Trivia - batting January 30, 2008

# Extreme batting - fastest and slowest innings in Tests

What are the fastest and slowest Test innings of all time
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What are the fastest and slowest Test innings of all time?

A simple question like this is actually tricky, thanks to the extreme range of possible scores. Comparing innings large and small, based on scoring speed alone, is unsatisfactory. For instance, Adam Gilchrist’s 102 off 59 balls in 2006 was considerably faster that Nathan Astle’s 222 off 168 balls in 2002; both were freakish innings, but which was the more remarkable?

One way to answer this is by measuring how far each innings deviates from normal innings of similar size. To do this, we take every innings of a given size – in terms of runs scored – calculate the average (or mean) balls faced, and then calculate the standard deviation, which is a measure of the spread or variability of the data. We can then give the most exceptional innings a z-score (the number of standard deviations from the mean) which becomes a measure of how extraordinary the innings were.

An example may help clarify this. Let’s look at all innings of exactly 76 runs in Test matches. We have balls faced data for 119 such innings. The average number of balls faced is 161 and the standard deviation of this data is about 49.

The fastest known innings of 76 in Tests was off 72 balls by Viv Richards in Adelaide in 1980. This is 1.75 standard deviations faster than the average, so the innings gets a z-score of -1.75. Likewise, the slowest innings of 76 was 315 balls by Glenn Turner in 1971, with a z-score of +3.2.

To compare many innings of different sizes, the process must be repeated for all possible scores. This process gives big innings a better rating than smaller innings of a similar speed, because it is more difficult to score rapidly for longer periods.

So which innings have the most extreme z-scores? At fast end of the scale, the results look like this:

The fastest innings
Batsman Runs Balls Match Venue & year z-score
Viv Richards 110 58 WI v Eng Antigua, 1986 -2.58
Nathan Astle 222 168 NZ v Eng Christchurch, 2002 -2.54
Adam Gilchrist 102 59 Aus v Eng Perth, 2006 -2.45
Chris Cairns 82 47 NZ v Eng Lord’s, 2004 -2.36
Jack Gregory 119 81 Aus v SA Johannesburg, 1921 -2.34
Jacques Kallis 54 25 SA v Zim Cape Town, 2005 -2.34
Kapil Dev 89 55 Ind v Eng Lord’s, 1982 -2.33

Recent innings are prominent in this list, a sign of the speed of the modern game. Still, no batsman has reached quite the extremes of Viv Richards in his record-breaking century in 1986. I wonder what it is about English bowling that has attracted so many extreme innings.

At the other end of the scale, we must go further back in time.

The slowest innings
Batsman Runs Balls Match Venue & year z-score
Hanif Mohammad 20 223 Pak v Eng Lord’s, 1954 7.90
Alec Bannerman 91 620 Aus v Eng Sydney, 1892 7.88
Herbie Collins 40 340 Aus v Eng Manchester, 1921 7.66
John Murray 3 100 Eng v Aus Sydney, 1963 7.15
Yashpal Sharma 13 159 Ind v Aus Adelaide, 1981 6.90
Geoff Allott 0 77 NZ v SA Auckland, 1999 6.80

It is interesting to see a wide range of scores, from 0 to 91, appearing on this list. Modern cricket watchers can only wonder at the extremes represented here. In terms of time, Hanif would have, going by modern-day over-rates, taken more than five hours for his 20 runs, while Alec Bannerman’s 91 would probably take more than two full days. Apart from Bannerman, every other batsman who has faced 620 or more balls in a Test innings has scored well over 200 runs, and the most balls faced (known) in reaching a century is 525 by Colin Cowdrey in 1957. Perhaps it is no wonder that Bannerman, unlike his more adventurous brother Charles, never scored a Test century.

Of course, there are quite a number of past innings for which balls faced are unknown, so we don’t know exactly where they may fit on the scale, but we can still make some estimates. Of particular interest is Dilip Sardesai’s 60 against the West Indies in Bridgetown in 1962. Sardesai was at the crease for 155 overs, and probably faced over 450 balls; if so, his z-score would be 7.93. His dismissal in that match started an extraordinary collapse that saw Lance Gibbs take eight wickets for six runs.

