A STALLED REFORM MOVEMENT & RENEWED IMPETUS – Part I

Peter Kettle | 6:26am GMT 14 November 2026

Nature & Appeal of Batting Averages

Cricket batting averages continue to be frequently referred to as a single statistic that is intended to encapsulate demonstrated ability over a whole career, or sometimes a particular phase of it or a particular series of matches. The traditionally formulated batting average – the number of runs made per completed (dismissal) innings – is the most commonly used indicator of a batsman’s, or batswoman’s, comparative ability.

An uncompleted innings – a Not Out – can arise in a variety of ways. Chiefly, due to rain or poor light ending play, a team declaration being made by the captain, an injury sustained while at the crease, and having attained a winning team target.

Batting averages are documented and published separately for different formats of the game, as these call for different approaches to the batting task and differ in the general degree of urgency of run-scoring: five-day Test matches, county/province/state matches which are scheduled over three or four days; one-day 50 Over matches and Twenty Overs a side matches of a certain specified standard of play. The illustrative material noted in this essay mainly relates to Test cricket.

To elaborate: an individual’s batting average is traditionally derived by taking their total runs scored off the bat (ignoring “extras”, chiefly in the form of byes, wides and illegitimate deliveries) and dividing this sum by the number of times dismissed. Hence, Not Out (unbeaten) innings don’t feature in the denominator. This remains the universally accepted convention, having the merit of simplicity and able to be readily cited for international and other “first-class” players throughout the entire history of the game. It still stands despite a number of weaknesses (noted shortly) and various proposals aired in journals for reform.

As to the raw material concerning the series of scores made at the point of a player being dismissed, these almost invariably take a shape akin to an exponential distribution – as sketched below (with the number of dismissal innings played being displayed vertically). An individual’s set of dismissals typically contains a sizeable proportion of very low scores and a considerable number, all told, of relatively high ones which brings up their dismissal average; whilst their collection of Not Out scores tends to take the form of a random scatter, a feature returned to later.

Purposes of this Essay

• to explain the lack of progress with potential reforms,
• bring out the main differences between proposals put forward in the literature,
• critique them and outline a way forward, and
• persuade those who determine and publish batting averages to adopt the recommendation being made.

Drawbacks of the Traditional Formulation

Given a context of wanting to use a single statistic to stand for comparative ability, notable drawbacks of the traditional formulation arise from the way that Not Out scores are dealt with. That is, by adding a player’s undefeated scores to their accumulated runs of their completed innings before dividing by the number of times having been dismissed.

Various reasons have been put forward for wanting an improved treatment of Not Out innings.

First: as illustrated by the following scenario, the traditional way that Not Outs are incorporated is, intuitively, over-generous to the player concerned.

Assume that a given player has scores upon being dismissed of 40, 15, 35; totalling 90, hence giving 30.0 as their batting average.

1. Then comes a Not Out innings of 9 runs. With the traditional treatment, Not Out scores are, in effect, spread across all of the player’s dismissal scores.

Spread over the three previous dismissal innings, this produces another 3 runs for each of them. So becoming 43, 18, 38: totalling 99. Dividing by 3 again for completed innings gives 33.0 as the updated traditional average.

2. This is equivalent to treating the Not Out score of 9 as the previous average of 30.0 and then adding 3 runs to itself – producing 33.0 runs as a notionally completed innings, the same as the new traditional average. The undefeated score of 9 has, in effect, become 33 – which is three and two-thirds times its original size.

Now, if your Mother were watching you bat, she may heartily approve of this way of processing a Not Out score, and tell the former Test star, Geoffrey Boycott, sitting in a deck chair alongside: “That innings of nine by my son were worth thirty-three… no less, Geoffrey. I know how to do the sums!” Boys (as he’s generally known) might well raise an eyebrow and deliver a sceptical word or two! Why: because, among other things, this treatment:

Implies that if allowed to continue that innings, you would have ended up reaching the greatly enhanced score of 33 runs before being dismissed and without there being a sound, laid down, basis for thinking it to be a plausible outcome, let alone a most likely outcome.

