Of all the analysis work I have done in these columns, none gives me greater pleasure than the Peer Analysis work I did in 2009. In summary, I set up a subset of Tests for each batsman from his first Test to his last and compared his performance with those of his team-mates and all the players who played during this period. It was a terrific idea and was very well received by the readers. Just for a recap, after incorporating the tweaks suggested by readers, Don Bradman had a peer factor of 2.42. A few players from the '50s and '60s had around 1.8. Most of the top current players were hovering around the 1.6 mark. Malcolm Marshall led the bowlers' list with a peer factor of 1.55, followed by Glenn McGrath and Muttiah Muralitharan with 1.53.

Another bit of work I loved was related to the HSI (High Scoring Index). This enables us to quantify the support received. It enabled us to recognise the value of a lone-ranger effort such as VVS Laxman's 167 at the SCG against that of his own 281 where he had excellent support from Rahul Dravid. The comparisons could not have been more dramatic. The 167 got a HSI of 4.65 and the 281, 0.70.

As I started working on my revised ratings work, primarily for my book, I realised that there was a need for exhaustive peer analysis within a match, to enable me to get a strong handle on how the batsman performed: in comparison with his team-mates and the other batsmen who played in the Test. There would thus be an equalising of conditions and bowler skills across the match. Both peer comparisons, within team and across match, are important. This whole concept came across so well that I have used these results in my ratings work.

What is a peer comparison? In simple terms, it is how the batsman has performed in comparison with his peers. When I did my peer comparison across a player's career, there was no way I could say that the conditions were similar across this period. Different pitches, different weather conditions, different types of bowlers, different quality of bowlers, even different laws in operation, and so on. I could say only that there was a reasonable degree of similarity in the peer comparisons with the team.

However, the peer comparisons across a match are much tighter. We can say with certainty that for player comparison within the team, the conditions are almost totally similar, and for comparisons across the match, the conditions are reasonably similar. If someone says that the conditions at 10am are different to those at 4pm, I will say "Thank you" and move on. We are talking common sense here.

The index will be called IPV (Innings Peer Value). IPV-Team will denote the team-based computations and IPV-Match will indicate match-based computations. And finally there is IPV-Final, what I ended with. In all cases the IPV will refer to a single batsman innings.

IPV-Team: Based on team scores
The basic premise is simple. The batsman has scored 120 runs in one innings. If his team-mates average 20 runs per innings, he has performed six times better. If they have scored 48 runs per innings, he has performed 2.5 times better. If his score was 10, he would have performed half as well (0.5) as his team-mates in the first case, and only around one-fifth as well (0.208) in the second case.

No issues with the numerator: the runs scored by the batsmen. The denominator is tricky. Let me first explain what I have used and then provide the reasons. I have taken the runs scored by the team in both team innings, subtracted the batsman's score and divided that by the number of innings minus 1. In other words

IPV-Team for Batsman = (Team runs for both team innings - Team extras for both team innings - Batsman score) / (No. of innings in both team innings - 1).

Why RpI and not RpW? I am in favour of recognising the not-outs only for the purpose of determining the batsman average, and that is mainly because that method has been ingrained in us: 99.94 means something to most cricket followers; no point in tampering with that. However, in most other instances I would prefer RpI. If the five scores in a team innings (Test No. 450) are 260, 25, 365*, 39 and 88*, the average is 155.4 (777/5). It would be quite silly to use 259.0 (777/3). Garry Sobers scored 365 and remained not out. That innings had better be considered. Recently, in Test No. 2049, the scores were 131, 0, 311* and 182*. The average is 156 (624/4) and certainly not 312 (624/2).

Thus it can be seen that the concept of peer comparison is implemented properly by excluding the subject innings from the total runs. What about the subject batsman's other innings? No problems at all. As far as this extensive analysis is concerned, the other innings is what it is: another innings made at a different time, albeit by the same batsman. There is no conflict.

The working out of IPV values is very easy. Just to show how easy, I will take the reader through the last Test played: Test No. 2202, Brendon McCullum's farewell Test, in Christchurch against Australia.

New Zealand scored 370 and 335. This total is 705 for 22 batsmen innings. The total extras for the two team innings are 26. So the batsmen total is 679 for 22 innings. For McCullum's first-innings score, the famous 145, the batsman-specific peer RpI value is (679-145)/21 which works out to 25.42. Hence the IPV-Team value for McCullum's 145 is 5.70 (145/25.42). Anyone can work this out in two minutes. For Corey Anderson, the IPV-Team value is 72/[(679-72)/21], which works out to 2.49.

