In addition, the mean variance for all these positions has been worked out. This is the average of the absolute variances from the BPI. Let me explain this. Let us say that a batsman batted at No. 4 in 10 innings. His BPI is 4.0. Let there be another batsman who batted 4 times at No 3, twice at No. 4 and four times at No. 5. His BPI is also 4.0. However the Mean Variance for the first batsman is 0.0, indicating no departure from the mean position while the Mean Variance for the second batsman is 0.8, indicating quite a bit of shuffling around of the batting position. I have adopted the mean variance rather than the standard variance since the distribution is not Standard or Gaussian. The distribution of batting positions does not have a pattern. A player tends to bat around his favourite (and/or) productive positions. There may be a case for using standard deviation. Since I do not have a master's degree in statistics, I have resorted to using my common sense which tells me that I should use the mean deviation rather than standard deviation. The other value which has been determined is the most often batted position for each batsman and the percentage of total for this position. There is a strong correlation between this % value and the Mean Variance. As this percentage value approaches 100, the Mean Variance approaches 0.0. A few notes about the Batting Position Index. A BPI of 1.0 indicates that the batsman has always played in the opening position, while a BPI of 11.0 indicates that the batsman has similarly always batted last for his team. These are the extremes, and fairly clear. On the other hand a BPA of 5.0 does not indicate that the batsman always batted at No.5, since he could have batted 25 times at No. 4 and 25 times at No. 6. However a BPI of 4.15 generally indicates that he might have batted more at No. 4 while a BPA of 4.90 indicates that he would have batted more often at No. 5. Keep this in mind while you look at the numbers. A number of interesting facts came to light and are presented below. Let us see some tables. '~' indicates a left-handed batsman. These are current upto Test no 1854, the third Test between Sri Lanka and England played at Galle. A minimum of 50 innings has been taken as the minimum qualification.

A minimum of 50 innings has been taken as the minimum qualification.

Batsman	L	Cty	Tests	Inns	BPTot	BPIdx	MeanDev	Freq	Batpos (%)
Border A.R	~	Aus	156	265	1245	4.7	0.98	89 @	4( 33.6)
Waugh S.R		Aus	168	260	1410	5.42	0.74	142 @	5( 54.6)
Stewart A.J		Eng	133	235	842	3.58	1.97	77 @	1( 32.8)
Lara B.C	~	Win	131	232	876	3.78	0.52	148 @	4( 63.8)
Tendulkar S.R		Ind	142	229	983	4.29	0.61	185 @	4( 80.8)
Gooch G.A		Eng	118	215	313	1.46	0.83	184 @	1( 85.6)
Gavaskar S.M		Ind	125	214	269	1.26	0.53	203 @	1( 94.9)
Atherton M.A		Eng	115	212	249	1.17	0.43	197 @	1( 92.9)
Waugh M.E		Aus	128	209	886	4.24	0.57	170 @	4( 81.3)
Gower D.I	~	Eng	117	204	812	3.98	0.72	91 @	4( 44.6)
Haynes D.L		Win	116	202	209	1.03	0.03	201 @	1( 99.5)
Inzamam-ul-Haq		Pak	120	200	930	4.65	0.91	98 @	4( 49.0)
Warne S.K		Aus	145	199	1649	8.29	0.82	113 @	8( 56.8)
Dravid R		Ind	115	197	640	3.25	0.78	146 @	3( 74.1)
Boycott G		Eng	108	193	199	1.03	0.16	191 @	1( 99.0)
Boon D.C		Aus	107	190	478	2.52	1.05	111 @	3( 58.4)
Kallis J.H		Saf	111	189	712	3.77	0.62	91 @	4( 48.1)
Javed Miandad		Pak	124	189	798	4.22	0.57	140 @	4( 74.1)
Cowdrey M.C		Eng	114	188	684	3.64	1.33	54 @	5( 28.7)
Jayasuriya S.T	~	Slk	110	188	367	1.95	1.77	152 @	1( 80.9)
Taylor M.A	~	Aus	104	186	186	1	0	186 @	1(100.0)
Ponting R.T		Aus	112	186	751	4.04	1.34	121 @	3( 65.1)
Walsh C.A		Win	132	185	1965	10.62	0.65	122 @	11( 65.9)
Richards I.V.A		Win	121	182	758	4.16	1.07	63 @	5( 34.6)
Sobers G.St.A	~	Win	93	160	807	5.04	1.2	57 @	6( 35.6)
Bradman D.G		Aus	52	80	292	3.65	1.24	56 @	3( 70.0)
Jones A.H		Nzl	39	74	223	3.01	0.11	70 @	3( 94.6)