Second: The traditional treatment of Not Outs substantially bumps up the career batting averages for those in the middle order and those just above the tail-enders who happen to stay around until the end of the team’s innings.

Third: In cases where a player’s Not Out innings are frequent and quite a number of their Not Out scores are relatively high, the resulting batting average often represents a major deviation from their evident, observed, capability.

Fourth: Some low-in-the-batting-order players make their reputation for having a decent average by being stranded as the last remaining – undefeated – batsman, yet rarely if ever making a substantial score.

What seemingly has happened is that compilers of batting averages in the early days were faced with a loose end and sought a convenient and quick way of tying it up. The solution that was adopted set a precedent, and the establishment has been mentally disturbed by any proposed departures from it. In part, because that would have meant having to “correct” all the past records and, worse, maybe prompting the need to reappraise the status of prominent individuals in the history of the game. And so a heavy layer of inertia has set in.

Not Out Innings: Weighty Enough for Discussion

To take two points by way of illustration:

• From the earliest times through to the present day (start of September 2026), taking England’s 100 Test batsmen who have played at least 20 innings and averaged at least 30.0 per completed innings: they each had to retire before being dismissed in at least 7%, and up to 19%, of all their innings; the average proportion working out at 11%.
• Concerning all Test matches up to the end of 2019: of the 312 batsmen who had scored more than 2,000 runs during their Test career, 25 of them were not dismissed in at least 16% of their innings.

Two Notable General Features

Drawing on Kartikeya Date’s work, the frequency of Test batsmen remaining undefeated not only increases systematically as one proceeds down the order; it also increases sharply for the middle and lower ranges – as team innings declarations takes effect, and compounded later on by batsmen being stranded without a partner when ten team wickets have fallen:

A corollary is a similarly shaped rise in the number of Not Out innings played per 1,000 runs scored as one works down the batting order. This being the combined effect of a lower frequency of dismissals and lower scoring abilities as one moves into the middle order and more so when coming to those low in the batting order (number 8 to 11).

Those low down in the batting order – “tail-enders” – for which Not Out frequencies are highest, often arouse considerable interest. Although, for England, Jimmy Anderson has achieved notoriety in his Number 11 position, 114 times undefeated in his 265 Test innings (2003–24), the King of Not Outs, in proportional terms, has been the gifted flighter of left arm spin and invariably the number 11 batsman, Phil Tufnell – with 49% of his 59 Test innings undefeated (1990 to 2001), which yielded 5.1 runs per completed innings.

Despite a good deal of interesting work by researchers, as things currently stand the future doesn’t look bright for improving the way that batting averages are determined. I begin with the person who has given the reform movement a good deal of momentum, before turning to the knotty current impasse and the specifics of what others have proposed in the literature.

Health Warning

All those who have proposed reforms which are reviewed here couch their articles in terms of male players and invariably use the term “batsman.” Rest assured, their discussions and recommendations apply equally to women which should dispel impending angst or mental stress among the female readership.

I note that the term “Batter” is in vogue for those commentators who are indoctrinated in the creeds of political correctness and gender-neutral terminology. Their enthusiasm for its use knows no bounds; many these days referring to “Batter” in the context of a men-only match as well as a women-only match. As one by-stander has commented:

Use of the term “batter” is more than just faintly ridiculous. For one thing, “batter” is redolent of fish and chips; and, for another, it conjures up a vision of the bat being applied to give the opposition bowlers a continual battering…sending them into oblivion and, perhaps, to their eventual death!

Also, the term “batswoman” (alternatively, “batwoman”) tends to give a woman with bat in hand somewhat more weight… more gravitas …than being merely a “batter.” So perhaps acquiring more importance than a counterpart man with bat in hand. Either of these two epithets for female players are in tune with the feminist movement that continues to sweep swathes of the Western world.