Now let us see the Australian batsmen. Australia scored 505 all out and 201/3w. This total is 706 for 16 batsmen innings. The total extras for the two team innings are 25. So the batsmen total is 681 for 16 innings. For Joe Burns' first innings of 170, remembering that the RpI is batsman-specific, the peer RpI value is (681-170)/15, which works out to 34.07. Hence the IPV-Team value for Burns' 170 is 4.99 (170/34.07). For Steven Smith, the batsman-specific peer RpI value is (681-138)/15 which works out to 36.2. Hence the IPV-Team value for Smith's 138 is 3.81 (138/36.2).

The IPV-Match values for these four innings are worked out later in the article. Let us now view the table.

Note: the Team Runs exclude extras

Jimmy Sinclair's spectacular innings at Newlands against England before the dawn of the 20th century, tops the table with an incredible IPV-Team value of 23.43. Let us digest this for a minute. In this innings, Sinclair scored 23.43 times the average runs per innings scored by the other ten batsmen in the two innings (and his own other innings). That is extraordinary. The South African second innings of 36 no doubt helped and also indicates the type of pitch in operation. Incidentally the figure of 23.43 can be arrived at from the numbers shown in the table: 23.43 = 106/[(201-106)/21]. The formula is self-explanatory.

Charles Bannerman's magnificent hundred, made 139 years ago on the first ever day of Test cricket, has weathered many a storm and comes in second place. It is still the hundred that accounts for the highest percentage of the team innings. The formula, repeated once more, is 165/[(333-165)/21]. The IPV-Team value is 20.62. These two are the only innings that have IPV-Team values exceeding 20.0.

In third place is a very recent innings. This is the match in which the middle two team innings were bizarre: 96 and 45. Michael Clarke's 151 was made out of 284 in the first innings. This innings has a IPV-Team value of 19.45. It is possible that for these innings, the IPV-Match, which uses as the base all the four team innings, might be the solution, since the pitch was neither a 96/47 pitch nor a 284/236-2 pitch.

The other famous innings in the top ten are Laxman's burning-deck classic of 167, Hanif Mohammad's 16-hour marathon of 337, Fawad Alam's 168 away in Sri Lanka, and Don Bradman's 334 at Headingley in 1930.

In positions 11-20, the famous innings are Nathan Astle's once-in-a-century masterpiece of 222 against England, Brian Lara's 213 in the famous 1999 series, Bradman's 299 against South Africa, and finally, Saeed Anwar's match-winning innings of 188 in Kolkata in 1999. The top quality of all these innings indicates that this measure is a tough one to crack.

Bradman is the leading batsman in this table, with two innings in the top 20. Though Lara has only one innings in the top 20, he has three more in the next 20 and leads with four in the top 40.

In the top 20, ten innings have been played in matches that ended in losses, five in drawn matches and only five in matches that were won. This single fact confirms that this measure is, if anything, not unduly influenced by wins and slanted towards the other results.

IPV-Match: Based on match scores
The basic premise is simple. The batsman has scored 120 runs in one innings. If the other batsmen in the match average 20 runs per innings, he has performed six times better. If they have scored at 40 runs per innings, he has performed three times better. If his score was 10, he would have performed half as well (0.5) as the other batsmen in the first case and only one fourth as well (0.25) in the second case.

No issues with the numerator: the runs scored by the batsmen. The denominator is tricky. Let me first explain what I have used and then provide the reasons. I have taken the runs scored by the teams in all four team innings, subtracted extras, subtracted the batsman's score and divided this value by the number of innings minus 1. In other words

IPV-Match for Batsman = (Team runs for all four team innings - Team extras for all four team innings - Batsman score) / (No. of innings in all four team innings - 1).

The IPV-Team has a few benefits against IPV-Match, and a few drawbacks. The comparison is listed below. The question of which measure to use is largely determined by the needs. I would be happy to receive comments from readers on this.

Let me make one thing clear to pre-empt some queries. These are tables of high IPV values, that is all. These are not ratings tables nor tables listing great innings. In these particular measures, these are the top 20 innings.