A postscript puzzle: innings of four runs, on average, involve fewer balls faced than innings of three runs. There is a logical reason for this (for readers to ponder).

[Notes for the statistically-minded: this process works quite well when we have data available for a very large number of innings. However, it does require some smoothing and trend-fitting at higher, rarer scores (above 120). Note also that the distributions are skewed, so z-scores of fast innings are different in magnitude to slow ones, and at the fast end of the scale the calculation is not very useful for innings of less than 40 runs. However, the process is still useful as long as we just compare fast with fast, and slow with slow.]

• Single Kontakt zum Flirten aus Berlin on February 6, 2011, 18:03 GMT

Dig the well before you are thirsty.

• Deepak on February 15, 2010, 11:57 GMT

how are slowest innings anything but the ducks?

• Andy on March 7, 2008, 8:54 GMT

Very interesting comments by Mick on Feb 1 and Madhan on Feb 11. :)

• Madhan on February 11, 2008, 14:18 GMT

One question. Would not the batsman's innings be better compared with how the other batsmen performed in the same match. An e.g., in a match of run rate 3, an innings with run rate 6 per over could be better rated than a match where everyone scored at 5.5 and one batsmen scored 6 per over. Just a thought.

• Bharat L. Sud on February 10, 2008, 7:07 GMT

Loved the use of z-stats, allows comparison across different scores thus making them comparable. Am using similar standardization approaches in my research in marketing! Maybe one thing that could add to this is to also use the approach suggested by Ritwik and use both measures, together, to come up with fastest and slowest innings. Another thing, maybe control for the average run/strike rates in that year, so in the 1950s a 50 of 70 balls might be fast but today it is average. Regarding Sarojini Kumar's point, yes cricket is simple and we might be complicating it, but this is just for fun beyond the cricket matches, not too serious stuff but allows another perspective at some great innings.

• anders on February 1, 2008, 17:11 GMT

Top work. The balls-per-innings distribution has the strong scent of a lognormal, to me (a wise man once said to me: given the choice of living in a world where everything was logged or a world where nothing could be, he'd choose Planet Logarithm every time!)

• caspian on February 1, 2008, 14:02 GMT

"Rate the innings at the milestones, ie rate both Gilly and Richards' innings at 100, and all the 300 scores would be rated at 100, 150, 200, 250 as well as the 300. Its obvious Gilchrist's and Richard's innings should be directly compared and not broken into 2 exclusive subsets. Statistics are so good to abuse."

This is a rather good idea. It would be a lot easier to judge the 300+ scores this way.

Great article, by the way. I'm curious as to what the average amount of balls faced for a duck is, as well.

• Somasekhar on February 1, 2008, 8:17 GMT

Nice article. If only it can be related to the total runs scored in the match so that a perspective of the degree of difficulty involved (read pitch and playing conditions) can be also be achieved.

• Srijoy on February 1, 2008, 7:23 GMT

Aaaaarrrgghhh!!! Numbers!!! Is this cricket or an algebra lesson? a+b-x+z-score = Boring! Let the bat do the scoring and Duck for all you're Worth!

• Sailesh on February 1, 2008, 4:50 GMT

Sarojini, it is okay to keep quiet once in a while!

@Charles: Interesting analysis. I have a suggestion to partially alleviate the problem of thin data. A score of, say, 55, and a score of 56 (or 54, for that matter) are not really different. So, when computing the mean and SD of a score of 55, you can include all innings with scores 54 or 56 (maybe even 53 and 57). This will not entirely solve the problem for higher scores, but it will atleast smooth the data for scores less than 100.

Also, instead of computing the z-score as a function of the deviation from mean, you can create probability distributions by fitting curves for each score (which would take care of the problem of asymmetric spreads) and compute a one-tailed likelihood for each innings. This will also make comparisions between different scores more meaningful, by eliminating the implicit assumption that the distributions for all scores are the same.