For those interested in gender in relation to cricket terminology, perhaps peruse a recent piece by Megan Maurice: A Batsman by Any Other Name: Why Language Matters in Sport (February 2025).

Kimber’s Seminal Work

The spur to work on reform has been Alan Kimber’s seminal article, published in the Journal of the Royal Statistical Society (Series A) in February 1993. He readily acknowledged it is very much in the spirit of G.R. White’s suggestion, made during discussion of a paper by George Wood delivered back in November 1944, to use a “life table” to obtain a batsman’s true average by adding to the runs actually made an expectation of life in each of the uncompleted innings. The “how” of this being left unresolved at the time.

Wood’s paper (Cricket Scores and Geometric Progression) was read before members of the Royal Statistical Society in London and published, along with an account of the ensuing discussion, in the Society’s journal the following year.

Currently with Emeritus status as a Reader at the UK’s University of Southampton, School of Mathematical Sciences, Dr Kimber’s major contribution to the debate reflects his specialist interest in the analysis of materials and equipment reliability, and the modelling of survival data relating to diseases and treatments. In his article, he applied the principles of what is known as the Product Limit Estimator (PLE) to project the scores of Not Out innings to a notional conclusion as a means of putting batting averages on a realistic basis.

(Note: a person of Emeritus status is a retired faculty member who receives an honorary title to recognise a distinguished academic career.)

The PLE methodology had been conceived and developed operationally, in a medical context, by two eminent USA researchers: the mathematician Edward L. Kaplan and the bio-statistician Paul Meier in their celebrated (merged) twenty-two page article of June 1958. It appeared in the Journal of the American Statistical Association under the title, Nonparametric Estimation from Incomplete Observations. Their path-breaking approach and mechanics was applied initially in follow-up studies of samples of patients to show how effective a particular treatment (or type of surgery) is compared with a different treatment or taking no action.

The role of the PLE being to establish an expected eventual outcome for patients who had, for whatever reason, dropped out of being monitored following treatment. The future course of their condition is, in principle, discoverable but remains unknown – hidden from view – and so they go missing from the overall data set. The data collected prior to a patient dropping-out is, rather oddly, termed “censored” data.

The term Censored Data is said to have been introduced and established by the Danish statistician Anders Hald in the late-1940s and early-1950s, having credited John E. Kerrich with suggesting the term for statistical use. This is a somewhat misleading label, being at odds with the everyday use of censorship in the political and media world, meaning the active, deliberate suppression or hiding of objectionable material. I feel that it should now be eliminated and replaced with something more suitable – though the term “truncation” is ruled out as this has been harnessed by statisticians to refer to cases when a sample is drawn from an incompletely observed sub-population where entire groups outside a certain range never enter the sample at all.

In essence, the solution found by Kaplan and Meier to the task of gap-filling is to base the survival prospects of a drop-out patient on the observed survival rates (elapsed times to relapse or death) for all those patients who continue to be monitored. A patient having an uncompleted observation during the follow-up process is regarded as having the same prospects of continued survival as those continuing to be followed-up.

Once the likely post-treatment survival rates of drop-out patients had been estimated, it enabled the creation of a comprehensive data set for the patient sample as a whole; and a reliable statement could then be made about the likely survival prospects of future patients to be treated. The PLE has become the standard tool used by medical researchers for determining the length of survival in a multitude of studies, ranging from cancer to cardiovascular disease to diabetes.

Estimation of the “true” batting average of those having Not Out innings can, likewise, be viewed as an unknown quantity and susceptible to treatment by the PLE’s inherently probabilistic approach to projecting incomplete data to notionally concluded outcomes. In applying the PLE and the framework of survival analysis to Not Out scores in cricket, Kimber makes the following substitutions:

1. The progression of runs made by a batsman, from zero onwards, is substituted for the progression of time in monitoring patients from a given starting point.
2. A shot that results in a single run is substituted for the interval of time in patient monitoring. If a player hits a two or a three, a four or a six, these represent the passing of two, three, etc time intervals.
3. The totality of dismissal innings played by a batsman is put in place of the group of patients continuing to be monitored until their relapse or death.
4. A batsman having to retire Not Out takes the place of a patient dropping out of being monitored. He is assumed to have the same prospects for continued survival as when he continues to bat on having already reached that particular score and ultimately completes those innings by being dismissed. Kimber considers this to be a fully reasonable assumption to make.

