### Extra runs saved per match - ERS/M

The bowling equivalent of the batting average: Quantitative evaluation of the contribution of bowlers in cricket using a novel statistic of ‘extra runs saved per match’ (ERS/M)

Bruce G Charlton

OR Insight. 2007; 20: 3-9.

School of Psychology, Newcastle University, UK, NE1 7RU

bruce.charlton@ncl.ac.uk

Abstract

In cricket, the specialist batsman’s ability may be evaluated by a single summary statistic: the batting average, but there is no equivalent measure for quantitative evaluation of the specialist bowler. I describe a method for calculating a novel bowling performance measure equivalent to the batting average: Extra Runs Saved per Match (ERS/M). The ERS/M is derived from the bowling average and the wickets per match statistic. It compares one bowler with another or with a standard ‘norm’ of bowling average 30 runs per wicket and 3 wickets per match. The value of ERS/M to selectors is that it measures differences in bowling ability between rival players, and combined with the batting average can calculate each player’s runs contributed per match. The business value of the ERS/M relates to its potential as an objective and comparative measure of a player's productivity.

***

Introduction

Cricket is the most ‘statistical’ of major UK sports due to its being based upon many repetitions of a basic unit of action - a bowler delivering a ball to a batsman in a highly-regulated manner. This makes cricket more amenable to an Operational Research approach than less stereotyped sports such as football in which only a few aspects (eg. the kick-off, corners, penalties) are sufficiently similar to enable statistical analysis.

Cricket has recently also experienced a highly-successful intervention from the field of Operational Research in the form of the ‘Duckworth-Lewis’ method of adjusting target scores in one day matches (Duckworth & Lewis, 1995) which was subsequently adopted by the International Cricket Council (Duckworth & Lewis, 2004).

There are 18 first class counties in England and Wales, each employing a squad of professional cricketers large enough to cover the needs of selection under different conditions and forms of the game (11 players allowed per game), plus inevitable injuries. For example, Lancashire Country Cricket Club (which is relatively large and successful) has an annual turnover (2006) of 11.7 million pounds Sterling with salaries accounting for 4.8 million pounds Sterling (Lancashire CCC, 2006). The England and Wales Cricket Board (ECB) presides over this system and manages the England national team – it has a turnover of 78.8 million pounds Sterling with a salary expenditure of 10.9 million pounds Sterling (ECB, 2006).

Performance measures in sport

Sport is ultimately a form of entertainment – and if a sport fails to engage attention and provide pleasure then the performance levels of players are irrelevant. Furthermore, sporting success is a zero-sum game, in the sense that winning is predicated on losing.

Both these considerations mean that the importance of sports player performance measures are circumscribed compared with the economic sector. Nonetheless, individual player performance probably underpins marginal income in cricket, as in most commercially-developed sports, in the sense that that individual player performance contributes to team wins (or to proxy measures of wins, such as runs or goals scored). A team's greater success in winning matches generally leads not only to greater sporting status but also to a significantly greater share of that sport’s total income (Szymanski & Zimbalist, 2005).

The use of statistical measures of player performance in cricket is a growing area (eg. Barr & Kantor, 2004; Lewis, 2006). However this field remains under-developed in cricket compared with that of its similarly-structured US cousin baseball which has been revolutionized by the ‘Sabermetric’ school of analysis (James, 2003) - a form of Operational Research. Initially used as a sophisticated form of appreciation among fans, Sabermetrics spread to fantasy baseball competitions, and more recently has been used in high level general management and coaching of teams such as the Oakland Athletics and Boston Red Sox – where it is usually termed the ‘Moneyball’ approach (Lewis, 2004).

The Moneyball approach attempts to maximize team performance within a given monetary budget, by discovering baseball skills which contribute significantly to wins but are currently undervalued by the market. This requires a variety of measures of players 'marginal productivity' (Mankiw, 1997) – ie. their contribution to the 'output' of the team, which (in baseball) can be measured in wins, or proxy measures of wins such as runs or 'outs'. Moneyball is therefore a form of ‘arbitrage’ which aims to take advantage of any difference in the valuation of a player in two markets: the sports value of a player in terms of contributing to team wins, and the economic value of the player in terms of salary. Statistical analysis is needed because the relative contribution of two rival individual players to team wins (ie. the difference between their productivities) is non-obvious.

The idea of measuring the size of contribution of specific players to team wins has hardly yet been addressed in cricket, perhaps because players salaries have been much lower in First Class cricket than Major League baseball (Smith, 2002); and because international cricketers cannot transfer between teams. This may explain why cricket fandom, journalism, selection and management still seems to be based on ‘traditional’ statistical performance measures of variable validity; and intuitive, qualitative evaluations.

Nonetheless, player performance remains of significant interest to cricket team selectors and coaches who are trying to build winning teams from a fixed pool of players or within a limited budget, and this imperative to win will probably lead to developments in the use of cricket statistics – at least among cricket professionals.

A bowler's summary average

In cricket, the specialist batsman’s ability may be evaluated by a single statistic: the batting average – the average of runs scored per times given 'out'. The batting average has limitations. For example, batting average does not take account of how quickly runs are scored – ‘strike rate’ – or how long the batter has survived without being dismissed - ‘balls faced’. Nevertheless, the batting average is generally regarded as a valid single summary statistic measure of a batters contribution to the team. The man regarded as the greatest-ever test match batsman – Don Bradman (Australia) has by-far the highest ever test batting average (99.94 – ie. 100).

However, until now, there has been no equivalent statistic to evaluate the specialist bowler, nor is the magnitude of difference between bowler's contributions very obvious. The usually quoted test match bowling statistics are the bowling average (runs conceded per wicket taken), the number of wickets a bowler has attained in their career (often omitting the number of matches played-in), and the career number of 5-wickets-per-innings analyses. But there is no accepted single summary measure of the ‘per match’ performance of test bowlers.

One major difficulty with statistical evaluation of bowlers stems from the fact that a test match bowler has two distinct (although in practice often related) jobs. Firstly, they should take wickets, since a test match can only be won if twenty wickets are taken. This emphasizes that bowling evaluation needs an average wickets per match (W/M) statistic. Secondly bowlers should be ‘economical’ and prevent runs being scored while taking these wickets – this is already adequately measured by the standard ‘bowling average’ (BowAv). However, at present there is no statistical evaluation method which combines both the number of wickets taken per match and the economy with which wickets are taken.

