	M Hafeez & S Malik	0.182	17 runs in 16 balls
4	S Malik & U Akmal	0.229	7 runs in 8 balls
5	S Malik & M Rizwan	0.273	0 run in 3 balls
6	M Rizwan & S Afridi	0.182	61runs in 37 balls
7	S Afridi & I Wasim	0.216	6 runs in 4 balls
8	I Wasim & A Ali	−1.123	58 runs in 27 balls
9	I Wasim & S Tanvir	0.063	2 runs in 2 balls
10	I Wasim & M Irfan	Incomplete innings

Table 3.

Values of pressure indices at the end of each over of the Pakistan innings.

Over	Runs	Wicket	Rate req.	Runs req.	Balls rem.	CI	Wickets down	∑w_i	PI ₃	BI
1	7	0	8.73	166	114	1.009		0	1.029	1.011
2	4	0	9.00	162	108	1.040		0	1.083	1.000
3	6	2	9.17	156	102	1.060	2, 1	2.65	1.306	0.920
4	2	2	9.62	154	96	1.112		2.65	1.394	0.896
5	13	2	9.40	141	90	1.087		2.65	1.387	0.938
6	2	3	9.92	139	84	1.147	3	4.05	1.618	0.876
7	6	4	10.23	133	78	1.183	5	5.43	1.847	0.842
8	4	5	10.75	129	72	1.243	4	6.88	2.167	0.802
9	4	5	11.36	125	66	1.313		6.88	2.320	0.776
10	13	5	11.20	112	60	1.295		6.88	2.320	0.807
11	12	5	11.11	100	54	1.284		6.88	2.338	0.836
12	15	5	10.62	85	48	1.228		6.88	2.273	0.899
13	7	5	11.14	78	42	1.288		6.88	2.431	0.885
14	12	6	11.00	66	36	1.272	6	8.06	2.616	0.905
15	6	7	12.00	60	30	1.387	7	9.04	3.089	0.850
16	7	7	13.25	53	24	1.532		9.04	3.466	0.793
17	21	7	10.66	32	18	1.232		9.04	2.837	1.018
18	12	7	10.00	20	12	1.156		9.04	2.722	1.130
19	14	9	6.00	6	6	0.694	9, 10	10.02	1.765	1.939
20	7	9	0.00	0	0	0.000		10.02	0.000	∞

Both graphs of PI₁ and PI₂ depict the situation appropriately with the pressure on Pakistan increasing at the fall of a wicket and the pressure decreasing after an over where many runs were scored. Table 1 provides the increases in the values of PI₁ and PI₂ at different situations of the match. The increase in both the indices after high and low scoring overs and after the fall of wickets provides interesting insight into the indices. It is seen that generally the change in the index PI₁ is less than in PI₂ both in the case of low and high scoring overs. Thus, PI₂ seems to be more sensitive than PI₁ in this regard. However, in the case of the fall of wickets PI₁ is more sensitive compared to PI₂ as can be seen from Table 1. This is obvious, as the factor CI is common to both PI₁ and PI₂. The only difference between PI₁ and PI₂ is in their exponents. The exponent changes after every ball in case of PI₂ because of a change in the value of RU, but the exponent of PI₁ changes only at the fall of a wicket. When a wicket falls, the task of the batting team becomes markedly more difficult because the new batsman has to play himself in and the scoring rate normally drops. PI₁ reflects this very well. But during a partnership PI₂ is better suited than PI₁ to reflect the use of resources. It is therefore logical to use the average of PI₁ and PI₂ as measure for the pressure index of the batting team. This is defined as

{PI}_{3} = (\frac{CRRR}{IRRR}) \times \frac{1}{2} [\exp (RU / 100) + \exp (Σ w_{i} / 11)]

(5)

The values of PI₃ were computed at the end of each over and are shown in Figure 2. Plotting at the end of each over, instead of ball-by-ball, makes the pressure curve smoother.