Accordingly, when applied to Not Out innings, the focus of applying the PLE is to project each of a player’s undefeated innings scores to a notional conclusion, assuming he were able to continue batting with a partner.

Whilst in its operation the PLE is a complex mechanism, the key steps when dealing with Not Out scores may be summarised in lay person terms as follows:

1. The prognosis for a Not Out innings is represented by the array of a player’s surviving innings with scores at, and beyond, his undefeated score. More specifically, it is based on the relative number of times he reaches different completed scores during the totality of his innings, with each of these completed scores being assigned an equal probability of eventuating. Accordingly, the most likely potential outcome for the Not Out innings rests on the mathematical notion of “expected value.”

Special treatment is not accorded to relevant dismissal scores that are well detached from the rest of the field (or the bulk of it), termed Outliers. They are retained because they form part of the observed real life experience and also because they become diluted in importance – usually much diluted for a whole career – owing to the presence of all the other dismissal scores.

3. Projection of Not Out scores is carried out as an iterative procedure, by factoring in other, larger, Not Outs when they have been projected to completion.

This nuance can be managed by initially projecting the highest Not Out score and then dealing with the rest of the Not Outs in descending order of size.

4. The projected-to-completion Not Out scores are then added to the sum of the actually completed (dismissal) scores to give the batsman’s total runs; and dividing by his total innings played gives his “truly representative” average.

As applied by Kimber – with inclusion of any Outliers, with iteration, and with use of the equi-probability rule, the PLE method amounts to projecting Not Out scores by taking the Mean value of the relevant dismissal and other projected Not Out scores. This distilled form of the PLE exactly replicates Kimber’s own findings and, in personal correspondence, has his approval.

There are two caveats to note:

1. The PLE’s estimate is undefined for those fairly occasional cases where a player’s highest score is a Not Out, requiring an assumption to be made about his missing upper tail-end of dismissal scores. Kimber had “some empirical evidence that the upper tail of the distribution (for first-class cricketers) is roughly geometric and so batsmen have an approximately constant risk of being dismissed in that region.” The degree of risk is therefore similar to that faced when they have negotiated the early playing-in phase of an innings of around 12 to 20 runs.

For this, Kimber proposes a modified version of the PLE method which converts the Not Out concerned to an estimated completed score by adding on the batsman’s overall dismissal average. Perhaps the average of scores beyond the playing-in phase would be a worthwhile refinement.

2. When the proportion of Not Out innings becomes “huge”, the PLE method is said by Kimber to “be no more sensible than the traditional approach” of incorporating them. Although he doesn’t explicitly indicate the region in which this begins to occur, he gives the example of Bill Johnston being Not Out in 16 of his 17 innings (94%) for 102 runs on the 1953 tour of England; and it seems Kimber has in mind a proportion that exceeds at least 40% or 50%, which occurs only rarely for a full first-class season, let alone a full career. More on this threshold matter shortly.

To explain why the PLE eventually loses its advantage over the traditional method: with a rising proportion of Not Out innings, unreliability of the outcomes starts to occur as the number of relevant dismissal scores on which to base projecting those Not Out scores becomes patchy and sparse. A further reduction in the presence of relevant dismissal scores then has a substantial and erratic impact on the resulting projections, causing the estimated “true” batting average to become highly unstable. For the example given of Bill Johnston, with a top score of 28 Not Out (vs Sussex in early-June) and his next highest being 27 Not Out (vs the Gentlemen of England in late-August) – the PLE results in a batting average of 81, while the traditional method results in 102. Both highly misleading of Johnston’s actual ability.