Average W/M is not a standard statistic, but can easily be calculated from the standard statistics of total number of test wickets divided by the total number of test matches played (Ganju, 2007). This makes the method applicable to historical bowlers, as well as current players. For example, SF Barnes (1873-1967 - England) is frequently discussed as perhaps the greatest ever bowler (Edmonds, & Berry, 1989): he took 7 W/M at a bowling average (BowAv) of 16 (W/M rounded to one decimal place, bowling average rounded to nearest integer, for clarity*). Of current long-serving internationals the highest number of W/M seems to be Muttiah Muralitharan (Sri Lanka) with 6.1 W/M BowAv 22 followed by Shane Warne (Australia) with 4.9 W/M at BowAv 25. From this we may infer that Barnes was a better bowler than Muralitharan who is better than Warne, but without further analysis we do not know how-much better.

Although these great test match bowlers averaged many wickets per match (high W/M) and also take wickets for relatively few runs conceded (low BowAv) there are seldom enough great bowlers to fill an international side, so teams must usually select from players who are stronger in one function than another. Indeed, two or three of the specialist bowlers selected are usually expected to be batting ‘all rounders’ at least to the extent of contributing a significant number of runs, or protecting their wicket for long enough to enable higher-order batters to add to the score or bat-out a draw. But the lack of quantification for bowling performance means that the relative value of better bowling cannot currently be balanced against the relative value of better batting.

Extra Runs Saved per Match (ERS/M) as a measure of bowling contribution

I describe a method of comparing two specialist bowlers in terms of their bowling contribution quantified in terms of Extra Runs Saved per Match (ERS/M) by the better bowler compared with the lesser bowler. The ERS/M is derived from the bowling average and the average wickets per match statistic for two bowlers who are being compared.

The ERS/M therefore gives an objective measure of test match bowling performance which can then be offset against any difference between the test batting average of the two players – batting average is total runs scored per number of times the batsman has been given out, and the difference between two players batting averages can be used to calculate the Extra Runs Contributed per Match (ERC/M) by the better batter (see Appendix).

I also describe a modified version of the method by which the ERS/M calculation can be used to quantify bowling ability objectively by comparing each bowler with a standard ‘norm’ bowler – the example chosen being a norm test match bowler who takes 3 W/M at an average of 30 runs per wicket.

Method

The extra runs saved ‘ERS/M’ measurement combines the average extra runs saved per match by having a lower bowling average plus the extra runs saved by taking more wickets per match.

Comparing bowler A and bowler B, where bowler A has both a lower bowling average (BowAv) than B and also takes more wickets per match (W/M).

Bowling average difference (BowAvD) = BowAv A – BowAv B

Wickets per match (W/M) = Total number of wickets taken/ Total number of matches played

Wickets per match difference (W/MD) = W/M A – W/M B.

1. To calculate extra runs saved (ERS/M) by lower bowling average:

BowAvD X W/M B

This describes how many extra runs would be saved by the lower bowling average of A assuming that he took the same number of wickets per match as B.

But A also takes more wickets per match than B, and needs to be given credit for the extra runs saved that these wickets represent.

2. To calculate extra runs saved per match (ERS/M) by taking more wickets/ match:

W/MD X BowAv B

This statistic describes how many extra runs per match would be saved by measuring the number of extra runs B would be expected to concede in taking the greater number of wickets per match that A is able to achieve.

3. To calculate Total ERS/M:

Total ERS/M = ERS/M by lower bowling average + ERS/M by taking more wickets

ERS/M A = BowAvD X W/M B + W/MD X BowAv B

This step entails adding the extra runs saved per match due to the lower bowling average of A with the extra runs saved per match due to the greater number of wickets/ match of A. (See Appendix for a worked example.)

(NOTE: where one bowler has a lower bowling average and the other bowler takes more wickets per match, the ERS/M is calculated for each measure separately. The smaller value deducted from the larger value to define which bowler is superior. The difference in number of ERS/M should be credited to the better bowler.)

ERS/M in performance evaluation

Comparison of two players

The major innovation of ERS/M is in quantification; especially in giving bowlers extra credit for the runs saved per match resulting from their taking a higher number of wickets per match. This extra credit is calculated from the number of runs the weaker bowler would concede in taking the extra wickets.

Because the ERS/M method measures bowling contribution using the same basic unit as the batting average (ie. ‘runs’), it is possible quantitatively to measure and combine both bowling and batting contributions. This can be used to evaluate pairs of players, whether these are specialist bowlers, or specialist all-rounders.

England’s selection dilemma of Monty Panesar versus Ashley Giles from the 2006-7 Australia versus England test match series can be used as a worked example of how the ERS/M method can provide an objective and quantified comparison of bowling contribution, which can then be offset against batting contribution (see Appendix). According to this calculation Panesar’s bowling would save about 42 runs extra runs compared with Giles (22 runs by greater economy, plus 20 runs by taking an extra 0.5 wickets), and Giles’s batting would add an extra 20 runs compared with Panesar. On balance, Panesar would therefore contribute about 22 extra runs compared with Giles.

Comparison with a norm

As well as comparing two players, it is also possible to use the ERS/M method to generate absolute measures of a players bowling contribution, and therefore to measure objectively the performance of current and historical players. This requires generating a bowling performance ‘norm’ for the purpose of comparison.

The norm could be derived from a consensus, a calculation of averages, or by an arbitrary but simple and plausible definition of the ‘expected’ performance of a test match bowler in an average match. For example the norm could be set at 3 wickets per match taken at an average of 30 runs per wicket, and an equivalent batting average norm could be set at 30 runs per dismissal; because the average of all Test match averages up to October 23 2007 was a batting average of 30 and a bowling average of 31 (Lynch, 2007). The number of 3 wickets per match is just my guess, and chosen for the elegance of keeping all the integers at 3.

Each actual bowler could then be compared with that 3-30 norm to calculate the value of ERS/M, which could be positive or negative – according to whether the actual bowler performed better or worse than the norm bowler. The ranking of bowlers would depend on the relative importance ascribed to the BowAv compared with W/M, and would be especially sensitive to the size of the W/M in the norm. Setting the W/M norm at a high level strongly penalizes bowlers with a significantly lower W/M statistic, and vice versa.

