Nathan Astle's 168-ball 222 against England ranks second in the list of fastest innings © Cricinfo Ltd

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:

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.

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.]

	Jacques Kallis averages 71.84 at No. 4 © Getty Images

The opening-position conundrum

A couple of readers have suggested that the two openers be allotted a number other than 1.00. The two suggestions offered are 1.5 or 2.0 for both openers. Both suggestions have their merits. 1.5 is more correct since the total for the two batsmen comes to 3.0 which is the sum of 1 and 2. However it does not look good as 1.0 or 2.0 would do. Allotting 2.0 to both batsmen is probably the better solution since it allows one to maintain continuity in numbers from 2.0 to 11.0. The other major benefit is that when an opener bats at 3.0, the variance will be a more correct 1.0 than the somewhat bloated 2.0 as is currently the case. Hence I have decided to allot both openers 2.0 and re-do the tables.

There will be no changes for the batsmen who have never opened. There will be no changes (other than a mean value of 2.0 as against the current 1.0) for the omni-present openers. For batsmen such as Boycott and Gavaskar there will be very little change. The change is significant only for those batsmen like Alec Stewart who have moved up and down the order quite frequently. Some of the key batsmen are compared below.

These are current up to the recently concluded second Test between New Zealand and Bangladesh.

Batsman            L Cty  Tests Inns BPTot  BPIdx MeanDev  Freq Batpos (%)
Revised
Stewart A.J          Eng   133  235    919   3.91   1.56    77 @  1( 32.8)
Previous
Stewart A.J          Eng   133  235    842   3.58   1.97    77 @  1( 32.8)
Revised
Jayasuriya S.T     ~ Slk   110  188    519   2.76   1.44   152 @  1( 80.9)
Previous
Jayasuriya S.T     ~ Slk   110  188    367   1.95   1.77   152 @  1( 80.9)
Revised
Langer J.L         ~ Aus   105  182    440   2.42   0.55   115 @  1( 63.2)
Previous
Langer J.L         ~ Aus   105  182    325   1.79   0.90   115 @  1 (63.2) 

Just for the record, the top 25 batsmen in the revised table are listed in order of innings played. As per Steve Procter's suggestion, the Standard Deviation has been calculated and shown.

Batsman         L Cty  Tests Inns  BPIdx  M Dev   Freq Batpos (%) StdDev


Border A.R      ~ Aus   156  265    4.70   0.98    89 @  4( 33.6)   1.14

Waugh S.R         Aus   168  260    5.42   0.74   142 @  5( 54.6)   0.95

Stewart A.J       Eng   133  235    3.91   1.56    77 @  1( 32.8)   1.79

Tendulkar S.R     Ind   144  233    4.29   0.60   189 @  4( 81.1)   0.71

Lara B.C        ~ Win   131  232    3.78   0.51   148 @  4( 63.8)   0.65

Gooch G.A         Eng   118  215    2.31   0.57   184 @  1( 85.6)   0.80

Gavaskar S.M      Ind   125  214    2.21   0.44   203 @  1( 94.9)   1.42

Atherton M.A      Eng   115  212    2.10   0.25   197 @  1( 92.9)   0.49

Waugh M.E         Aus   128  209    4.24   0.56   170 @  4( 81.3)   0.67

Gower D.I       ~ Eng   117  204    4.00   0.71    91 @  4( 44.6)   0.87

Haynes D.L        Win   116  202    2.03   0.03   201 @  1( 99.5)   0.42

Dravid R          Ind   117  201    3.29   0.78   146 @  3( 72.6)   1.00

Inzamam-ul-Haq    Pak   120  200    4.66   0.91    98 @  4( 49.0)   1.15

Warne S.K         Aus   145  199    8.29   0.82   113 @  8( 56.8)   1.18

Kallis J.H        Saf   114  194    3.77   0.61    96 @  4( 49.5)   0.79

Boycott G         Eng   108  193    2.02   0.13   191 @  1( 99.0)   0.80

Boon D.C          Aus   107  190    2.85   0.61   111 @  3( 58.4)   0.84

Ponting R.T       Aus   114  190    4.02   1.34   125 @  3( 65.8)   1.46

Javed Miandad     Pak   124  189    4.24   0.57   140 @  4( 74.1)   0.71

Jayasuriya S.T  ~ Slk   110  188    2.76   1.44   152 @  1( 80.9)   1.72

Cowdrey M.C       Eng   114  188    3.84   1.09    54 @  5( 28.7)   1.28

Taylor M.A      ~ Aus   104  186    2.00   0.00   186 @  1(100.0)   0.00

Walsh C.A         Win   132  185   10.62   0.65   122 @ 11( 65.9)   0.63

Greenidge C.G     Win   108  185    2.03   0.13   182 @  1( 98.4)   0.72

Vengsarkar D.B    Ind   116  185    3.67   0.81    74 @  3( 40.0)   1.03

Highest averages at different batting positions

This is another interesting request. It would be of considerable interest to see the highest averages at different batting positions, with a qualification of at least 1000 runs at that position. Some surprises are in store for us.

Opening: Sutcliffe H        Eng  4522      61.11

No.3:    Bradman D.G        Aus  5078     103.63

No.4:    Kallis J.H         Saf  5675      71.84

No.5:    Worrell F.M.M      Win  1189      59.45

No.6:    Chanderpaul S    ~ Win  2087      63.24

No.7:    McMillan B.M       Saf  1051      58.39

No.8:    Vettori          ~ Nzl  1136      42.07

No batsman has scored 1000 runs at Nos.9, 10 and 11. Just for the record, Shaun Pollock has scored 534 runs @ 41.08 at No. 9 (min 500 runs). W. Oldfield has scored 263 runs at 26.30 at No.10 (min 250 runs). Brian Statham has scored 348 runs @ 13.92 at the No.11 position (min 250 runs).

Just to complete the analysis and in anticipation of reader demands, I have given below the best batting position, runs and average for a few key batsmen, with the proviso that a minimum of 1000 runs should have been scored in that position.

Tendulkar       4  9573  57.32 
Dravid          3  7444  57.26
Laxman          6  2130  48.41
Ponting         3  7062  66.00
S.R.Waugh       5  6754  56.28
Lara            3  3749  60.47
Richards        3  3508  61.54
Sangakkara      3  5557  61.07
Inzamam-ul-haq  4  4867  52.90
...
Warne           8  2005  19.10
Vaas            8  1703  24.33
Kumble          7  1087  20.13
and
CS Martin      11    64   2.46

Chris Martin had an average of 2.00 (that is the BPI of an opener) until his last innings against Bangladesh when he reached his best ever score of 12 not out, increasing the average to 2.46. In the current Test he remained unbeaten on 0, thus remaining at the majestic figure of 2.46. The million-dollar question is if he will ever get 100 Test runs and reach a double-digit score again.