The working out of IPV-Match values is very easy. Just to show how easy, I will take the reader through the last Test played: Test #2202, McCullum's farewell Test in Christchurch against Australia.

In the four team innings the two teams scored 1411 runs for 38 batsmen innings. The total extras for the four team innings are 51. So the batsmen total is 1360 for 38 innings. For McCullum's first innings, the 145, the batsman-specific peer RpI value is (1360-145)/37 which works out to 32.84. Hence the IPV-Match value for McCullum's 145 is 4.42 (145/32.84). For Corey Anderson, the IPV-Match value is 72/[(1360-72)/37] which works out to 2.07.

Now the Australian batsmen. For Joe Burns' first innings of 170, remembering that the RpI is batsman-specific, the peer RpI value is (1360-170)/37 which works out to 32.16. Hence the IPV-Match value for Burns' 170 is 5.29 (170/32.16). For Steven Smith, batsman-specific peer RpI value is (1360-138)/37, which works out to 33.02. Hence the IPV-Match value for Smith's 138 is 4.18 (138/33.02).

If the reader goes back to the IPV-Team values, it can be seen that McCullum gets a higher IPV-Team value than IPV-Match value (5.70 against 4.42) because the other team's batsmen have averaged more (similar runs for lower number of innings). This indicates that the pitch is slightly better than was indicated by the peer team performances.

On the contrary, Burns gets a lower IPV-Team value than IPV-Match (4.99 against 5.29) because the other teams' batsmen have averaged less (similar runs for more innings). This indicates that the pitch is slightly worse than was indicated by the peer team performances. Overall I feel the IPV-Match values are more indicative of the overall pitch condition, across four or five days.

This difference will be a lot more pronounced in matches where the two teams performed totally differently. Two examples come to mind, from Trent Bridge and Sharjah. Root's 130 at Trent Bridge scores 8.35 in the IPV-Match calculation and 5.44 in the IPV-Team calculations. In Sharjah, Matthew Hayden's 119 scores 14.48 in the IPV-Match calculation and 7.12 in the IPV-Team calculations. In both cases, this difference indicates the way the two teams batted. As I say often, the pitches were neither 59/53/61 pitches nor 310/391 pitches.

Hammond tops the IPV-Match table with an IPV-Match value of 19.99 for his 336 against the hapless and inexperienced New Zealand team in 1933. This was a drawn match with only one innings being played by the two teams. New Zealand's low total, combined with Hammond's dominating score of 336 out of 548, helped Hammond get a huge IPV-Match value of 19.99. This value can be derived from the numbers displayed: 19.99 = 336/[(689-336)/21]. Even though this innings is at the top, I would say that it is a combination of numbers and circumstances that have put it there. I am wary of any Test in which only half the usual number of innings are played.

The second-placed match is slightly better in some ways. A huge score followed by two low scores and an innings win. Inzamam-ul-Haq scored 329 out of the 586 runs scored while he was at the crease. His IPV-Match value is 17.63. This is derived by the formula 329/[(926-329)/32].

The third-placed innings is Bradman's 185 made against India in 1947-48. Bradman was no doubt helped by the two sub-100 Indian scores. His IPV-Match value of 16.87 can be verified by the formula 185/[(525-185)/31].

Bannerman's 165 is in seventh place, with a IPV-Match value of 15.22. Hayden's 119 is in tenth place, with 14.48. Jonathan Trott's 184 in the Lord's Test in 2010 against Pakistan has an IPV-Match value of 13.72. The other innings of interest is Sanath Jayasuriya's 253 against Pakistan in Faisalabad, an all-time classic. This performance has made rapid strides and has moved into the top ten of the Innings Ratings table.

Hammond's innings at No. 1 and Bradman's 304 at No. 20 are the only matches ending in draws. Most matches are huge innings wins or wins by huge run margins. The exception is Bannerman's match (Test No. 1), won by Australia by only 45 runs. In the top 55 innings, Clarke's 151 is the only innings played in a losing cause, with an IPV-Match value of 11.30. Bradman is the leading batsmen in this table with 6 innings in the top-24.

IPV-Final: Based on top-20 scores

The IPV-Team is an excellent measure with true peer analysis at its core. How a player measured against his colleagues is a fascinating area of analysis. However, this measure seems to favour great batting efforts in losing matches. The number of low scores by team members act to increase the IPV-Team values of the bigger-scoring batsmen. The opposition bowling strength goes out of the situation since all the batsmen face the same bowlers. There is nothing wrong intrinsically but it is something we have to remember.