The batting average resulting with Kimber’s application of the PLE method will usually be smaller than a batsman’s traditional average, a typical case being a batsman with a relatively high proportion of Not Outs who hasn’t made many very high scores. A reverse outcome can sometimes occur, depending on the batsman’s scoring profile.

Kimber gives the following findings, including four Surrey players in the 1983 English County Championship (noted last):

Player	Traditional Ave	Kimber (PLE)
Mohinder Amarnath: 113 Test inns, 8.8% not out	42.50	41.45
Don Bradman: 80 Test inns, 12.5% not out	99.94	98.98
Don Bradman: 338 first-class inns, 12.7% not out	95.14	94.76
David Gower: 200 Test inns (to mid-1992), 8.0% not out	44.31	44.30
Victor Trumper: 395 first-class inns, 5.3% not out	46.58	47.36
Monte Lynch, middle order: 39 inns, 25.6% not out	53.72	49.11
Alec Stewart, number three: 16 inns, 18.8% not out	31.30	30.71
Jack Richards, lower middle order: 34 inns, 23.5% not out	27.61	27.46
Sylvester Clarke, tail-ender: 24 inns, 16.7% not out	14.25	14.15

Kimber’s adjusted averages are lower in nearly all cases than the traditional average, whilst being a little higher for Trumper in his first-class matches. Apart from Lynch (down 8.6%), the changes are quite small. As seen, a relatively high proportion of Not Out doesn’t, by itself, necessarily lead to a substantially lower average when using the PLE.

A curiosity is that whilst Bradman’s adjusted average is only 0.96% lower than his traditionally-derived average, it implies that instead of having to score four runs in his last Test match innings (at the Kennington Oval in August 1948) to attain an average of 100.0, he would have needed to make 89 runs. Nearly as tantalising a prospect!

Shubhabratha Das (article of 2011) has computed the impact of using the PLE to revise the “official” batting averages for the careers of 37 Test players (post-WW1), all with traditional averages of 50.0 plus runs and having played at least 20 matches. For most of them, the batting average changes relatively little. Exceptions are Jacques Kallis and Mohammad Yousuf, both down by 2.2%; and Wally Hammond, Garry Sobers, Andy Flower and Javed Miandad, respectively up by 4.5%, 5.6%, 3.4% and 2.5%. The reason for these moderate to high upward changes is the batsmen have a substantial number of low Not Out scores relative to their traditional average.

Small changes predominate in this sample of Das because, being generally high in the order batsmen, their proportion of Not Out is generally low. The great majority lying in the 6% to 12% range, with only a few (five) reaching as high as 16% to 18%.

What neither Kimber nor Das report on is the resulting uplifts on a batsman’s Not Outs scores, being of interest when expressed as an overall multiplier – so showing this specific impact of applying the PLE method which doesn’t affect anything else directly relating to the batting average. The uplift to Not Out scores for Kimber’s seven players in Tests and County matches works out, overall, at 1.76: five of them being clustered in the 1.75 to 1.99 range:

• Amarnath: 1.49
• Bradman: 1.82
• Gower: 1.55
• Lynch: 1.75
• Stewart: 1.80
• Richards: 1.99
• Clarke: 1.92

So the PLE has had a substantial effect where it actually bears, although the multiplier is well below the three and two-thirds of the example given earlier for applying the traditional method.

Lemmer’s Perspective and his Estimator

Professor Hermanus (“Hoffie”) Lemmer of Johannesburg University takes a different perspective on the matter mentioned earlier of the Not Out innings threshold. In an often cited (referred) article of 2008, he argues that for batting in Tests, and other unlimited overs matches (notably ODIs), both the PLE and the traditional method become unreliable for producing a “true” batting average when the proportion of a player’s Not Out innings is in the vicinity of 20% and get increasing unreliable beyond that. He derives an “Estimator” (an equation) to address those cases. For a whole Test (and perhaps a one-day international) career and taking a minimum of 35 innings, 20% plus of Not Outs is very largely confined to those low down in the batting order – particularly numbers 9 to 11.