• Single Kontakt zum Flirten aus Berlin on February 6, 2011, 18:03 GMT

Dig the well before you are thirsty.

• Deepak on February 15, 2010, 11:57 GMT

how are slowest innings anything but the ducks?

• Andy on March 7, 2008, 8:54 GMT

Very interesting comments by Mick on Feb 1 and Madhan on Feb 11. :)

• Madhan on February 11, 2008, 14:18 GMT

One question. Would not the batsman's innings be better compared with how the other batsmen performed in the same match. An e.g., in a match of run rate 3, an innings with run rate 6 per over could be better rated than a match where everyone scored at 5.5 and one batsmen scored 6 per over. Just a thought.

• Bharat L. Sud on February 10, 2008, 7:07 GMT

Loved the use of z-stats, allows comparison across different scores thus making them comparable. Am using similar standardization approaches in my research in marketing! Maybe one thing that could add to this is to also use the approach suggested by Ritwik and use both measures, together, to come up with fastest and slowest innings. Another thing, maybe control for the average run/strike rates in that year, so in the 1950s a 50 of 70 balls might be fast but today it is average. Regarding Sarojini Kumar's point, yes cricket is simple and we might be complicating it, but this is just for fun beyond the cricket matches, not too serious stuff but allows another perspective at some great innings.

• anders on February 1, 2008, 17:11 GMT

Top work. The balls-per-innings distribution has the strong scent of a lognormal, to me (a wise man once said to me: given the choice of living in a world where everything was logged or a world where nothing could be, he'd choose Planet Logarithm every time!)

• caspian on February 1, 2008, 14:02 GMT

"Rate the innings at the milestones, ie rate both Gilly and Richards' innings at 100, and all the 300 scores would be rated at 100, 150, 200, 250 as well as the 300. Its obvious Gilchrist's and Richard's innings should be directly compared and not broken into 2 exclusive subsets. Statistics are so good to abuse."

This is a rather good idea. It would be a lot easier to judge the 300+ scores this way.

Great article, by the way. I'm curious as to what the average amount of balls faced for a duck is, as well.

• Somasekhar on February 1, 2008, 8:17 GMT

Nice article. If only it can be related to the total runs scored in the match so that a perspective of the degree of difficulty involved (read pitch and playing conditions) can be also be achieved.

• Srijoy on February 1, 2008, 7:23 GMT

Aaaaarrrgghhh!!! Numbers!!! Is this cricket or an algebra lesson? a+b-x+z-score = Boring! Let the bat do the scoring and Duck for all you're Worth!

• Sailesh on February 1, 2008, 4:50 GMT

Sarojini, it is okay to keep quiet once in a while!

@Charles: Interesting analysis. I have a suggestion to partially alleviate the problem of thin data. A score of, say, 55, and a score of 56 (or 54, for that matter) are not really different. So, when computing the mean and SD of a score of 55, you can include all innings with scores 54 or 56 (maybe even 53 and 57). This will not entirely solve the problem for higher scores, but it will atleast smooth the data for scores less than 100.

Also, instead of computing the z-score as a function of the deviation from mean, you can create probability distributions by fitting curves for each score (which would take care of the problem of asymmetric spreads) and compute a one-tailed likelihood for each innings. This will also make comparisions between different scores more meaningful, by eliminating the implicit assumption that the distributions for all scores are the same.

• rohit on February 1, 2008, 4:42 GMT

I am encouraged to see people so enthuasiatic about the 'z-value'. Just 3 words were forgotten that is gonna break the party....central limit theoram.....you cannot just apply a z-value to any set of data.....it has to be normal ( bell shaped). So the kind of data you may wanna look at is annual averages of balls for scoring '76' runs over the past 30 years ( atleast) and then apply z-value to that....I agree with some of the others...keep cricket simple....and please don't use stats wisely....

[response: please don't get too worked up about z-scores. Z-scores are simply defined as the number of standard deviations from the mean for a piece of data. While often applied to normal distributions, they don't have to be, so think of the z-scores in this article in those terms.