The top five batsmen have batted in middle order positions. Allan Border has batted at No. 4 for most of his career while Steve Waugh has batted a position lower. Alec Stewart and Brian Lara have batted higher up the order with Stewart getting his numbers lower because of his frequent opening stints. Lara's most batted position is No. 4. Sachin Tendulkar has batted slightly down the order, while Dravid has batted at No. 3 most of the times in his career. Now we get to the batsmen specialising in opening positions. Graham Gooch has batted lower down more often than Sunil Gavaskar or Michael Atherton. Desmond Haynes' index is interesting. He has batted only once other than the opening slot, and that was in the last innings he ever played in his career. He batted at No. 8 and scored 15 against England in 1994. Why he batted at that position is quite a mystery. Boycott batted at lower positions twice in a single Test. Most of the top batsmen in this group have mean variance values below 1, indicating a reasonably settled existence in the opening or middle order slots. The only exception is, of course, Alec Stewart, with a high value of 1.97. Note also the somewhat high variance value for Jayasuriya, indicating long tenure at positions 1 and 2, and a few innings way down the order. Richards, surprisingly, has batted at No. 5 most. This explains his slightly higher BPI. Walsh averages 10.62, indicating a near-permanent residency at No. 11. Andrew Jones makes an interesting examples. He has the highest occupancy % of a single batting position among the middle-order batsmen, 94.6% at No. 3, only four times batting away from this position. He is the only batsman with a 90+% occupancy of a batting position among the non-opening, and non-No. 11 batsmen.

Now a table of Opening batsmen who have never vacated their assigned slots.

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Taylor M.A	~	Aus	104	186	1.00	0.00	186 @	1( 100.0)
Hayden M.L	~	Aus	91	162	1.00	0.00	162 @	1( 100.0)
Slater M.J		Aus	74	131	1.00	0.00	131 @	1( 100.0)
Lawry W.M	~	Aus	67	123	1.00	0.00	123 @	1( 100.0)
Strauss A.J	~	Eng	43	81	1.00	0.00	81 @	1( 100.0)
Hunte C.C	~	Win	44	78	1.00	0.00	78 @	1( 100.0)
Srikkanth K		Ind	43	72	1.00	0.00	72 @	1( 100.0)
Imran Farhat	~	Pak	27	51	1.00	0.00	51 @	1( 100.0)

These are die-hard opening batsmen who have frowned at doing anything else. The fact that this list is headed by four Australians - and three of them from recent times - seems to indicate the importance they place on the opening position in particular, and a sense of permanence in their batting orders in general. Even when Taylor struggled, they did not try to push him down to No. 4 or 5. What about those other opening stalwarts like Gavaskar, Boycott, Hutton, Hobbs, Sutcliffe, Haynes and Greenidge? Well, Gavaskar batted at No. 4 in quite a few matches. Sutcliffe batted at No. 6 in a single match. Hobbs batted low down in a few matches. Boycott batted at No. 4 a few times. Similarly Haynes and Greenidge went off their opening positions for no more than a couple of innings. Note Mark Taylor's perfect index value of 1.00. He has batted in the opening slot in each of the 186 innings he has played for Australia. This is the highest number of innings by an opener without vacating his slot.

A list of the real rabbits.

