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July 6, 2008

Trivia - batting

Extrapolating high scores in Tests

Charles Davis
Sanath Jayasuriya on his way to a run-a-ball 154, Cricket Australia Chairman's XI v Sri Lankans, Adelaide, 1st day, October 27, 2007
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When comparing the biggest team scores in Tests, the results can be a bit messy. This is because cricket often does not allow teams to carry their innings to completion, and big innings are often truncated by declaration or lack of time. We know for sure that the highest innings in a Test match is Sri Lanka’s 952 for 6 in 1997, but an interesting side question would ask if this is also the most ‘extraordinary’ score in Tests. For example, we know that the West Indies once made a score of 790 for 3. Where might such an innings have gone if it had continued? Can we compare it to Sri Lanka’s record?

While we can never know for sure, it is possible to make a statistical estimate. The approach is to look at the way that innings naturally progress over a wide range of scores. Of course, there is plenty of variation between innings [part of cricket’s appeal], but there are statistical patterns. A team that is, say, five wickets down, will on average add a certain number of runs if the innings is played to completion.

This average number of runs added also depends on the starting point. A team on, say, 50 for 5, can be expected to add fewer runs than a team on 500 for 5 before being bowled out. But there is a surprising result to be found here. Contrary to expectation, the number of runs at the starting point is not very important, with only a limited effect on the future progress of the innings. This is shown in the following table, calculated from the outcomes of all relevant Test innings, which gives the average number of runs added by teams with five wickets down, at different starting points.

Average runs added when five down
Starting score Runs added (average) Projected all-out score
50 for 5 85 135
100 for 5 91 191
200 for 5 99 299
300 for 5 114 414
400 for 5 116 516
500 for 5 114 614
600 for 5 110 710

What we see here is that above a certain level, in this case about 300 runs, there is very little change in the potential scoring of a team. This is surprising, but it probably comes down to the fact that a batsman coming in at a score of 600 for 5 is likely to bat in a riskier manner, or with less intensity, than one who comes in at 300 for 5. This would appear to balance out any advantage from tired bowling or benign conditions. This pattern is also seen at 6, 7, 8 or 9 wickets down.

It should be stressed that these runs added will often be theoretical in practice. For example, the projected all-out score for teams that reach 600 for 5 is 710, but in practice most such innings will not reach 700, often because of declarations. What the projected all-out score gives us is an estimate of where the innings was headed if the limits of time and tactics had been removed – its trajectory if you will.

With modern computer power, the result of this process is an “Innings Projector” that can give a projected estimate for any score. (In practice, it only works for innings with two or more wickets down.) Estimates for extreme innings must remain provisional because of the rarity of the situations, but the fact that trends are so stable, as illustrated by the first table, adds confidence to the results.

So what are the most extreme projected scores? Here is a list of the results:

Most extreme projected scores
Team Opponent Venue, year Score Projected score
Sri Lanka India Colombo (RPS) 1997 952-6 1028
West Indies Pakistan Kingston, Jamaica 1958 790-3 996
England Australia The Oval 1938 903-7 951
Sri Lanka Zimbabwe Bulawayo 2004 713-3 919
Sri Lanka South Africa Colombo (SSC) 2006 756-5 866
West Indies England St John’s, Antigua 2004 751-4 861
England West Indies Kingston, Jamaica 1930 849-10 849
New Zealand Sri Lanka Wellington 1991 671-4 821
India Bangladesh Dhaka (Mirpur) 2007 610-3 816
Australia Zimbabwe Perth (WACA) 2003 735-6 810
Pakistan India Lahore 1989 699-5 809
South Africa Zimbabwe Harare 2001 600-3 806
Australia England Lord’s 1930 729-6 804
England India Lord’s 1990 653-4 803
Australia England Leeds (Headingley) 1993 653-4 803
Australia England The Oval 2001 641-4 791
Australia West Indies Kingston, Jamaica 1955 758-8 788
Pakistan India Hyderabad (Pak) 1983 581-3 787
India Pakistan Multan 2004 675-5 785
Australia England Lord’s 1993 632-4 782
England South Africa Lord’s 1924 531-2 779
West Indies New Zealand Wellington 1995 660-5 770
England South Africa Durban 1939 654-5 764
Pakistan Sri Lanka Faisalabad 1985 555-3 761
South Africa England Lord’s 2003 682-6 757
Pakistan Bangladesh Multan 2001 546-3 752
India Australia Sydney 2004 705-7 752
India Australia Sydney 1986 600-4 750

So Sri Lanka retains the No. 1 position under this calculation. However, the West Indies 790 for 3 moves up to second place, while England’s 849 all out in the Timeless Test of 1930 moves down to seventh.

Another aspect to these scores is that the distribution of the scores around these projections can be calculated, which means that the probability of a specific score can also be calculated. For example, the probability of a score of 790 for 3 actually exceeding the 1028 assigned to Sri Lanka’s record is about 24%.