Charles Davis]

• John B on February 1, 2008, 4:12 GMT

Interesting so far as it goes, but like any analysis based on just the figures it has to be remembered that this doesn't take account of obvious factors like the ground played at (polo field or postage stamp), pitch conditions, match situation or standard of opposition, or the less obvious but still significant factors of the changes in rules and equipment over time.

• bob on February 1, 2008, 1:59 GMT

Astles innings was amazing, just madness. Cairns at one end batting one handed and Astle whacking it out of the ground, some of the expressions on his face were classic. Good article.

• mick on February 1, 2008, 0:24 GMT

Perhaps you could try making it a regression problem (a line of best fit), i.e. predicted runs = f(balls). Then the fastest and slowest innings would be the greatest distance from the line. It would then be able to interpolate (and extrapolate) at the higher numbers of balls faced, and give maybe more meaningful results (if there is a good fit). Any good statistics package can handle the more complex (non-linear) regression for you.

• True cricket lovers everywhere on January 31, 2008, 22:03 GMT

Philip John Joseph - I'm guessing you gleaned your information about Bradman from having played with and against him on a cricket field. Armed with such fabulous knowledge of his trickery (imagine treating each ball or bowler on his merits) I'm surprised I don't see your name in the record books with a test average of 100.94!! The only lame joke here is you in your pathetic and mean spirited attempt to demean the greatest batsman the world will ever know.

You're also forgetting that in Bradman's day, the pitches weren't covered which introduces a massive range of batting (and bowling conditions) which we don't see today. This partially explains the high standard deviations of performance at the time.

• Sudeep on January 31, 2008, 20:08 GMT

I belive generating such statistics are at best misleading. Mr. Davis himself alludes to one of the problems due to the rarity of higher scores, e.g., only 5 scores of exactly 222 vs. 54 such of 110. Reducing test cricket, or even test innings to a single z-score for purposes of comparison is pointless because it is such a nuanced game. Every innings should be judged taking into consideration the situation of the game. Case in point - Allott's 77-ball 0 is slower (since its No. 6 on the list) than a recent one not on the list - Dravid's 96-ball 12 last year. Allott's team was trying to save (unsuccessfully) a follow-on, and Dravid was leading his team to set a 2nd inn. target with a 300+ 1st inn. lead already. I am sure all would agree Dravid's innings is the more tragic one. So, please do not try to reduce cricket, specially test cricket, to strike rates and z-scores. If one is interested in best/worst lists, select the top few, and peruse the scorecards - thanks to Cricinfo.

• Arvind on January 31, 2008, 16:43 GMT

Do you weight by number of innings for a particular score, in order to arrive at the z-score? Else it could be a problem if, say, there are 100 instances of a score of 117 in test cricket and 2000 instances of a score of 27, making the comparison of z-scores a case of comparing apples with oranges.

• Philip John Joseph on January 31, 2008, 14:50 GMT

Be careful what you do with statistics. Mathematically, you could end up with the conclusion that Bradman was overrated, which of course is true. I challenge the statisticians to own up to the fact that the standard deviation in Bradman's time was much greater than it is today for a specific country when comparing specific points in the batting orders. If they can admit this fact, then they would also have to admit that Bradman lived and played in a time when the quality of play was extremely erratic, proved by the ridiculously high standard deviations of the time; all of which conclusively proves that Bradman just got lucky by hitting the bad bowlers of which there were many, and then played defense with a capital D against the few good bowlers, waiting them out pending the arrival of the aforementioned "many" bad bowlers. Statistics and standard deviation will prove that Bradman was a lame joke who benefited from the enormously erratic standards of play of his day and age.

• Vineet Meghrajani on January 31, 2008, 13:17 GMT

I really feel that Rahul Dravid has worked hard enough throughout his career to earn a position in the top in the list of worst batting strike rates. He actually deserves an award for being the most pathetic batsman in the modern era, simply because of the fact that invariably almost all the batsman including the tailenders in the team have a strike rate considerably more than Dravid's.

• zinc on January 31, 2008, 12:36 GMT

Hi Charles, Any chance of a nice plot of mean score at a number versus number of balls, that would be quite useful to know for determining the speed of innings.