In Section 2 (Possible Estimators) of his article, titled General Measures of Batting Performance (South African Statistical Journal, Issue No. 1, 2008, pages 65–87), Lemmer provides an intuitively satisfactory rationale, in high level (very general) terms, for expecting an approximate doubling of a batsman’s Not Out scores if allowed to bat on until he gets dismissed:

Due to various reasons a batsman’s innings can be curtailed, leaving him with a not-out score. If it is assumed that external factors that can lead to a curtailment are random and independent of the batsman’s potential score…

Lemmer then shows, by use of equations, that the score the batsman will obtain, if allowed to bat on until he gets out, can be expected to be double his Not Out score.

The rationale in lay terms:

Depending on when a batsman’s uncompleted innings is halted (permanently), he could be just starting it and so be on nought or, at the other extreme, have played virtually a “full” innings as can be represented by the arithmetic average of his dismissal scores. It will readily be seen that the average of a random series of uncompleted scores will lie half-way between these two extremes, and hence projecting that half-way score to a notionally completed one (at dismissal) will imply a doubling of it.

For Test and other unlimited overs matches, Lemmer’s “most suitable” estimator of the “true” batting average is specified as:

Sum of Dismissal Scores plus (f8 times the Sum of Not Out Scores),
divided by Total Number of Innings Played:

in which f8 = 2.2 minus (0.01 times the average of Not Out Scores)
plus (0.15 times the proportion of Not Out Scores)

The proportion of Not Out Scores being expressed as 0.15 (for 15%), 0.35 (for 35%), etc.

Some readers will no doubt be curious as to how Lemmer has derived this rather fancy formula for calculating the value of his key f8 parameter – giving the multiplier to be applied to a batsman’s Not Out scores. Bear in mind that he was seeking a generally applicable estimator, and which is to have special relevance to batsmen with around 20% plus of Not Out innings. He arrives at it along the following lines (given here without any unexplained jargon):

• For reliability of the eventual formula, Lemmer takes a substantial random sample of then current Test batsman, numbering 20; all of whom have played a large number of innings – ranging from just over 100 to close to 180.
• As none of these batsmen met the required threshold of 20% Not Outs, he ingeniously simulates a sufficient number of additional uncompleted innings. Doing this by utilising each individual batsman’s ratios of Not Out scores to their projected to notional dismissal scores. He (randomly) selects a number of these ratios to “downsize” (or deflate) randomly selected dismissal scores to become equivalent Not Out scores. So providing him with enough Not Outs to work with for each batsman. These are then projected to a notional completion – dubbed an “augmented score” – in the same way that the PLE method does it.
• The deflated dismissal scores are “upsized” in this way to become completed innings scores again, although with slightly altered magnitudes from the original size as the base of dismissal scores for doing this has been somewhat reduced in number.
• Now Lemmer is nearly home and dry! The final steps being to pool (or merge) all the scores of a sub-set of the 20 batsmen – 9 being chosen on technical statistical grounds – to form the totality of scores which are representative of a generalised, composite, batsman. Finally, he tests a number of equations to find that which gives the “best fit” to the aggregate data. That is, the one that best captures the statistical relationship between the actual and simulated Not Out scores on the one hand and, on the other, their projected to completion counterparts.
• Applying the f8 formula requires two readily available (or easily determined) player specific performance statistics to be plugged in: the average value of all his Not Out scores, and the proportion of his total runs that are due to his Not Out scores.

I have applied both Lemmer’s estimator and the distilled version of the PLE to the Not Out scores of a structured sample of Test batsmen – fifty players of the modern era, as listed in Appendix I – to see how the findings differ for those attaining the 20% of Not Outs threshold. I have also done this for those batsmen who come below his threshold to see if there is generally a sizeable difference.