As an example, for the 3-30 norm suggested above, the extra runs saved per match for bowlers previously mentioned would be as follows: Warne 72 ERS/M; Muralitharan 117 ERS/M; and Barnes 162 ERS/M. This demonstrates and quantifies the enormous added-value of a great bowler compared with a 'norm' bowler.

Bowlers and batters could also be compared. For instance, using the 3-30-30 norm, Don Bradman – with an average of 100 - would have contributed 140 extra runs per match compared with the norm batter’s two innings of 30 each. Since SF Barnes saved 162 extra runs compared with Bradman adding and extra 140 runs it seems that the ‘best-ever bowler’ was an average of 22 runs per match more valuable to his team than the ‘best-ever batter’. So the best-ever cricketer was an Englishman!

Conclusion

ERS/M combines bowling average with the statistic of wickets per match (W/M) to provide a method for quantifying the contribution of a bowler in terms of a single summary statistic: extra runs saved per match by comparison with another player, or compared with a standard norm.

Selectors usually know which of two rival players is the better bowler, but there is currently no statistical method for quantifying just how much better. The potential benefit of ERS/M for selectors is that it measures the difference in bowling contribution between players, and enables selectors to measure both batting and bowling contributions. The next step in validating player performance measures such as ERS/M and ERC/M would be to seek statistical correlations between the sum of individual player's measures within a team, and team outcome measures such as wins, runs-scored and wickets-taken.

It is my impression that the two major aspects of bowling performance which constitute the ERS/M – ie. wickets per match and bowling average – are probably both proxy measures of a single underlying ability as a bowler. In particular, I have not been able to discover a long established Test Match bowler who took a large number of wickets per match but at a high (ie. expensive) bowling average over a significant length of time.

It therefore looks as if the ability to take large numbers of wickets per match either entails a low bowling average, or else requires the level of bowling control which is implied by a good economy rate. Simply measuring the number of wickets per match gives a reasonable approximation of ERS/M (Ganju, 2007). Nonetheless, there are significant differences in bowling average even between those elite bowlers who took more than 4.5 wickets per match, and the ERS/M statistic measures the advantage of taking many wickets at a lower average.

In sports, it is a general rule (or, at least, assumption) that team wins are, associated with improved marginal income – a greater share of that sports income (Szymanski & Zimbalist, 2005). But the supply of good players is limited and the amount which can be paid in player salaries is constrained. The problem for those managing cricket is therefore how to build the most effective (ie. most winning) cricket team, given the money and players available. The ERS/M provides a way to measure two players relative productivity (ie. the difference between their bowling contributions) which can be used along with information on the difference between their salaries to calculate an efficiency measure (such as ERS/M per 10 000 pounds salary).

As with the Moneyball approach in baseball, which is supported by new forms of statistical analysis, a major potential value of the ERS/M statistic for the ‘business’ of cricket would therefore be in ‘arbitrage’; ie. to discover and employ those overlooked players whose productivity and contribution to team wins is under-valued by the business salary market. Significant under-valuation (and, of course, over-valuation) of players is postulated to be almost inevitable due to the current deficiency of valid quantitative statistical information concerning a player’s bowling performance. This is due to the lack of an adequate summary statistic for bowling ability. The ERS/ M is proposed as such a performance measure.

*Note: Player performance statistics are derived from www.cricinfo.com ‘Statsguru’ on 13 October 2006.

References

Barr GDI and Kantor BS (2004). A criterion for comparing and selecting batsmen in limited overs cricket. Journal of the Operational Research Society 55: 1266-1274

Duckworth FC and Lewis AJ (1998). A fair method for re-setting the target in interrupted one-day cricket matches. Operational Research Applied to Sports 29: 220-227.

Duckworth FC and Lewis AJ (2004). A successful operational research intervention in one-day cricket. Journal of the Operational Research Society 55: 749-759.

Edmonds P, Berry S (1989). One hundred greatest bowlers. London: Queen Ann Press.

England and Wales Cricket Board. ECB annual report and accounts 2005. www.ecb.co.uk. Accessed 19 March 2006.

Ganju K. Looking for the greatest bowler. Cricket Trivia. www.crictrivia.com. Accessed March 19 2007.

James B (2003) The new Bill James Historical Baseball Abstract. New York: Free Press.

Lancashire County Cricket Club. Lancs announce record profit for 2006. www.lccc.co.uk. Accessed 19 March 2006.

Lewis AL (2006) Towards fairer measures of player performance in one-day cricket. J Opl Res Soc 56: 804-815.

Lewis M (2004). Moneyball: the art of winning an unfair game. New York: WW Norton.

Lynch S (2004). The overall Test average, and the most outs without. www.cricinfo.com. 23 October 2007. Accessed October 23 2007.

Mankiw NG. Principles of Economics. Fort Worth, TX, USA: Harcourt Brace, 1997.

Smith ET. Playing hardball: A Kent County Cricketer's Journey Into Big League Baseball. London: Abacus, 2002.

Szymanski S, Zimbalist A. National Pastime: How Americans Play Baseball and the Rest of the World Plays Soccer. Washington, DC: Brookings Institution, 2005.

Appendix

Worked example of ERS/M method – Panesar versus Giles

Here I use a recent England test match selection dilemma, which seems typical of the problem facing selectors.

Approaching the 2006-7 ‘Ashes’ series against Australia, England had two candidates for the role of left arm orthodox slow bowler: Ashley Giles and Monty Panesar. The dilemma occurred because Panesar was probably a better bowler (having a lower bowling average and taking more wickets per match) while Giles was probably a better batter (having a higher batting average; BatAv). The question was whether Panesar’s better bowling would compensate for the fewer runs he would contribute with the bat.

Panesar’s statistics were less precise estimates of his ability than were Giles’s, because in late 2006 Panesar had played only 10 test matches compared to Giles 52. However, Panesar’s statistics were more recent since he replaced Giles during 2005-6 while Giles was injured. For the sake of clarity, all statistics are rounded to the nearest integer except wickets/match which is rounded to one decimal place.