Also I bet there is quite a difference in a the number of balls it takes say an opener to score on average 4 runs, and for a tailender, you could make it a 3 dimensional plot by including the batting position from which that score was made!

• Saurabh Somani on January 31, 2008, 9:41 GMT

@ charles: great analysis... innovative use of the z score... @ sarojini kumar: do u even have a point???

• Chewie on January 31, 2008, 5:51 GMT

If this was done for ODI's I wonder where Brendon McCullum's 29 ball 80 would rank...rather high I should presume.

• Demon Rob on January 31, 2008, 5:43 GMT

Rate the innings at the milestones, ie rate both Gilly and Richards' innings at 100, and all the 300 scores would be rated at 100, 150, 200, 250 as well as the 300. Its obvious Gilchrist's and Richard's innings should be directly compared and not broken into 2 exclusive subsets. Statistics are so good to abuse.

• RS on January 31, 2008, 5:19 GMT

This is a great article and while these may sound like personal requests, it would be great to know the corresponding measures for innings that were quite famous e.g. Hanif Mohammad's 337, Mudassar Nazar's first century, couple of Gavaskar centuries - against England in Bangalore in 1981-82. What would be also interesting is to see if there are scores with significant positive as well as negative z scores in the same innings. e.g. Mohinder Amarnath's 80 odd in the run chase against WI, when India scored 400+ to win. This is fascinating analysis. Would disagree that this distracts from the simplicity of the game and would suggest that it adds to the wealth of such analysis that cricket can produce. Compared to baseball (blasphemy!) it does not produce the same volume of statistical analysis. Let us have more ! What about some clustering & discriminant analysis around the WI in the 80s and the Australians under Taylor, Waugh and Ponting to see how they compare on these lines ?

• mick on January 31, 2008, 3:56 GMT

I agree with ritwik. The balls should be the constant and the runs the variable. Otherwise, good article - now maybe try it for ODIs.

On a more statistical note, the Z-score is an estimate of P(runs|balls), but only for a normal distribution. I wonder if results would be different for a GMM (Gaussian Mixture Model). All it requires is an estimation of P(runs,balls), then P(runs|balls) = P(runs,balls) / P(balls), and hopefully the least probable innings will emerge.

• poor old bowler on January 31, 2008, 2:01 GMT

the whole point of batting is to score runs. with gilchrist,astle,richards and cairns the captain would be baffled setting a field.

with these slow scorers just bring the field up have 4-5 slips a bat pad and short cover and really put the pressure on.in 500-600 balls at the wicket surely the would mistime one defensive stroke.most captains would have a field day with players like these.

look at dravid and kaif compared sewag and pathan opening the batting for india,because dravid and kaif refuse to play attacking cricket they could be attacked.i would rather see someone getting out trying to score than some one getting out playing a defensive stroke.

plus aggresive attacking cricket is good for the viewer at home.

thats why adam gilchrist was the best batsmen of his generation,he scored 17 test centries at a fast rate and australia declared on him another 10 times when he was bound to score a century,that would have close to 25 centurys in the short time he played cricket

• PaulEvans on January 31, 2008, 1:49 GMT

Another superb article - thanks guys.

It would be brilliant if you could concieve of a means of comparing scores above 300. I guess the number is so small that it is possible to set them next to one another. I dare say the variance in the z-scores is marginal at these heady heights.

• John on January 31, 2008, 0:13 GMT

I think it is better if you smooth the curve by averaging out scores in ranges. Like 0-5, 5-10 instead of taking each individual score. The final table could possibly look the same but you can get over spikes, if they exist.

• Karthick on January 30, 2008, 23:35 GMT

The z-score is a very good idea. The slowest scores tend to be in the pre-1990s era before the exploits of one-day cricket. What would be really interesting is to see the slowest knocks since 1990s. Those would be the real slowest knocks since everybody else was going hammers. Another interesting stat would be the batsmen who stone-walled in the decisive test match of a series (Sachin-Dravid partnership at ape Town 2007 is an example). That list will give the guys who preferred the snail approach even when the series was up for grabs.