Figure 2.

The pressure curve based on the pressure index (PI₃) values at the end of each over.

The pressure indices can be used for a number of purposes. Some of these are highlighted here.

Finding the best partnership of an innings

The most successful partnership of the match can be identified as the one in which the pressure on team B was decreased to the largest extent.

The exercise shows (see Table 2) that the performance of I Wasim and A Ali as pair has decreased the pressure on the batting team the most. The negative value of the increase in pressure index PI₃ indicates the lowering of the pressure on Pakistan due to this pair of batsmen.

Finding the turning point of the match

The pressure curve will sometimes indicate a clear turning point of the match if team B wins, as can be seen in Figure 2. The curve shows that Pakistan lost wickets at regular intervals and the pressure level kept on increasing. But things changed in the 17th over, where 21 runs were scored. After a six from the second ball the pressure decreased and the trend continued in the remaining overs—cf. Table 6. Thus, the 97th ball (the first of the 17th over) of the Pakistan innings can be considered as the turning point of the match. In some matches, there can be more than one possible turning point. In such a case, the peak or trough, which is closest to the end of the match, may be considered as the ultimate turning point.

Measuring batting performance

The batting performance of the players in a single innings can be compared in terms of runs scored or/and strike rate. But here we propose a new batting performance measure in a single innings.

Ave(PI₃) denotes the average pressure of the entire innings. PI_3, _i denotes the pressure level at the end of the ith ball of the second innings of the match. R_ij denotes the runs scored by batsman j in the ith ball of the match. Then, the runs scored in the ith ball is transformed to the adjusted runs scored (R_ij*) using the following relation

R_{ij}^{*} = R_{ij} \times \frac{{PI}_{3, i - 1}}{Ave ({PI}_{3})}

(6)

That is, the adjusted runs scored in the ith ball is the actual runs scored multiplied by the ratio of the pressure index at the end of the previous ball to the average pressure level of the innings. Summing up all the adjusted runs scored by a batsman (jth batsman, say) renders his adjusted score. The adjusted score is a function of the actual runs scored and the pressure level under which these runs were scored. We call this his batting contribution ${AR}_{j}^{*}$ where

{AR}_{j}^{*} = \sum_{i} R_{ij}^{*}

(7)

where the summation extends over those balls that the jth batsman faced. The significance of his batting contribution can be put into perspective by taking into account how many balls he used, say B_j. Define his batting achievement by

{BA}_{j} = {AR}_{j}^{*} / B_{j}

(8)

The batting achievement can be considered as a measure of batting performance in a single innings where runs are chased. Table 4 shows the batting achievements of the Pakistan innings based on this new measure of batting performance. The batsmen are arranged in descending order of the magnitude of their BA_j values.

Table 4.

Batting and bowling performance values based on pressure indices.

Batsman	Runs scored (balls)	Batting performance (BA_j)	Bowler	Bowling figure	Bowling performance (BPI_j)
Anwar Ali	46 (17)	3.4264	TAM Siriwardana	17/1 in 2 overs	0.1156
Shahid Afridi	45 (23)	2.2428	JDF Vandersay	19/0 in 2 overs	−0.0019
Imad Wasim	24 (14)	2.0311	SL Malinga	40/1 in 4 overs	−0.0955
Mohammad Rizwan	17 (17)	1.1221	DSNFG Jayasuriya	37/1 in 4 overs	0.0046
Umar Akmal	4 (4)	0.8575	NLTC Perera	27/1 in 4 overs	0.0602
Shoaib Malik	8 (12)	0.8503	B Fernando	33/2 in 3.2 overs	0.0751
Mohammad Irfan	1 (1)	0.7372
Md. Hafeez	11 (11)	0.7049
Sohail Tanvir	1 (1)	0.702
Ahmed Shehzad	7 (8)	0.4973
Mukhtar Ahmed	4 (7)	0.3281

Table 5.