One other possible calculation here is a re-appraisal of the most one-sided innings victories in Tests. Using the projected score, the margin of victory can be re-calculated and compared more evenly. The most one-sided Tests in this analysis are:

Most one-sided Tests
Venue, year Team Opponent Score Projected score Original margin Projected margin
The Oval 1938 England Australia 903-7 951 Inng and 579 Inng and 627
Multan 2001 Pakistan Bangladesh 546-3 752 Inng and 264 Inng and 470
Bulawayo 2004 Sri Lanka Zimbabwe 713-3 919 Inng and 254 Inng and 460
Kolkata 1958 West Indies India 614-5 724 Inng and 336 Inng and 446
Dhaka (Mirpur) 2007 India Bangladesh 610-3 816 Inng and 239 Inng and 445
Wellington 1995 West Indies New Zealand 660-5 770 Inng and 322 Inng and 432
Johannesburg (New Wanderers) 2002 Australia South Africa 652-7 699 Inng and 360 Inng and 407

(Please, no comments that the ‘highest’ does not mean the ‘greatest’. No one is claiming that it does. We are just looking at extremes here.)

[Technical note: the trajectory at large scores must be calculated with care, because teams that continue with great success from a high starting point rarely complete their innings. This must be allowed for in the calculation. The way to do this is through an iterative process, where big innings that are declared closed are themselves calculated through to completion, firstly for innings that are nine wickets down, then eight, seven, and so forth, and these results are then fed back into the calculation for end points starting from fewer wickets down.

For example, take a score of 500 for 3. This has occurred 37 times in Test matches. The projected score in this case is 705 all out. However, only three of the 37 teams have actually reached or exceeded a score of 705, while nine have been bowled out for less than 700. The reason that the projected score is above 700 is that many teams continue to do well but declare before reaching 700. Careful iterative analysis of these declared scores produces the average estimate of 205 runs added, or 705 all out for a projected score.]

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Posted by Ron on (October 19, 2008, 10:17 GMT)

Interesting bit of analysis. However, it is not enough to look at the expected value of the extrapolated score. What is the variance? My guess is that it'd be fairly high.

Posted by Alex on (August 10, 2008, 11:49 GMT)

PS to my earlier comment: earlier commenters were even more confused than I thought (even managed to confuse me too). Jayasuriya's innings in the big total against India was only 340, nowhere near a record. The 374 by Jayawardene was after Haydn's 380 AND after Lara's 400*.

Posted by Alex on (August 10, 2008, 11:39 GMT)

Re Jayawardene "eclipsing" Lara. According to Cricinfo's best batting performances, he only made 374, so he certainly did not eclipse Lara. That feat was achieved by Matthew Hayden with his 380 against Zimbabwe. Of course, as you mentioned in your response to Slog Unlimited, Lara later regained the record with his 400 not out.

Posted by santhosh kudva on (August 2, 2008, 16:56 GMT)

... i think one more factor needs to be considered in this analysis. the team composition. a team like australia is likely to amass more runs for the last five wickets than the west indies team would during its entire innings...or better still give a teamwise breakup...

Posted by Ananth on (July 12, 2008, 0:50 GMT)

Charles, an excellent analysis. I know a few people have already commented on this. I personally think the most dominating innings of all time is West Indies 790 for 3 because Sobers and Walcott were still batting and they still had two batsmen yet to come. Non-statistically, I am convinced that that would have been the only innings capable of reaching 1000. I had done a similar study on ODI innings and there the Australian total of 434 for 4 (a losing score) stood at the pinnacle.

Posted by Chris on (July 11, 2008, 12:04 GMT)

To Azad Ahmad.. Do you think SL will score more than 952.. were Aus able to score thatmuch against the same bowling - NO!. And i find this record quite selfish, if it were the Australians... they would have forced the second innings by declaring at 700 odd.

Posted by Abhay Goyal on (July 11, 2008, 8:15 GMT)

I believe you may be right till some extent but I beleive there is a certain patience of a batsman playing in the middle. In these days,its very difficult for batsmen to play a long test like knock and stay in the middle for long hours. And see if each batsman scores over a 100 in the team then the score reaches somewhere near 1000 mark. Thats a very difficult mark and also one needs to be very bold not to go for declaring. Remember, Captain Dravid once declared Indian innings at a stage when tendulkar was at the crease on 194*. Wow, 952 was a very good score, Hats off to Srilanka. 790 runs scored by westindies was still far from the 1000 mark and it was clas of lara that more than 50% of the runs were his.

Posted by SRAM on (July 10, 2008, 21:47 GMT)

Were the scores considered at the time of declaration or at the time of the last wicket to fall? For instance a 600/5 decl might have been a 300/4 when the 4th wicket fell. The fall of the 5th wicket might be just round the corner and one could consider the score to be 600/6.

[Response: It cannot be predicted statistically whether a wicket is 'just around the corner'. There is no particular level of a partnership, once it is past 20 runs or so, where a fall of wicket suddenly becomes more or less likely.]

Posted by Amer - Canada on (July 10, 2008, 18:31 GMT)

Charles, Interesting. I remember in 86/76 Pakistan posting a huge total vs England at the Oval. I think it was around 600+. I was expecting to see that innings here.

[response: Pakistan were all out for 708 at the Oval in 1987.]

Posted by Slog Unlimited on (July 10, 2008, 16:49 GMT)

Seriously, I just read a comment in which someone said that in that 952/6 by SL, Jayasuriya went all over the park to eclipse Lara! Man, that's impossible! Lara's 400* knock was after Jayasuriya's, which means Lara actually eclipsed Jayasuriya! Learn simple cricket history, then talk Statistics!

[Response: Lara did hold the world record at the time, with his 375.]

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