• Chris on January 30, 2008, 23:07 GMT

This was a fascinating article. Keep up the good work!

• Jdog on January 30, 2008, 22:25 GMT

Crikey, there's some real statistics! However, I think Geoff Allot really deserves better than sixth place for his duck off of 77 balls. I think that in practice this would be harder to achieve than Hanif Mohammad's 20 from a couple of hundred balls.

• Michael on January 30, 2008, 22:01 GMT

The batsmen who generally score four are the hot heads who slash wildly at the ball, run for everything and risk all on the chancy shots. Theirs is a "death or glory" innings.. The batsmen who go for 3 are the stoics. Diligently crafting an innings run by run. Turning the ball to all points of the field but never risking the fieldsman's arm. Masters of the the defensive shots they take body blows and smother the ball rather than throw their wicket away playing speculative shots at the bouncer or turning ball. They know they are the backbone of their team's totals so they resolve to build a spine of solid willow and pure gumption. Always ready to let others take the 4-run glory - glory built on the shoulders of his Gibraltar-like 3-run foundation. The unsung innings that unlocks victory. Stout yeomen, those.

• Venkattraman on January 30, 2008, 20:23 GMT

I dont really understand your notion of comparing different sizes. I feel that you calculate z-score of a particular score based on history of batters scoring that particular score. But after that you talk of repeating the process. I dont really understand tat part

• Qaiser Saify on January 30, 2008, 20:10 GMT

This seems like a very useful exercise when you want to find out the greatest innings played by batsmen. Up until now, people only had a qualitative idea of the greatest innings, and often enough commentators mentioned Adam Gilchrists and Viv Richards innings. Now we have a rigorous method for doing this, and the results are similar. Excellent! Now, how about Inzamam's 329, Jayasuriya's 340, or Lara's 400. How would you handle those? Wouldn't 400 be the best ever, but can't ever be compared? Maybe by comparing with first class scores, only slightly, though. I do think the scores of 100s and 200s will find the best expression using this method, but it is difficult for scores above 300. As for the 4 runs taking fewer balls than 3 runs, it is simply because those four runs were scores off a single boundary, which requires only a single ball as a minimum, while the 3's were painstakingly grinded out with singles or so before the batsman was dismissed. Over and out!

• Nirmal on January 30, 2008, 20:04 GMT

Interesting analysis. You could do the same with career averages and strike rates to establish players who are good in ODIs for instance too! This might for example, be useful to compare players like Shahid Afridi and Adam Gilchrist, both of whom have high strike rates, but vastly different averages

• Samir Chopra on January 30, 2008, 19:26 GMT

"A postscript puzzle: innings of four runs, on average, involve fewer balls faced than innings of three runs. There is a logical reason for this (for readers to ponder)."

Because innings of four runs might only need one scoring stroke, while an innings of three runs might need as many as three.

• Jeff on January 30, 2008, 18:46 GMT

re the postcript puzzle - there are more permutations leading to 4 runs in the same number of balls than for 3, due to the fact that one can score a boundary. i.e. there are only 2 ways to each 3 runs off 2 balls: 1+2 and 3+0, but there are 3 ways to reach 4 runs off 2 balls, 4+0, 3+1, and 2+2

• Christopher Lucky on January 30, 2008, 18:23 GMT

It should be obvious that innings of 4runs can contain a boundary while innings of 3runs cannot and therefore the reason for fewer balls

• Aanand on January 30, 2008, 17:47 GMT

To score 4 runs, a batsman would take a single shot to the boundary. This could be the logical reason for 4 runs in fewer balls than 3 runs.

• Kaushik on January 30, 2008, 17:32 GMT

Hi,

Great analysis. Loved the use of the z-stat. Innings of four probably have a high ratio of a single scoring stroke (a boundary) compared with innings of three runs and hence on average are likely to involve fewer balls faced, right?