Wicket weights of different batting positions as in Lemmer.¹⁵

Batting position (i)	Wicket weight w_i
1	1.30
2	1.35
3	1.40
4	1.45
5	1.38
6	1.18
7	0.98
8	0.79
9	0.59
10	0.39
11	0.19

Table 6.

The ball-by-ball calculation of the values of pressure indices.

Ball	Runs	Fall of wickets (batting position)	CRRR	CI	RU	PI ₁	PI ₂	PI ₃
1	1		1.445	1.003	0.71	1.003	1.010	1.007
2	0		1.458	1.011	1.24	1.011	1.024	1.018
3	0		1.470	1.020	1.94	1.020	1.040	1.030
4	4		1.448	1.005	2.65	1.005	1.032	1.019
5	2		1.443	1.001	3.18	1.001	1.034	1.018
6	0		1.456	1.010	3.89	1.010	1.050	1.030
7	0		1.469	1.019	4.42	1.019	1.065	1.042
8	0		1.482	1.028	5.12	1.028	1.082	1.055
9	0		1.495	1.037	5.83	1.037	1.100	1.069
10	2		1.491	1.034	6.54	1.034	1.104	1.069
11	1		1.495	1.037	7.07	1.037	1.113	1.075
12	1		1.500	1.040	7.77	1.040	1.125	1.083
13	0		1.514	1.050	8.48	1.050	1.143	1.097
14	0	A Shehzad (2)	1.528	1.060	11.66	1.199	1.191	1.195
15	4		1.505	1.044	12.37	1.180	1.181	1.181
16	1		1.510	1.047	12.90	1.184	1.191	1.188
17	0	M Ahmed (1)	1.524	1.057	16.96	1.345	1.253	1.299
18	1		1.529	1.061	17.49	1.350	1.264	1.307
19	0		1.545	1.071	18.02	1.363	1.283	1.323
20	0		1.560	1.082	18.73	1.377	1.305	1.341
21	0		1.576	1.093	19.26	1.391	1.325	1.358
22	0		1.592	1.104	19.79	1.405	1.346	1.376
23	1		1.598	1.108	20.49	1.410	1.361	1.386
24	1		1.604	1.113	21.02	1.416	1.373	1.395
25	2		1.600	1.110	21.73	1.412	1.379	1.396
26	1		1.606	1.114	22.26	1.418	1.392	1.405
27	2		1.602	1.111	22.97	1.414	1.398	1.406
28	1		1.609	1.116	23.50	1.420	1.411	1.416
29	6		1.560	1.082	24.20	1.377	1.379	1.378
30	1		1.567	1.087	24.73	1.383	1.392	1.388
31	1		1.573	1.091	25.44	1.388	1.407	1.398
32	0		1.591	1.104	26.15	1.404	1.433	1.419
33	0	M Hafeez (3)	1.609	1.116	30.21	1.613	1.510	1.562
34	1		1.616	1.121	30.74	1.620	1.525	1.573
35	0		1.635	1.134	31.27	1.639	1.551	1.595
36	0		1.655	1.148	31.98	1.659	1.580	1.620
37	1		1.663	1.153	32.51	1.667	1.596	1.632
38	3		1.646	1.142	33.04	1.650	1.589	1.620
39	0		1.667	1.156	33.75	1.671	1.620	1.646
40	1		1.675	1.162	34.28	1.679	1.637	1.658
41	1	U Akmal (5)	1.684	1.168	39.05	1.913	1.726	1.820
42	0		1.705	1.183	39.58	1.938	1.757	1.848
43	0		1.727	1.198	40.11	1.963	1.789	1.876
44	0	S Malik (4)	1.750	1.214	46.29	2.269	1.928	2.099
45	1		1.760	1.221	46.82	2.282	1.950	2.116
46	1		1.770	1.228	47.17	2.295	1.968	2.132
47	1		1.781	1.235	47.70	2.309	1.990	2.150
48	1		1.792	1.243	48.06	2.323	2.010	2.167
49	1		1.803	1.251	48.59	2.337	2.033	2.185
50	1		1.814	1.258	49.12	2.352	2.057	2.205
51	0		1.841	1.277	49.47	2.386	2.094	2.240
52	1		1.853	1.285	50.00	2.402	2.119	2.261
53	0		1.881	1.304	50.53	2.438	2.162	2.300
54	1		1.894	1.314	50.88	2.455	2.185	2.320
55	1		1.908	1.323	51.41	2.473	2.213	2.343