Monty Panesar (MP)

Test Matches – 10

Test wickets – 32

W/M – 3.2

BowAv – 32

BatAv - 10

Ashley Giles (AG)

Test Matches – 52

Test wickets – 140

Wickets/ Match (W/M) – 2.7

BowAv – 40

BatAv - 20

BowAvD = 40 - 32 = 8

W/M difference = 0.5

Step 1: To calculate extra runs saved (ERS/M) by lower bowling average:

BowAvD X W/M AG = 8 X 2.7 = 21.6 ERS/M by MP by lower bowling average

Step 2: To calculate extra runs saved per match (ERS/M) by taking more wickets/match:

W/MD X BowAv AG + 0.5 X 40 = 20 ERS/M by MP by more W/M

Step 3: To calculate Total ERS/M: -Total ERS/M by MP

BowAvD X W/M AG + W/MD X BowAv AG

8 X 2.7 + 0.5 X 40 = 21.6 + 20

= ERS/M by MP of 41.6 runs per match.

The two stage calculation assumes that Giles can take his average of 2.7 W/M at only 8 runs per wicket more expense than Panesar (ie. the bowling average difference), but for Giles to take the extra 0.5 W/M which Panesar on average contributes, Giles would on average concede an extra 20 runs.

The conclusion is that MP will save an average of 21.6 runs by greater economy, plus 20 runs by taking an extra 0.5 wickets. At this point the superiority of Panesar’s bowling has been quantified as an ERS/M of 41.6 runs per match.

Calculating Extra Runs Contributed per Match (ERC/M) with bat

The extra runs saved per match (ERS/M) by Panesar can be offset against the extra runs contributed per match (ERC/M) by Giles when batting.

AG’s batting average (BatAv) is 20 compared with MP’s of 10. Since a player may bat twice in a test match, a simple approximation of batting contribution would be double the difference in batting average (BatAvD). The BatAvD for AG and MP is 10 runs.

BatAvD = BatAv A – Bat Av B

ERC/M = 2 X BatAvD

ERC/M AG = 2 X 10 = 20

Therefore Giles would be expected to contribute 20 extra runs per match, compared with Panesar.

(Note: If a more precise measure of batting contribution is required, the batting average difference could be multiplied by the average number of innings batted per test match, which is 1.5 for Giles and 1.3 for Panesar; the difference arising from the fact that Panesar bats number 11 and therefore less often than Giles who bats at number 8. Players may be injured during the match, and therefore fail to bat; and in the case of England winning or drawing, not all England players would necessarily be required to bat twice. However, the statistic for innings batted per test match is not always calculable from standard cricketing statistical tables.)

According to this calculation Panesar’s bowling saves 41.6 runs extra runs compared with Giles, and Giles batting adds an extra 20 runs compared with Panesar. Based on this averaged and slightly approximated data, and all else being equal between the two players (which is seldom the case), Panesar's better bowling contributes 22 more runs per match than Giles better batting contributes, and it would therefore be better to select Panesar.

Bruce G Charlton

OR Insight. 2007; 20: 3-9.

School of Psychology, Newcastle University, UK, NE1 7RU

bruce.charlton@ncl.ac.uk

Abstract

In cricket, the specialist batsman’s ability may be evaluated by a single summary statistic: the batting average, but there is no equivalent measure for quantitative evaluation of the specialist bowler. I describe a method for calculating a novel bowling performance measure equivalent to the batting average: Extra Runs Saved per Match (ERS/M). The ERS/M is derived from the bowling average and the wickets per match statistic. It compares one bowler with another or with a standard ‘norm’ of bowling average 30 runs per wicket and 3 wickets per match. The value of ERS/M to selectors is that it measures differences in bowling ability between rival players, and combined with the batting average can calculate each player’s runs contributed per match. The business value of the ERS/M relates to its potential as an objective and comparative measure of a player's productivity.

***

Introduction

Cricket is the most ‘statistical’ of major UK sports due to its being based upon many repetitions of a basic unit of action - a bowler delivering a ball to a batsman in a highly-regulated manner. This makes cricket more amenable to an Operational Research approach than less stereotyped sports such as football in which only a few aspects (eg. the kick-off, corners, penalties) are sufficiently similar to enable statistical analysis.

Cricket has recently also experienced a highly-successful intervention from the field of Operational Research in the form of the ‘Duckworth-Lewis’ method of adjusting target scores in one day matches (Duckworth & Lewis, 1995) which was subsequently adopted by the International Cricket Council (Duckworth & Lewis, 2004).

There are 18 first class counties in England and Wales, each employing a squad of professional cricketers large enough to cover the needs of selection under different conditions and forms of the game (11 players allowed per game), plus inevitable injuries. For example, Lancashire Country Cricket Club (which is relatively large and successful) has an annual turnover (2006) of 11.7 million pounds Sterling with salaries accounting for 4.8 million pounds Sterling (Lancashire CCC, 2006). The England and Wales Cricket Board (ECB) presides over this system and manages the England national team – it has a turnover of 78.8 million pounds Sterling with a salary expenditure of 10.9 million pounds Sterling (ECB, 2006).

Performance measures in sport

Sport is ultimately a form of entertainment – and if a sport fails to engage attention and provide pleasure then the performance levels of players are irrelevant. Furthermore, sporting success is a zero-sum game, in the sense that winning is predicated on losing.

Both these considerations mean that the importance of sports player performance measures are circumscribed compared with the economic sector. Nonetheless, individual player performance probably underpins marginal income in cricket, as in most commercially-developed sports, in the sense that that individual player performance contributes to team wins (or to proxy measures of wins, such as runs or goals scored). A team's greater success in winning matches generally leads not only to greater sporting status but also to a significantly greater share of that sport’s total income (Szymanski & Zimbalist, 2005).

The use of statistical measures of player performance in cricket is a growing area (eg. Barr & Kantor, 2004; Lewis, 2006). However this field remains under-developed in cricket compared with that of its similarly-structured US cousin baseball which has been revolutionized by the ‘Sabermetric’ school of analysis (James, 2003) - a form of Operational Research. Initially used as a sophisticated form of appreciation among fans, Sabermetrics spread to fantasy baseball competitions, and more recently has been used in high level general management and coaching of teams such as the Oakland Athletics and Boston Red Sox – where it is usually termed the ‘Moneyball’ approach (Lewis, 2004).