• Kris Nayak on January 30, 2008, 16:01 GMT

Think innings of four generally come from tailenders, who manage to get a ball to boundary (edges, mostly!) and get out soon after. inns of three, normally, would need something like 3 singles or one 1 and a 2. hard to get, generally :o)

• prasad on January 30, 2008, 15:34 GMT

excellent

• Sarojini D Kumar on January 30, 2008, 15:34 GMT

Mr. Davis' article is data mining and it is complicating a game known for its simplicity. Statistics has it splace but standard deviations and Z-scores? What's next, trinomials and quadratic equations? While you are at it can you figure out the number of hundreds scored when the temperature was between 77.39 and 99.94 degrees (F). Lets leave the complicated mathematics alone and enjoy a beautiful sport for waht it is: its simplicity.

• Gibran Lalani on January 30, 2008, 15:34 GMT

Innings of 4 generally involve a boundary or 2 twos in them while an innings of 3 most likely involves 3 singles or perhaps a 2 and a single. 1 ball or 2 balls scored of vs. 3 balls or 2 balls.

• rsingh on January 30, 2008, 15:13 GMT

Two Indo-Guyanese cum West Indians come to mind: Chandrapaul's 61-ball century against Australia at Bourda and Leonard Baichan's painful century (his one and only in his first test) somewhere in Pakistan in the '70s..I wonder what their relative z-scores were?

• Ritwik on January 30, 2008, 15:12 GMT

a similar effort with the number of balls faced a constant and the the number of runs scored for all known innings involving those many balls as the variable can be performed. It would be interesting to see if the results of that match with these results.

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• Ritwik on January 30, 2008, 15:12 GMT

a similar effort with the number of balls faced a constant and the the number of runs scored for all known innings involving those many balls as the variable can be performed. It would be interesting to see if the results of that match with these results.

• rsingh on January 30, 2008, 15:13 GMT

Two Indo-Guyanese cum West Indians come to mind: Chandrapaul's 61-ball century against Australia at Bourda and Leonard Baichan's painful century (his one and only in his first test) somewhere in Pakistan in the '70s..I wonder what their relative z-scores were?

• Gibran Lalani on January 30, 2008, 15:34 GMT

Innings of 4 generally involve a boundary or 2 twos in them while an innings of 3 most likely involves 3 singles or perhaps a 2 and a single. 1 ball or 2 balls scored of vs. 3 balls or 2 balls.

• Sarojini D Kumar on January 30, 2008, 15:34 GMT

Mr. Davis' article is data mining and it is complicating a game known for its simplicity. Statistics has it splace but standard deviations and Z-scores? What's next, trinomials and quadratic equations? While you are at it can you figure out the number of hundreds scored when the temperature was between 77.39 and 99.94 degrees (F). Lets leave the complicated mathematics alone and enjoy a beautiful sport for waht it is: its simplicity.

• prasad on January 30, 2008, 15:34 GMT

excellent

• Kris Nayak on January 30, 2008, 16:01 GMT

Think innings of four generally come from tailenders, who manage to get a ball to boundary (edges, mostly!) and get out soon after. inns of three, normally, would need something like 3 singles or one 1 and a 2. hard to get, generally :o)

• Kaushik on January 30, 2008, 17:32 GMT

Hi,

Great analysis. Loved the use of the z-stat. Innings of four probably have a high ratio of a single scoring stroke (a boundary) compared with innings of three runs and hence on average are likely to involve fewer balls faced, right?

• Aanand on January 30, 2008, 17:47 GMT

To score 4 runs, a batsman would take a single shot to the boundary. This could be the logical reason for 4 runs in fewer balls than 3 runs.

• Christopher Lucky on January 30, 2008, 18:23 GMT

It should be obvious that innings of 4runs can contain a boundary while innings of 3runs cannot and therefore the reason for fewer balls

• Jeff on January 30, 2008, 18:46 GMT

re the postcript puzzle - there are more permutations leading to 4 runs in the same number of balls than for 3, due to the fact that one can score a boundary. i.e. there are only 2 ways to each 3 runs off 2 balls: 1+2 and 3+0, but there are 3 ways to reach 4 runs off 2 balls, 4+0, 3+1, and 2+2