56	1		1.922	1.333	51.94	2.492	2.241	2.367
57	4		1.889	1.310	52.47	2.449	2.214	2.332
58	6		1.823	1.264	53.00	2.363	2.148	2.256
59	1		1.836	1.274	53.45	2.380	2.173	2.277
60	0		1.867	1.295	53.89	2.420	2.219	2.320
61	2		1.864	1.293	54.42	2.417	2.228	2.323
62	6		1.793	1.244	54.95	2.325	2.155	2.240
63	1		1.807	1.253	55.48	2.343	2.183	2.263
64	1		1.821	1.263	56.01	2.361	2.212	2.287
65	1		1.836	1.274	56.71	2.381	2.246	2.314
66	1		1.852	1.285	57.24	2.401	2.277	2.339
67	1		1.868	1.296	57.77	2.422	2.309	2.366
68	6		1.788	1.241	58.30	2.319	2.222	2.271
69	1		1.804	1.251	58.83	2.339	2.254	2.297
70	1		1.820	1.262	59.54	2.360	2.290	2.325
71	6		1.735	1.203	60.07	2.249	2.194	2.222
72	0		1.771	1.228	60.60	2.296	2.252	2.274
73	1		1.787	1.240	61.31	2.317	2.289	2.303
74	2		1.783	1.236	61.84	2.311	2.295	2.303
75	2		1.778	1.233	62.54	2.305	2.305	2.305
76	1		1.795	1.245	63.07	2.328	2.340	2.334
77	0		1.837	1.274	63.78	2.382	2.412	2.397
78	1		1.857	1.288	64.49	2.408	2.455	2.432
79	4		1.805	1.252	65.02	2.340	2.399	2.370
80	1		1.825	1.266	65.72	2.366	2.443	2.405
81	1	Md. Rizwan (6)	1.846	1.281	69.43	2.665	2.564	2.615
82	1		1.868	1.296	69.96	2.697	2.609	2.653
83	1		1.892	1.312	70.49	2.731	2.656	2.694
84	4		1.833	1.272	71.02	2.646	2.587	2.617
85	0	S. Afridi (7)	1.886	1.308	75.80	2.975	2.791	2.883
86	2		1.882	1.306	76.15	2.970	2.796	2.883
87	1		1.909	1.324	76.68	3.012	2.851	2.932
88	0		1.969	1.366	77.03	3.106	2.950	3.028
89	1		2.000	1.387	77.47	3.156	3.010	3.083
90	2		2.000	1.387	77.92	3.156	3.024	3.090
91	4		1.931	1.339	78.45	3.047	2.935	2.991
92	0		2.000	1.387	78.98	3.156	3.056	3.106
93	0		2.074	1.439	79.51	3.272	3.186	3.229
94	1		2.115	1.467	80.04	3.338	3.267	3.303
95	1		2.160	1.498	80.57	3.408	3.353	3.381
96	1		2.208	1.532	81.10	3.484	3.447	3.466
97	1		2.261	1.568	81.63	3.567	3.547	3.557
98	6		2.091	1.450	82.16	3.299	3.298	3.299
99	1		2.143	1.486	82.86	3.381	3.404	3.393
100	3		2.100	1.457	83.39	3.313	3.354	3.334
101	6		1.895	1.314	84.10	2.989	3.047	3.018
102	4		1.778	1.233	84.63	2.805	2.874	2.840
103	1		1.824	1.265	85.34	2.877	2.969	2.923
104	0		1.938	1.344	86.04	3.057	3.177	3.117
105	6		1.667	1.156	86.75	2.630	2.753	2.692
106	4		1.500	1.040	87.46	2.367	2.495	2.431
107	0		1.615	1.120	88.16	2.549	2.706	2.628
108	1		1.667	1.156	89.05	2.630	2.816	2.723
109	6		1.273	0.883	89.75	2.008	2.166	2.087
110	2		1.200	0.832	90.64	1.893	2.060	1.977
111	4		0.889	0.617	91.34	1.402	1.537	1.470
112	0	A Ali (9)	1.000	0.694	92.93	1.665	1.757	1.711
113	1		1.000	0.694	93.64	1.665	1.769	1.717
114	1	Sohail Tanvir (10)	1.000	0.694	95.58	1.725	1.804	1.765
115	1		1.000	0.694	96.11	1.725	1.814	1.770
116	6		0.000	0.000	96.64	0.000	0.000	0.000