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Chatfield E.J		Nzl	43	54	10.96	0.06	52 @	11 ( 96.3)
Chandrasekhar B.S		Ind	58	82	10.93	0.16	75 @	11 ( 93.8)
McGrath G.D		Aus	124	138	10.92	0.21	128 @	11 ( 92.8)
Valentine A.L		Win	36	51	10.90	0.14	46 @	11 ( 90.2)
Malcolm D.E		Eng	40	58	10.84	0.32	49 @	11 ( 84.5)
Danish Kaneria		Pak	51	69	10.78	0.43	55 @	11 ( 79.7)
Tufnell P.C.R		Eng	42	59	10.75	0.48	44 @	11 ( 74.6)
Alderman T.M		Aus	41	53	10.62	0.76	42 @	11 ( 79.2)
Walsh C.A		Win	132	185	10.62	0.65	122 @	11 ( 65.9)
Willis R.G.D		Eng	90	128	10.61	0.54	78 @	11 ( 60.9)

Not a very surprising list. All confirmed rabbits. Is there a batsman who has ALWAYS batted at No.11? If a player has a long enough career, he inevitably bats at least once or twice in some other position, and thus has a BPI different from 11. The nearest we get to a perfect rabbit is in the form of Chatfield, who has batted at No.11 a total of 52 times and at No.10 twice, both times because of injuries to other batsmen. Chatfield has scored 18 zeros (dismissed or otherwise) out of these 54 innings. McGrath has batted at positions earlier than No.11 a few times. Similarly Chandrasekhar has achieved promotion a couple of times. Valentine and Malcolm complete the perfect list of rabbits. Chandrasekhar's batting average is the lowest in this collection. Chris Martin of New Zealand, almost the perfect no.11, just misses the cut, having played 49 innings. he has batted at no.11 a total of 46 times, gaining promotion 3 times to no.10. Oh! that is wrong. As David Barry has pointed out, Martin was moved upto no.10 (probably very reluctantly) because Vettori, Bond and Cummings were absent in the three innings respectively. A true no.11, then Martin is.

Batsmen who have a Mean Deviation of 0.00 (All opening batsmen!)

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Taylor M.A	~	Aus	104	186	1.00	0.00	186 @	1( 100.0)
Hayden M.L	~	Aus	91	162	1.00	0.00	162 @	1( 100.0)
Slater M.J		Aus	74	131	1.00	0.00	131 @	1( 100.0)
Lawry W.M	~	Aus	67	123	1.00	0.00	123 @	1( 100.0)
Strauss A.J	~	Eng	43	81	1.00	0.00	81 @	1( 100.0)
Hunte C.C	~	Win	44	78	1.00	0.00	78 @	1( 100.0)
Srikkanth K		Ind	43	72	1.00	0.00	72 @	1( 100.0)
Imran Farhat	~	Pak	27	51	1.00	0.00	51 @	1( 100.0)

and one very close to 0.00

Chatfield E.J

Nzl

43

54

10.96

0.06

52 @

11 ( 96.3)

This is the same list of omni-present opening batsmen already presented, re-displayed here to emphasise the fact that the opening slots have a sense of permanency attached to them. We have already talked of Chatfield.

Non-opening batsmen who have a Mean Deviation of 0.00 (Below 50 innings played)

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Jayawardene H.A.P.W		Slk	17	19	7.00	0.00	19 @	7 ( 100.0)
Hayward M		Saf	16	17	11.00	0.00	17 @	11 ( 100.0)
Aamer Nazir		Pak	6	11	11.00	0.00	11 @	11 ( 100.0)
Mohsin Kamal		Pak	9	11	11.00	0.00	11 @	11 ( 100.0)
Renneberg D.A		Aus	8	13	11.00	0.00	13 @	11 ( 100.0)
Mpofu C.B		Zim	6	12	11.00	0.00	12 @	11 ( 100.0)
Owens M.B		Nzl	8	12	11.00	0.00	12 @	11 ( 100.0)

and

Patterson B.P

Win

28

38

10.97

0.03

37 @

11 ( 97.4)

Prasanna Jayawardene, the current Sri Lankan wicket-keeper leads this list with 19 batting stints, all at position 7 (including the test which finished now). The maximum number of innings played by a batsman whose entire career was spent in the No. 11 position is 17, by Nantie Hayward of South Africa. West Indies’ Pattrick Patterson has had one promotion, out of 38 attempts, from 11 to 10.