On the other hand the IPV-Match measure seems to favour powerful batting efforts in matches won by that team by a big margin. Here the low scores scored by both teams' batsmen raise the IPV-Match of the bigger-scoring batsmen. There is also an intrinsic problem in that a number of these dismissals have been effected by the team's own bowlers. In other words, the strong bowling efforts by team members improve the IPV-Match values of batsmen.

This benefit accrues to all the batsmen but the bigger scores benefit a lot. However, the real benefit is that this measure finds an in-between solution to the poor batting performances by the losing teams and the very good batting performances by winning teams. Take Test No. 2175 (Broad's match): It was neither a 313/20w pitch (11.77) nor a 391/9w pitch (33.54). Something in between (21.66). IPV-Team does provide this solution and leads to a more refined evaluation of pitch conditions.

Thus I conclude that both options leave something to be desired. So I spent days looking at alternatives. The solution did not come in a flash. I had considered it and rejected it because of some inherent problems. Then I set out to iron out these wrinkles. The result is something I find acceptable. It was very useful that I had done something similar earlier in a different context.

The idea is to take the top X scores in the match. Get the RpI for these innings and use this value as the denominator to determine the IPV-Final value. The problem is in determining the number X. I got it working well after trying out the following different options.

- First 6/7 of the team innings: Might be failures. Also the late order might score big runs. Gilchrist may bat at No. 8 because of a nightwatchman.
- Top 6/7 of the team innings: There might be higher scores in other team innings than the one selected.

So I decided to take the best 20 scores (or fewer) in a match as the base. Why 20? I did not want to exceed 20 innings for the match. Already the 20th innings was turning out to be a small one and anything beyond would be still smaller and would lower the Top-Inns-RpI. Twenty also represents just below half the number of maximum batsmen innings, which is 44. How is X determined? For 43/44 innings, it is 20 and then mapping is done using a linear scale. If only two complete innings are played, the value of X is 11. Common sense tells us that these numbers will include all the relevant innings of the match and make the Top-Inns-RpI value meaningful.

The rest of the calculations are as explained already. Let us now see the relevant table. Just to complete the cycle, McCullum's 145 in Test No. 2202 gets an IPV-Final value of 2.34 points, Burns' 170, 2.81, and Smith's 138, 2.21. The values are much lower since the comparisons are with the much higher scores.

Hammond's massacre of the innocents leads the table again. Let us do the verification. Hammond's IPV-Final can be calculated through the formula 336/[(626-336)/9]. This gives us a value of 10.43. I will also explain how the number ten was arrived at. In the three team innings, 22 batsmen batted. As per the formula I have already explained, the top 10 scores were taken.

Bill Ponsford's match-winner at the SCG in 1931 against West Indies is in second place. Ponsford scored 183 while the next best score was 58 and the third best 31. Ponsford's IPV-Final can be calculated through the formula 183/[(449-183)/13]. This gives us a value of 8.94. In the three team innings, 31 batsmen batted. As per the formula I have already explained, the top 14 scores were taken.

Next in the table is Inzamam's 329 against New Zealand. Inzamam's IPV-Final can be calculated through the formula 329/[(865-329)/14]. This gives us a value of 8.59. In the three team innings, 33 batsmen batted. As per the formula I have already explained, the top 15 scores were taken.

Bradman's 185, in fourth place, is almost identical to Ponsford's innings, with even the second best innings being 58. Bannerman's 165 is in fifth place. This is one of two matches in the top 20 in which all 44 innings were played, leading to the top 20 scores being considered.

Bradman has five innings in the top 20. Hayden is the only other batsman with multiple innings in the top 20, with two. The most recent occurrence is Hayden's 380. As in the IPV-Team table, no Indian batsman appears in the top 20; the highest positioned innings by an Indian batsman is Dravid's 270 against Pakistan, with an IPV-Final value of 6.51. This is in the 28th place.

The fourth alternative
How about taking the batsman's scores for the match together and weighing it against the match figures?