The Moneyball approach attempts to maximize team performance within a given monetary budget, by discovering baseball skills which contribute significantly to wins but are currently undervalued by the market. This requires a variety of measures of players 'marginal productivity' (Mankiw, 1997) – ie. their contribution to the 'output' of the team, which (in baseball) can be measured in wins, or proxy measures of wins such as runs or 'outs'. Moneyball is therefore a form of ‘arbitrage’ which aims to take advantage of any difference in the valuation of a player in two markets: the sports value of a player in terms of contributing to team wins, and the economic value of the player in terms of salary. Statistical analysis is needed because the relative contribution of two rival individual players to team wins (ie. the difference between their productivities) is non-obvious.

The idea of measuring the size of contribution of specific players to team wins has hardly yet been addressed in cricket, perhaps because players salaries have been much lower in First Class cricket than Major League baseball (Smith, 2002); and because international cricketers cannot transfer between teams. This may explain why cricket fandom, journalism, selection and management still seems to be based on ‘traditional’ statistical performance measures of variable validity; and intuitive, qualitative evaluations.

Nonetheless, player performance remains of significant interest to cricket team selectors and coaches who are trying to build winning teams from a fixed pool of players or within a limited budget, and this imperative to win will probably lead to developments in the use of cricket statistics – at least among cricket professionals.

A bowler's summary average

In cricket, the specialist batsman’s ability may be evaluated by a single statistic: the batting average – the average of runs scored per times given 'out'. The batting average has limitations. For example, batting average does not take account of how quickly runs are scored – ‘strike rate’ – or how long the batter has survived without being dismissed - ‘balls faced’. Nevertheless, the batting average is generally regarded as a valid single summary statistic measure of a batters contribution to the team. The man regarded as the greatest-ever test match batsman – Don Bradman (Australia) has by-far the highest ever test batting average (99.94 – ie. 100).

However, until now, there has been no equivalent statistic to evaluate the specialist bowler, nor is the magnitude of difference between bowler's contributions very obvious. The usually quoted test match bowling statistics are the bowling average (runs conceded per wicket taken), the number of wickets a bowler has attained in their career (often omitting the number of matches played-in), and the career number of 5-wickets-per-innings analyses. But there is no accepted single summary measure of the ‘per match’ performance of test bowlers.

One major difficulty with statistical evaluation of bowlers stems from the fact that a test match bowler has two distinct (although in practice often related) jobs. Firstly, they should take wickets, since a test match can only be won if twenty wickets are taken. This emphasizes that bowling evaluation needs an average wickets per match (W/M) statistic. Secondly bowlers should be ‘economical’ and prevent runs being scored while taking these wickets – this is already adequately measured by the standard ‘bowling average’ (BowAv). However, at present there is no statistical evaluation method which combines both the number of wickets taken per match and the economy with which wickets are taken.

Average W/M is not a standard statistic, but can easily be calculated from the standard statistics of total number of test wickets divided by the total number of test matches played (Ganju, 2007). This makes the method applicable to historical bowlers, as well as current players. For example, SF Barnes (1873-1967 - England) is frequently discussed as perhaps the greatest ever bowler (Edmonds, & Berry, 1989): he took 7 W/M at a bowling average (BowAv) of 16 (W/M rounded to one decimal place, bowling average rounded to nearest integer, for clarity*). Of current long-serving internationals the highest number of W/M seems to be Muttiah Muralitharan (Sri Lanka) with 6.1 W/M BowAv 22 followed by Shane Warne (Australia) with 4.9 W/M at BowAv 25. From this we may infer that Barnes was a better bowler than Muralitharan who is better than Warne, but without further analysis we do not know how-much better.

Although these great test match bowlers averaged many wickets per match (high W/M) and also take wickets for relatively few runs conceded (low BowAv) there are seldom enough great bowlers to fill an international side, so teams must usually select from players who are stronger in one function than another. Indeed, two or three of the specialist bowlers selected are usually expected to be batting ‘all rounders’ at least to the extent of contributing a significant number of runs, or protecting their wicket for long enough to enable higher-order batters to add to the score or bat-out a draw. But the lack of quantification for bowling performance means that the relative value of better bowling cannot currently be balanced against the relative value of better batting.

Extra Runs Saved per Match (ERS/M) as a measure of bowling contribution

I describe a method of comparing two specialist bowlers in terms of their bowling contribution quantified in terms of Extra Runs Saved per Match (ERS/M) by the better bowler compared with the lesser bowler. The ERS/M is derived from the bowling average and the average wickets per match statistic for two bowlers who are being compared.

The ERS/M therefore gives an objective measure of test match bowling performance which can then be offset against any difference between the test batting average of the two players – batting average is total runs scored per number of times the batsman has been given out, and the difference between two players batting averages can be used to calculate the Extra Runs Contributed per Match (ERC/M) by the better batter (see Appendix).

I also describe a modified version of the method by which the ERS/M calculation can be used to quantify bowling ability objectively by comparing each bowler with a standard ‘norm’ bowler – the example chosen being a norm test match bowler who takes 3 W/M at an average of 30 runs per wicket.

Method

The extra runs saved ‘ERS/M’ measurement combines the average extra runs saved per match by having a lower bowling average plus the extra runs saved by taking more wickets per match.

Comparing bowler A and bowler B, where bowler A has both a lower bowling average (BowAv) than B and also takes more wickets per match (W/M).

Bowling average difference (BowAvD) = BowAv A – BowAv B

Wickets per match (W/M) = Total number of wickets taken/ Total number of matches played

Wickets per match difference (W/MD) = W/M A – W/M B.

1. To calculate extra runs saved (ERS/M) by lower bowling average:

BowAvD X W/M B

This describes how many extra runs would be saved by the lower bowling average of A assuming that he took the same number of wickets per match as B.

But A also takes more wickets per match than B, and needs to be given credit for the extra runs saved that these wickets represent.

2. To calculate extra runs saved per match (ERS/M) by taking more wickets/ match:

W/MD X BowAv B

This statistic describes how many extra runs per match would be saved by measuring the number of extra runs B would be expected to concede in taking the greater number of wickets per match that A is able to achieve.