The sample taken is, in effect, a random one for each of the four divisions specified of the batting order as there existed no prior indication of what would be the outcome: ie the extent of uplifts of the players’ Not Out scores. Most of those at numbers 9–11 in the batting order had Not Outs of at least 20.0% plus, while none of those higher up in the order approached that threshold.

Findings on Average Uplifts to Not Out Scores (“multipliers”)

• Four openers (all of India):
  PLE 2.44, HL 1.82 – being the lower by 25.6%.
• Six batsmen at numbers 3–4 (all of South Africa):
  PLE 1.64, HL 1.49 – being the lower by 8.7%.
• Twenty at numbers 5–8 (ten of England, ten of Australia):
  PLE 1.93, HL 1.75 – being lower by 9.4%.
• Eight at numbers 9–11 (six of England, two of West Indies), all with Not Outs of at least 20.0%:
  Five with Not Outs in the early/mid 20s, the others at 33%, 43% and 47%.
  PLE 3.03, HL 2.15 – being lower by 29.0%.
• Both estimators were also applied to five other batsmen at numbers 9–11 (two of England, three of West Indies), all with Not Outs materially below 20%.
  Collectively, producing an uplift for the PLE of 2.22, and for HL 2.10 – being the lower by 5%.

Lemmer’s estimator was also applied to:

• Another seven batsmen at numbers 9–11, all with more than 20% Not Outs (four of England, three of the West Indies): two of 33% Not Outs, one of 42%, two of 43%, one of 49% and one of 59%.
  Collectively, yielding an uplift of 2.22 (ranging from 2.19 to 2.26).

Observations on the Sample’s Findings

• When the PLE and Lemmer’s Estimator (HL) are applied for a batsman with at least 20.0% of Not Out innings, the resulting uplifts to his Not Out scores are substantially greater for each of them than for batsmen below that threshold.

The multiplier for the PLE averages 3.0 and for HL averages 2.2 for 20.0% plus Not Outs, while respectively averaging 2.0 and 1.8 for batsmen with under 20.0%.

• Lemmer’s findings are consistently lower than the PLE throughout the different segments of the batting order; and when the few outlying batsmen for the PLE are excluded (for both estimators), the differential becomes no more than moderate for the openers at 11.2% and reduces to only 2.1% for numbers 5 to 8 in the order. The differential is halved for the tail-enders, yet remains substantial at 14.8%. (There are no outliers in respect of those at numbers 3 and 4.)
• As to the PLE’s occasional outlying multipliers (four in a total of forty-three results): these arise owing to a batsman’s Not Out scores being predominantly low, in absolute terms, allied to having generally relatively very high dismissal scores. Although the impact of these outliers on the players’ traditional batting averages is, in each case, no more than 0.6 of a run – equating to a change of less than 6% – the acid test of an estimator’s suitability can be judged by the relative or absolute size of the resulting multiplier when projecting a player’s Not Out scores.

A good test of an estimator’s unsuitability is the presence of a “strong” Outlier among the findings for any particular group of batsmen. Alternatively, the presence of a high multiplier value can be used as the test, for which a value of three and higher is suggested. For such cases that arise with use of the PLE method, Lemmer’s estimator should be employed instead.

• A final point is that, using the PLE method, my findings show that 20% plus of Not Out innings does not necessarily lead to a high multiplier. Thus, Edmonds’ 23% of Not Outs produces a multiplier of 2.40 and Hendrick’s 43% of Not Outs produces a multiplier of 2.65, whereas Lehmann’s meagre 5% of Not Outs results in a considerably greater multiplier of 5.01.
• With respect to batsmen in One Day International matches, it is noted that 20% plus proportions of Not Out innings are largely (though not wholly) confined to those low down in the batting order. Taking England players as an example, there are 34 who meet that threshold whilst also having 35 or more innings under their belt (from a total of 282 players up to late-October this year, these matches beginning in 1971). Of the 34 qualifying players, only 7 have usually batted above number eight in the order.