Yes, that is possible. That means adding the two innings played by the batsman and dividing this by a suitable match RpI value. However when I saw the results I was not happy. The adding of the two innings together meant that I was consolidating the batsmen's efforts, made in the context of different pitch conditions, different team objectives, and possibly bowler combinations with differing assistance and skill levels. I would be combining a first-day innings and a fourth-day innings. That did not seem to be correct to me. In any case, all innings analysis work has to be done based on the innings-centric IPV values. The correct match-level computation had to be the addition of the two standalone IPV/Ratings values rather than this method.

Now for a final comparison of the three alternatives.

IPV-Team: Possibly the clearest peer analysis within the match. The comparisons are with the other batsmen of the team. As such, the bowlers faced by these batsmen and the pitch conditions are comparable, barring abrupt changes because of weather. Excellent tool for performance evaluation within the team. However, the inclusion of many low-scoring late-order innings dilutes the value a lot. The alternative of considering, say, the top 10 innings across the match for each team, an extension of the final option, is fraught with many pitfalls because in 1400 of the 4400 team innings, the number of batsmen who batted is lower than 15. Taking ten out of these will bring in too many small scores. This option is not too influenced by the result. Hence it is fine.

IPV-Match: The conditions across the match are considered for the base. As such, it truly solves the conundrum of whether the pitch is 313/20w or 391/9w variety. This is the fairest of the three calculations. However, the inclusion of many low-scoring late-order innings dilutes the value a lot. More so since many of these could be from the other team. This makes the own bowlers' strength come into the picture. Not a great idea since Richards/Ponting/Smith should not benefit from the quality of the West Indian/Australian/South African bowlers.

IPV-Final: This is an extension of the IPV-Match measure and removes the low-scoring innings, usually played late in the innings. Other than in matches with match RpI of around ten, most of the single-digit innings are excluded. As such, this is an excellent peer comparison, based on what the other batsmen (from both teams) achieved. The bowlers do not come in. Arguably the best of the three options. This measure also encompasses the HSI value. If the higher limit of 20 is reduced to, say, 15, the comparisons become still tougher. However I am quite comfortable with the 45% selection.

Conclusion
On balance, and considering all things, I would tend to go with the IPV-Final values for my ratings work. That is primarily because as peer comparisons go, this measure is a comparison with the better scores in the match and the own bowlers' skills do not influence the measure. Of course, the other IPV measures have their own use in player performance evaluation scenarios and other related analyses.

Martin Crowe: the legend
It is indeed with a heavy heart and tears in my eyes that I write this. I started writing for ESPNcricinfo about eight years ago. Martin emailed me about one of my articles about five years back. I paraphrased his wonderful comment for the readers. We were in regular touch afterwards. He was an outstanding student of the game and one of the most erudite and knowledgeable of players.

The last I wrote to him was after the World Cup 2015 final. He replied in his inimitable manner. His words were clear. He did not expect to live to see the World Cup in 2019. I replied that all the prayers of his fans, followers and friends would allow him to see New Zealand defeat South Africa in the final at Lord's. There was a short "thank you" note. After that I did not hear from him.

I cannot forget that Marty was on the field at the MCG on the final day. He had got over his acute disappointment with the Ross Taylor captaincy controversy and was completely behind the New Zealand side. When I mentioned in my mail that I wished Brendon had taken a couple more balls to play himself in, his words were very diplomatic and astute. "That is the way Brendon plays and he was true to his self." McCullum's penultimate Test innings reflected that free spirit.

Let us not forget, with the World T20 underway, that Martin was the brain behind the first attempt at such a format. A true visionary indeed.

His loss will be truly a great one for New Zealand cricket and his absence would be felt by the cricketing fraternity around the world. A man like Marty is a once-a-hundred-years phenomenon. Marty: Teach the guys up there how to bat beautifully, on the ground and off it. May your soul rest in peace. May God be with your family. A posthumous knighting would be appropriate.

Do not miss Mike Selvey's wonderful tribute to Marty in the Guardian and Mark Nicholas' moving piece on ESPNcricinfo. I am no Selvey or Nicholas. So as a tribute to the wonderful batsman he was, I am giving below the IPV-Final values for some of his most famous innings.

- 3.92 for 299 against Sri Lanka in Wellington, 1991.
- 2.66 for 188 against West Indies in Georgetown, 1985.
- 3.30 for 188 against Australia in Brisbane, 1985.
- 3.05 for 174 against Pakistan in Wellington, 1988.
- 2.87 for 142 against England at Lord's, 1994.
- 3.10 for 137 against Australia in Christchurch, 1986.