3. To calculate Total ERS/M:

Total ERS/M = ERS/M by lower bowling average + ERS/M by taking more wickets

ERS/M A = BowAvD X W/M B + W/MD X BowAv B

This step entails adding the extra runs saved per match due to the lower bowling average of A with the extra runs saved per match due to the greater number of wickets/ match of A. (See Appendix for a worked example.)

(NOTE: where one bowler has a lower bowling average and the other bowler takes more wickets per match, the ERS/M is calculated for each measure separately. The smaller value deducted from the larger value to define which bowler is superior. The difference in number of ERS/M should be credited to the better bowler.)

ERS/M in performance evaluation

Comparison of two players

The major innovation of ERS/M is in quantification; especially in giving bowlers extra credit for the runs saved per match resulting from their taking a higher number of wickets per match. This extra credit is calculated from the number of runs the weaker bowler would concede in taking the extra wickets.

Because the ERS/M method measures bowling contribution using the same basic unit as the batting average (ie. ‘runs’), it is possible quantitatively to measure and combine both bowling and batting contributions. This can be used to evaluate pairs of players, whether these are specialist bowlers, or specialist all-rounders.

England’s selection dilemma of Monty Panesar versus Ashley Giles from the 2006-7 Australia versus England test match series can be used as a worked example of how the ERS/M method can provide an objective and quantified comparison of bowling contribution, which can then be offset against batting contribution (see Appendix). According to this calculation Panesar’s bowling would save about 42 runs extra runs compared with Giles (22 runs by greater economy, plus 20 runs by taking an extra 0.5 wickets), and Giles’s batting would add an extra 20 runs compared with Panesar. On balance, Panesar would therefore contribute about 22 extra runs compared with Giles.

Comparison with a norm

As well as comparing two players, it is also possible to use the ERS/M method to generate absolute measures of a players bowling contribution, and therefore to measure objectively the performance of current and historical players. This requires generating a bowling performance ‘norm’ for the purpose of comparison.

The norm could be derived from a consensus, a calculation of averages, or by an arbitrary but simple and plausible definition of the ‘expected’ performance of a test match bowler in an average match. For example the norm could be set at 3 wickets per match taken at an average of 30 runs per wicket, and an equivalent batting average norm could be set at 30 runs per dismissal; because the average of all Test match averages up to October 23 2007 was a batting average of 30 and a bowling average of 31 (Lynch, 2007). The number of 3 wickets per match is just my guess, and chosen for the elegance of keeping all the integers at 3.

Each actual bowler could then be compared with that 3-30 norm to calculate the value of ERS/M, which could be positive or negative – according to whether the actual bowler performed better or worse than the norm bowler. The ranking of bowlers would depend on the relative importance ascribed to the BowAv compared with W/M, and would be especially sensitive to the size of the W/M in the norm. Setting the W/M norm at a high level strongly penalizes bowlers with a significantly lower W/M statistic, and vice versa.

As an example, for the 3-30 norm suggested above, the extra runs saved per match for bowlers previously mentioned would be as follows: Warne 72 ERS/M; Muralitharan 117 ERS/M; and Barnes 162 ERS/M. This demonstrates and quantifies the enormous added-value of a great bowler compared with a 'norm' bowler.

Bowlers and batters could also be compared. For instance, using the 3-30-30 norm, Don Bradman – with an average of 100 - would have contributed 140 extra runs per match compared with the norm batter’s two innings of 30 each. Since SF Barnes saved 162 extra runs compared with Bradman adding and extra 140 runs it seems that the ‘best-ever bowler’ was an average of 22 runs per match more valuable to his team than the ‘best-ever batter’. So the best-ever cricketer was an Englishman!

Conclusion

ERS/M combines bowling average with the statistic of wickets per match (W/M) to provide a method for quantifying the contribution of a bowler in terms of a single summary statistic: extra runs saved per match by comparison with another player, or compared with a standard norm.

Selectors usually know which of two rival players is the better bowler, but there is currently no statistical method for quantifying just how much better. The potential benefit of ERS/M for selectors is that it measures the difference in bowling contribution between players, and enables selectors to measure both batting and bowling contributions. The next step in validating player performance measures such as ERS/M and ERC/M would be to seek statistical correlations between the sum of individual player's measures within a team, and team outcome measures such as wins, runs-scored and wickets-taken.

It is my impression that the two major aspects of bowling performance which constitute the ERS/M – ie. wickets per match and bowling average – are probably both proxy measures of a single underlying ability as a bowler. In particular, I have not been able to discover a long established Test Match bowler who took a large number of wickets per match but at a high (ie. expensive) bowling average over a significant length of time.

It therefore looks as if the ability to take large numbers of wickets per match either entails a low bowling average, or else requires the level of bowling control which is implied by a good economy rate. Simply measuring the number of wickets per match gives a reasonable approximation of ERS/M (Ganju, 2007). Nonetheless, there are significant differences in bowling average even between those elite bowlers who took more than 4.5 wickets per match, and the ERS/M statistic measures the advantage of taking many wickets at a lower average.

In sports, it is a general rule (or, at least, assumption) that team wins are, associated with improved marginal income – a greater share of that sports income (Szymanski & Zimbalist, 2005). But the supply of good players is limited and the amount which can be paid in player salaries is constrained. The problem for those managing cricket is therefore how to build the most effective (ie. most winning) cricket team, given the money and players available. The ERS/M provides a way to measure two players relative productivity (ie. the difference between their bowling contributions) which can be used along with information on the difference between their salaries to calculate an efficiency measure (such as ERS/M per 10 000 pounds salary).

As with the Moneyball approach in baseball, which is supported by new forms of statistical analysis, a major potential value of the ERS/M statistic for the ‘business’ of cricket would therefore be in ‘arbitrage’; ie. to discover and employ those overlooked players whose productivity and contribution to team wins is under-valued by the business salary market. Significant under-valuation (and, of course, over-valuation) of players is postulated to be almost inevitable due to the current deficiency of valid quantitative statistical information concerning a player’s bowling performance. This is due to the lack of an adequate summary statistic for bowling ability. The ERS/ M is proposed as such a performance measure.