Pressure index for bowling

Definition and functioning

If the batting team succeeds to decrease the value of CI, the value of its inverse 1/CI will increase and also the pressure on the bowling team. In order to lower the pressure on the bowling team, it will help if wickets are taken at a fast rate, i.e. the value of ∑ w_i must increase rapidly. The ratio (11 – ∑w_i)/11 reflects the decrease in resources due to wickets being taken, and therefore a decrease in pressure on the bowling team. However, as resources are used the pressure on the bowling team will increase, especially if wickets are not taken, so the factor exp(RU/100) must also be included. By using the average of these two counteracting factors, the sudden effect of a wicket falling as well as the depletion of overs is taken into account. The bowling pressure index is defined as

BI = (1 / CI) \times [(11 - \sum w_{i}) / 11 + exp (RU / 100)] / 2

(9)

The starting value of BI will always be 1.

The functioning of BI is illustrated by using the same data set as before, namely the second Twenty20 match of Pakistan versus Sri Lanka, played on 1 August 2015 (see Table 3).

When a wicket falls, the value of BI should normally decrease, but the number of runs scored in that over is also of importance. In over number 19, fourteen runs were scored and two wickets were taken. The joint weight of these two low order wickets was so small that the value of BI increased substantially despite two wickets falling, which was quite understandable in the context of the match situation. It must be kept in mind that if CI decreases, BI should normally increase. BI and CI are not mirror images of each other, but they will generally move in opposite directions.

Measuring bowling performance

The pressure index PI₃ can be used to define a measure of bowling performance because it reflects the pressure exerted on the batting team by the bowling team. The difference I_ij between the end and starting values of PI₃ in the ith over bowled by the jth bowler indicates the increase in pressure on the batting team created while bowler j bowled his ith over. The average increase in pressure created while bowler j was bowling is given by

{BPI}_{j} = \frac{\sum_{i = 1}^{n_{j}} I_{ij}}{n_{j}}

(10)

where n_j is the number of overs bowled by the jth bowler. This can be used as a measure of the bowling performance of the bowler. A good bowler is expected to bowl well and increase the pressure on the batting team by restricting runs and taking wickets. The index includes the change in pressure through dot balls and the fall of wickets. The higher the value of BPI_j, the better is his performance. Thus, the bowler with the largest value of BPI_j is the best bowler of the innings. One advantage of BPI_j over other traditional measures of bowling performance is that it is a positive measure. The other bowling performance measures, like economy rate, bowling strike rate and bowling average, have an inverse dimension in relation to bowling performance.

Table 4 shows that the best bowling performance amongst the Sri Lankan bowlers was by TAM Siriwardana followed by B Fernando. The best batting performance amongst the Pakistani batsmen was Anwar Ali followed by Shahid Afridi.