Returning briefly to Kimber’s reported findings with the PLE method, there is nothing unusual in his Not Out uplifts for the three batsmen with Not Outs in the vicinity of 20% and higher (19%, 24% and 26%). So there’s no reason for wanting to make any revisions.

Reasons for the Present Reform Impasse

Alan Kimber’s article – written with Alan Hansford, a former first-class player for Sussex County who sourced the cricket statistics – has spawned a number of proposals by those who apply what is, in effect, a distillation of his application of the PLE method mentioned earlier – ie a reduction down to its bare essentials. Yet there continues to be a lack of progress with getting any actual reform accomplished.

There are a variety of reasons for this impasse. One is that very few (only two) of the five main followers of Kimber explicitly acknowledge his fundamental contribution, and only one of them provides a brief overview of the underlying concepts of survival analysis (doing so in an article of six pages). For readers wishing to follow-up on this topic, two very approachable articles are:

• I. Etikan, S. Abubakar and R. Alkassim: The Kaplan-Meier Estimate in Survival Analysis. Biometrics & Biostatistics International Journal, February 2017 (pp 55–59).
• M.K. Goel, P. Khanna and J. Kishore: Understanding Survival Analysis: Kaplan-Meier Estimator. International Journal of Ayurveda (medical) Research, October 2010 (pp 212–16).

A related point is that Kimber’s exposition is not only highly compressed, it is well beyond the grasp of those with no more than a basic grounding in statistical methods. And so it is most likely beyond those involved in compiling and publishing batting averages, who are the targets for persuasion.

Third, those proposing improvements to establishing batting averages have, collectively, been beset with deficiencies of effective communication with those who have to be persuaded to act – to implement reforms. The following nugget appeared in the introductory to an article of 2006 by Uday Damodaran. He must be speaking to fellow statisticians, whilst plain English would have got the job just as neatly!

“The batting average of player i, Ri, is computed as:

Ri = ΣRit / (n−k)

Where:

• Rit is the number of runs scored by the i th player in the t th innings;
• n is the total number of innings in which the i th player has batted, and
• k is the number of innings in which the i th player has remained Not Out.”

It is something out of the theatre of the absurd, standing as an example that the likes of Samuel Beckett and Eugène Ionesco would have been delighted to parade. Many, or indeed most, of us have an inherent (if subconscious) desire to change the world in some way, and Damodaran talking exclusively to other qualified statisticians won’t, of course, do that.

Professional statisticians are blighted with the taunt: “There are three kinds of lies: lies, damned lies, and statistics.” This being commonly attributed to the former British Prime Minister, Benjamin Disraeli (who died in 1881), and popularised in the USA by Mark Twain in his autobiography of 1907. Moreover, mathematicians of the “pure” branch take a rather disdainful, lofty attitude towards statisticians with their generally pragmatic, and often scientifically unprovable, rules of thumb and guidelines.

Before getting into the heart of his article, Damodaran would already have turned off non-statisticians, including those who decide on how batting averages should be determined for publication – Cricinfo, HowStat, Wisden Cricketers’ Almanack and the International Cricket Council (ICC), among others. And he is far from being alone among the reform proposers in this respect.

One cannot, realistically, expect to persuade the kind of interest groups just mentioned to alter long established tradition – even to the extent of experimenting with publishing two sets of batting averages in parallel for each format of the game – if one doesn’t lay out the rationale for doing so in an easy to comprehend way for the lay person. Both Kimber’s and Kaplan & Meier’s papers are a tough read even for those with an undergraduate grounding in statistical methods. And nor are these interest groups likely to go scouting around to find a suitably pitched account of the underlying theory of survival analysis that these authors apply.

Fourth, Kimber’s followers have made choices about potential variants of his application of the PLE that, in general, have not been rationalised in a satisfactory way.