*Note: Player performance statistics are derived from www.cricinfo.com ‘Statsguru’ on 13 October 2006.

References

Barr GDI and Kantor BS (2004). A criterion for comparing and selecting batsmen in limited overs cricket. Journal of the Operational Research Society 55: 1266-1274

Duckworth FC and Lewis AJ (1998). A fair method for re-setting the target in interrupted one-day cricket matches. Operational Research Applied to Sports 29: 220-227.

Duckworth FC and Lewis AJ (2004). A successful operational research intervention in one-day cricket. Journal of the Operational Research Society 55: 749-759.

Edmonds P, Berry S (1989). One hundred greatest bowlers. London: Queen Ann Press.

England and Wales Cricket Board. ECB annual report and accounts 2005. www.ecb.co.uk. Accessed 19 March 2006.

Ganju K. Looking for the greatest bowler. Cricket Trivia. www.crictrivia.com. Accessed March 19 2007.

James B (2003) The new Bill James Historical Baseball Abstract. New York: Free Press.

Lancashire County Cricket Club. Lancs announce record profit for 2006. www.lccc.co.uk. Accessed 19 March 2006.

Lewis AL (2006) Towards fairer measures of player performance in one-day cricket. J Opl Res Soc 56: 804-815.

Lewis M (2004). Moneyball: the art of winning an unfair game. New York: WW Norton.

Lynch S (2004). The overall Test average, and the most outs without. www.cricinfo.com. 23 October 2007. Accessed October 23 2007.

Mankiw NG. Principles of Economics. Fort Worth, TX, USA: Harcourt Brace, 1997.

Smith ET. Playing hardball: A Kent County Cricketer's Journey Into Big League Baseball. London: Abacus, 2002.

Szymanski S, Zimbalist A. National Pastime: How Americans Play Baseball and the Rest of the World Plays Soccer. Washington, DC: Brookings Institution, 2005.

Appendix

Worked example of ERS/M method – Panesar versus Giles

Here I use a recent England test match selection dilemma, which seems typical of the problem facing selectors.

Approaching the 2006-7 ‘Ashes’ series against Australia, England had two candidates for the role of left arm orthodox slow bowler: Ashley Giles and Monty Panesar. The dilemma occurred because Panesar was probably a better bowler (having a lower bowling average and taking more wickets per match) while Giles was probably a better batter (having a higher batting average; BatAv). The question was whether Panesar’s better bowling would compensate for the fewer runs he would contribute with the bat.

Panesar’s statistics were less precise estimates of his ability than were Giles’s, because in late 2006 Panesar had played only 10 test matches compared to Giles 52. However, Panesar’s statistics were more recent since he replaced Giles during 2005-6 while Giles was injured. For the sake of clarity, all statistics are rounded to the nearest integer except wickets/match which is rounded to one decimal place.

Monty Panesar (MP)

Test Matches – 10

Test wickets – 32

W/M – 3.2

BowAv – 32

BatAv - 10

Ashley Giles (AG)

Test Matches – 52

Test wickets – 140

Wickets/ Match (W/M) – 2.7

BowAv – 40

BatAv - 20

BowAvD = 40 - 32 = 8

W/M difference = 0.5

Step 1: To calculate extra runs saved (ERS/M) by lower bowling average:

BowAvD X W/M AG = 8 X 2.7 = 21.6 ERS/M by MP by lower bowling average

Step 2: To calculate extra runs saved per match (ERS/M) by taking more wickets/match:

W/MD X BowAv AG + 0.5 X 40 = 20 ERS/M by MP by more W/M

Step 3: To calculate Total ERS/M: -Total ERS/M by MP

BowAvD X W/M AG + W/MD X BowAv AG

8 X 2.7 + 0.5 X 40 = 21.6 + 20

= ERS/M by MP of 41.6 runs per match.

The two stage calculation assumes that Giles can take his average of 2.7 W/M at only 8 runs per wicket more expense than Panesar (ie. the bowling average difference), but for Giles to take the extra 0.5 W/M which Panesar on average contributes, Giles would on average concede an extra 20 runs.

The conclusion is that MP will save an average of 21.6 runs by greater economy, plus 20 runs by taking an extra 0.5 wickets. At this point the superiority of Panesar’s bowling has been quantified as an ERS/M of 41.6 runs per match.

Calculating Extra Runs Contributed per Match (ERC/M) with bat

The extra runs saved per match (ERS/M) by Panesar can be offset against the extra runs contributed per match (ERC/M) by Giles when batting.

AG’s batting average (BatAv) is 20 compared with MP’s of 10. Since a player may bat twice in a test match, a simple approximation of batting contribution would be double the difference in batting average (BatAvD). The BatAvD for AG and MP is 10 runs.

BatAvD = BatAv A – Bat Av B

ERC/M = 2 X BatAvD

ERC/M AG = 2 X 10 = 20

Therefore Giles would be expected to contribute 20 extra runs per match, compared with Panesar.

(Note: If a more precise measure of batting contribution is required, the batting average difference could be multiplied by the average number of innings batted per test match, which is 1.5 for Giles and 1.3 for Panesar; the difference arising from the fact that Panesar bats number 11 and therefore less often than Giles who bats at number 8. Players may be injured during the match, and therefore fail to bat; and in the case of England winning or drawing, not all England players would necessarily be required to bat twice. However, the statistic for innings batted per test match is not always calculable from standard cricketing statistical tables.)

According to this calculation Panesar’s bowling saves 41.6 runs extra runs compared with Giles, and Giles batting adds an extra 20 runs compared with Panesar. Based on this averaged and slightly approximated data, and all else being equal between the two players (which is seldom the case), Panesar's better bowling contributes 22 more runs per match than Giles better batting contributes, and it would therefore be better to select Panesar.

## 8 Comments:

Strewth! You've gone a bit overboard there mate!

Very interesting.

I can see what you're trying to do, but I'm not sure it works.

What you are trying to say is that the selectors can pick AG, who will get 2.7 wickets at 40 runs, so conceding 108 runs in total.

They can also pick MP, who will get 2.7 wickets at 32 runs, so conceding only 86 runs in total.

So far, so easy, and you can just use the averages. But you want to somehow quantify the fact that MP is (on average) going to get another wicket, and AG isn't.