Now for the list of batsmen who have been tossed around a lot (High Mean Deviation)

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Rhodes W		Eng	58	98	5.06	3.39	43 @	1 ( 43.9)
Abid Ali S		Ind	29	53	4.85	3.17	21 @	1 ( 39.6)
Prabhakar M		Ind	39	58	4.09	3.05	30 @	1 ( 51.7)
Mongia N.R		Ind	44	68	4.84	3.00	30 @	7 ( 44.1)
Engineer F.M		Ind	46	87	3.77	2.96	48 @	1 ( 55.2)
Mankad M.H		Ind	44	72	3.56	2.94	40 @	1 ( 55.6)
Blackham J.M		Aus	35	62	7.45	2.45	16 @	8 ( 25.8)
de Villiers A.B		Saf	32	58	3.17	2.30	34 @	1 ( 58.6)
Macartney C.G		Aus	35	55	4.07	2.23	28 @	3 ( 50.9)
Grout A.T.W		Aus	51	67	8.57	2.20	29 @	9 ( 43.3)
Stewart A.J		Eng	133	235	3.58	1.97	77 @	1 ( 32.8)

Wilfred Rhodes is one of the very few who has batted in positions 1 to 11. Ravi Shastri, another 1 to 10 batsman, has a BPI of 4.93 and a Mean Variance of 1.74. Abid Ali has batted consistently in the opening slots and in the late order. Note the presence of five Indians among the top six, indicating a propensity for the Indian selectors to have stop-gap opening combinations more often than others. Alec Stewart is a special case worthy of separate discussion. Having played 235 innings, he has batted 77 times at No. 1 (only a third of the innings). Otherwise he has been shunted around to almost all the top-order batting positions, with No. 6 being the next highest. He has batted in 3, 4 and 5 quite frequently. He is the only batsman who has played over 130 Tests and 230 innings and has a mean variance of nearly 2. There is no doubt that he would have achieved more than an average of 39.56 if he had a settled batting slot.

Batsmen who have batted in the same position for many an innings

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Gavaskar S.M		Ind	125	214	1.26	0.53	203 @	1 ( 94.9)
Haynes D.L		Win	116	202	1.03	0.03	201 @	1 ( 99.5)
Atherton M.A		Eng	115	212	1.17	0.43	197 @	1 ( 92.9)
Boycott G		Eng	108	193	1.03	0.16	191 @	1 ( 99.0)
Taylor M.A	~	Aus	104	186	1.00	0.00	186 @	1 ( 100.0)
Tendulkar S.R		Ind	142	229	4.29	0.61	185 @	4 ( 80.8)
Gooch G.A		Eng	118	215	1.46	0.83	184 @	1 ( 85.6)
Greenidge C.G		Win	108	185	1.05	0.17	182 @	1 ( 98.4)
Waugh M.E		Aus	128	209	4.24	0.57	170 @	4 ( 81.3)
Hayden M.L	~	Aus	91	162	1.00	0.00	162 @	1 ( 100.0)
Jayasuriya S.T	~	Slk	110	188	1.95	1.77	152 @	1 ( 80.9)

Gavaskar and Haynes head this list having batted more than 200 times in their respective positions. Tendulkar has had a settled tenure at No.4 and Mark Waugh a similar comfort zone.

Barring these two, the other batsmen are all opening batsmen

Batsman	L	Cty	Tests	Inns	BPIdx	MeanDev	Freq	Batpos (%)
Dravid R		Ind	115	197	3.25	0.78	146 @	3 ( 74.1)
Waugh S.R		Aus	168	260	5.42	0.74	142 @	5 ( 54.6)
Botham I.T		Eng	102	161	6.23	0.69	94 @	6 ( 58.4)
Marsh R.W	~	Aus	96	150	6.91	0.46	123 @	7 ( 82.0)
McGrath G.D		Aus	124	138	10.92	0.21	128 @	11 ( 92.8)

Dravid, as expected, has played at No. 3 a total of 146 times, often walking in during the first 5 overs. Steve Waugh has batted at No. 5 a total of 142 times. Not so surprisingly Glenn McGrath has batted at No. 11 the most number of times.
PS: Quite a few readers have expressed that the Runs Per Innings value could also be used as an alternate measure for the Batting Average as compared to EBA. I had used this measure in most of my earlier simulation exercises, especially ODI. It presents a simple computation methodology as compared to EBA. The best thing to do is to leave the batting average as the main measure for analysis and leave the other two as alternative measures which could be used to complement the established measure.