Discussion

The methods defined here are equally applicable and valid in One-day matches. The indices were calculated per over for many matches and the results were similar to those of Twenty20 matches. For partnership calculations it is necessary to use ball-by-ball figures only for the overs concerned.

The pressure indices developed in this paper reflect how a team batting second, chases the target, and how it reacts to the changing match situations like tight bowling or the fall of wickets. There is obviously scope for improvement. One can develop player-specific weights and include these in the index instead of w_i, where i refers to the specific batsman and not to the batting order. The measures in this study are formally based on match statistics. Factors of a physical and psychological nature are also very important and it can be a challenge to take these into account too.

Some applications of the pressure indices were addressed in this paper. If the pressure indices of a team are studied for several matches it can help to identify batsmen who perform well in partnerships and using this to change the batting order, and also to study performances in power-plays. Studying the pressure curves of the opposing team can help to decide on a counter strategy. Future researchers are also expected to find several other applications of the pressure indices.

An interesting challenge for future research is to examine to what extent the methods of this study can be applied or modified for use in the first innings of limited overs matches. A possible starting point can be to use the average first innings score on the specific field to determine the “target score”.

Conclusion

The approach of Lemmer^11,12 of working with the product of performance measures has found various applications in the form of pressure indices for batting and bowling. It was shown how various aspects, like partnerships, turning points, and individual performances in a match, can be calculated and illustrated. It is hoped that these methods will be useful for practitioners. The availability of computers and other electronic media can be used to make cricket viewing even more enjoyable than before.

Footnotes

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.

References

Paccagnelia M. Performing under pressure, 2012. http://www.ausport.gov.au/sportscoachmag/psychology2/performing_under_pressure2.

Quinn E. Overcoming performance anxiety with sports psychology, 2013. http://sportsmedicine.about.com/cs/sport_psych/a/aa010603a.htm.

Mesagno

Marchant

Morris

. A pre-performance routine to alleviate choking in “choking-susceptible” athletes. Sport Psychol 2008; 22: 439–457.

Duckworth

Lewis

. A fair method of resetting the target in interrupted one-day cricket matches. J Oper Res Soc 1998; 49: 220–227.

Jayadevan

. A new method for the computation of target scores in interrupted limited-over cricket matches. Curr Sci 2002; 83: 577–586.

de Silva R. A method to calculate the target score in rain interrupted limited over cricket matches. In: International conference on mathematics and science (ICOMSc), Surabaya, Indonesia, 2011, pp. 1–8.

Preston

Thomas

. Rain rules for limited overs cricket and probabilities of victory. J Roy Stat Soc Ser D 2002; 51: 189–202.

Bhattacharya

Gill

Swartz

. Duckworth-Lewis and Twenty20 Cricket. J Oper Res Soc 2011; 62: 1951–1957.

Bailey

Clarke

. Predicting the match outcome in one-day international cricket matches, while the game is in progress. J Sports Sci Med 2006; 5: 480–487.

10.

de Silva

Pond

Swartz

. Estimation of magnitude of victory in one-day cricket. Aust NZ J Stat 2001; 43: 259–268.

11.

Lemmer

. A method to measure choking in cricket. Int J Sports Sci Coach 2015; 10: 461–477.

12.

Lemmer HH. A method to measure choking in cricket. In: South African statistical association annual conference, Polokwane, November 2013.

13.

Lemmer

. A method to measure strangling, a dramatic form of choking in cricket. Int J Sports Sci Coach 2015; 10: 715–726.

14.

Shah

. Pressure index in Cricket. IOSR J Sports Phys Educ 2014; 1: 9–11.

15.

Lemmer

. A method for the comparison of the bowling performances of bowlers in a match or series of matches. SA J Res Sport Phys Educ Recreat 2005; 27: 91–103.

16.

Cricinfo: Sri Lanka v. Pakistan, T20I Series, 2015. http://www.espncricinfo.com/sri-lanka-v-pakistan-2015/engine/match/860281.html.