You do this by noting that to get that wicket AG would cost another 20 runs.

The problem is though, wouldn't MP cost another 16 runs? And so the 'saving' of MP over AG is simply 3.2 wickets x (AG average - MP average), or 25.6

Now as I understand it you think this is unfair to MP because there's no reason to believe AG can simply scale up his 2.7 wickets to get 3.2. Which is correct. But I'm not sure your way of adjusting works either.

Thanks for your comment Matthew. You are correct that the ERS/M statistic hinges on the question of how to credit Panesar for the extra wickets (or fractions of wickets) taken.

I do this by assuming that the bowler who took fewer wickets would need to take the extra wickets at a cost in runs proportionate to their bowling average.

This is the key assumption in calculating the ERS/M - and like all such assumptions it is not self-evident, nor is it immune to criticism. Which is why the ERS/M needs testing in practice, just as the batting average has been tested in practice before being accepted as the best single batting summary statistic for test matches.

I'm sorry, but I couldn't follow your line of argument beyond this. I think maybe you have not accurately represented the way that ERS/M is calculated.

Interestingly, baseball sabermetrics seems to have shown that the traditional baseball single summary measure of baseball batting - the baseball batting average - does *not* have good validity as a performance measure; and other measures such as on-base-percentage or the slugging percentage work better as explainers and predictors.

"I do this by assuming that the bowler who took fewer wickets would need to take the extra wickets at a cost in runs proportionate to their bowling average."

But you ignore the extra runs that the bowler who takes more wickets needs to get those too?

ie MP would get 3.2 at a cost of 32 each, ie 100, and for AG to get that he'd need 128, so the differnce is 28, not 41?

Hi, Bruce,

Popping in from the conversation over at Tim Worstall's ...

In essence, I like the idea of including a bowler's penetrative power as well as merely his economy (a bowler with a low average could just be one who is very hard to score from and eventually just bores the batsman out, after all).

Also, providing a system which rates bowlers in "runs equivalent for the team" is good as it enables bowler-to-batsman comparisons.

My primary quibble is with the "zero-sum-game" aspect of the raw wickets/match statistic (if you are a spinner and the opening seam bowlers rip through the opposition, you're out of luck, for example; if you are in a team with superb bowlers, you are going to be fighting for a smaller share of wickets in comparison to a hypothetical clone of yourself in another team - although the runs scored might increase, as you have pointed out (conversely, if you are seen as the toughest bowler, the batsmen may try to shut up shop against you and score from the others, improving your figures)).

Therefore, using the strike rate against the average number of balls bowled per match for a front-line bowler to extrapolate an independent wickets/match value (what you would have taken if you'd always bowled the average number of balls in a team with exactly average bowlers) would slot in well to your existing method.

My other quibbles were not substantive, really, merely:

1- Refining the "norm" - I got figures from an "ask Bearders" column in 2005 where he had the total numbers of each type of dismissal up to the Trent Bridge 2005 test (Test 1726) and used those to find that about 14.7 wickets fell to each bowling line-up per match, which gave me the ~3.3 wickets/match figure.

2 - Adjusting the "norm" for different eras of cricket - but this could be so subjective. off the top of my head, significant changes which would/could cause a change of "era" would be:

- Changes to the lbw rule

- Covered pitches

- Protective gear for batsmen

If we could find a reasonably objective method of identifying changes in "era", then the batting average could be adjusted to get a time independent version as well (I'd suggest using the era in which Bradman batted is the default basis, simply because the 100 (near as dammit) average is just too good to lose :-) )

We'd then be able to show how many runs above the average were contributed by each batsman and how many runs-equivalent above average each bowler was worth.

Thanks to both Matthew and andy cooke for such intelligent comments.

First Matthew - I understand you now, sorry I didn't get the point at first.

You are correct to point out that the way I chose to reward the bowler who took more wickets is not the only one; and the way you suggest sounds equally rational. In the end, I was guided by the 'feel' of the statistics.

I think your alternative way of calculating the effect on runs saved of getting extra wickets does not give sufficient weight to the benefits of extra wickets. This is a matter of opinion/ judgment and there is certainly room for disagreement. I have tried out ERS/M over a range of situations and it seems to give palusible results - but obviously it would be better for other, more skeptical, people to do this kind of testing, and see whether an objective evaluation of ERS/M gives it a thumbs-up.

All sporting measures have an element of judgment which may or may not turn out to be useful in practice. For example, the baseball batting average did not include 'walks'/ base-on-balls - because this was assumed to be the pitcher's fault (for missing the strike zone four times) rather than a measure of batting ability. In practice, it turns out that the best batters get more walks, and it is therefore substantially an indirect measure of batting ability.

My feeling is that using the bowling average as a perfomance measure underestimates the benefits of wicket's taken.

Getting onto Andy's points. All good points, and I especially wish I had known about the data on average wicket per test match from Bearders - I was relying on nothing more than personal impression and a desire for numerical simplicity in examples.

As for the zero sum game - my rationale is that if the hypothestical spinner was playing in a team in which almost all the wickets were being taken by the fast bowlers, then indeed there would be no way that spinner could express their ability. In fact, that spinner would not be making a very large contribution to the side. I don't see any way around the fact that a single summary statistic perfomance measure on that spinner is inevitably going to underestimate their ability.

I guess this is at root a basic deficiency of statistics in team games.

Matthew (if you are still reading this?) - I've been thinking some more.

It seems that the calculation method you suggest only rewards a bowler for greater economy - is that correct?

For example, if two bowlers both have an identical bowling average of 30, but bowler A took average 4 wickets/ match while bowler B only took 3 wickets/ match - then how is bowler A to be credited for the fact that they actually take more wickets per match?

In practice, this is a pretty rare comparison - but it may establish (in a reductio ad absurdum kind of way) the need for some secondary and separate method for crediting better wickets/ match.

(Of course, my suggestion is only one such - there could be others.)

Yes, that's the problem with what I'm suggesting, it basically comes down to the average. And why I was suggesting your main aim is to reward the bowler who on average takes more wickets.

I'll do some thinking about how I would try to do it. Thanks for the reply - I didn't realise this post was so old until I tried to find it on your homepage!

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