December 24, 2011

Bowling Quality Index re-visited: incorporating home/away and recent form

Analysing Bowling quality using career-to-date home/away performances and recent form

Makhaya Ntini: superb at home but ordinary away © Getty Images

This is the first of three very important and significant articles on batting performances against differing conditions and players.

About six months back I had come out with an article on Batting performances against different Bowling groups based on BQI (Bowling Quality Index). Notwithstanding the fact that it was a rough unpolished stone, it was one of the best received of all my articles and I came out with a follow-up article after doing some amount of polishing. However there were so many valid suggestions and great ideas that there is a need for me to re-visit that theme, this time incorporating improvements and new ideas. These tweaks would define this very important theme once and for all.

I have summarized below the ideas and suggestions. This is the extract from hundreds of comments.

1. The Bowling quality was determined based on Ctd (Career-To-Date) figures. That was very essential. However there is a clear need to do this work based on Ctd-Home and Ctd-Away figures depending on where the Test was played. As has been shown later, bowlers are poor travellers and these changes will make a significant difference.

2. The initial phase of the bowlers, during the period when the bowler is yet to take the first 50 wickets, has to be done correctly with no ambiguity. Most bowlers who had captured quite a few wickets in their Test career were nearly as good from day 1 and this fact has to be recognized.

3. Almost all bowlers go through widely varying form swings in their career. This important factor of Recent Form has to be recognized, based on their performance during their last 10 Tests, usually an year or so.

4. Bowling quality is not enough to determine the value of batting. The type of pitch plays a very important fact. Scoring a 200 at Lahore during 2006 is equivalent to a 100 while scoring a 100 at Hobart last week is the equivalent to scoring a 200. It is also essential that this should not be considered as a stand-alone measure. It is necessary to combine the two measures. Facing Marshall and Ambrose at Faisalabad on a batting wicket is quite different to facing Prabhakar and Prasad on the same pitch.

All this means that this coverage cannot be completed in one article. It would require three articles. In this one, the first, I will talk about the revised Bowling Quality Index based on Ctd-H/Ctd-A values and adjusted by Recent Form values. In the second one I will cover in depth how the Pitch Index will be calculated, using both Top-7 RpW (Runs-per-wicket) and BpW (Balls-per-wicket) values. Once the stage is set, the third article will revert to the Batting performances against a combination of these two measures. Thus there will be opportunity to come with your comments on the whole process. All three articles will follow in one unbroken sequence so as not to lose the threads of discussion.

Streamlining of Bowling Quality Index:

First, the handling of the first 50 wickets has been streamlined. Until a bowler, who has captured 87 or more wickets in his career, reaches 50 wickets, he is protected to the level of his career average. If he has done better than his career average, like Brett Lee, that is considered. This is very fair to established bowlers. Of course the other bowlers will have notional figures during their first 50 wicket-period. Why 87, why not 100 ? The number 87 has been considered to get world-class bowlers like Mailey, Spofforth, Richardson, Patterson, Hawke, Bond, de Villiers et al into this group. If this is termed as arbitrary it is no more arbitrary than setting the cut-off at 100 wickets. What is the sanctity about 100 wickets anyhow?

Most of the top batsmen travel well. In other words their home and away batting averages do not show very high degree of variation, as described below. I have selected 6 batsmen who represent the best in Test batting from different angles.

Average comparison for selected batsmen

BatsmanCareer AvgeHome AvgeAway Avge

Even for Lara and Ponting their variation away from career average seems to be only around 10%. Bradman and Dravid have in fact performed better away to a significant extent.

Now let us look at a similar tables for bowlers. They seem to be poor travellers. I have selected the top four wicket-takers and two recent bowlers of different types and from different countries.

Average comparison for selected bowlers

BowlerCareer AvgeHome AvgeAway Avge
Harbhajan Singh32.2228.3639.65

Warne and McGrath have bowled with the same level of effectiveness home and away. For Muralitharan there is significant variation. Kumble, Harbhajan and Ntini have very significant variations. Please note that I have selected Harbhajan and Ntini only to show the type of variation which can exist.

It took me quite some time to compile the CTD Home and Away values. Determining the values on a continuous basis and storing them in the match data base presented its own problems. Once I completed that I have re-done the BQI and this has come out very well now. I am not going to show the BQI tables now since more work has to be done.

I would give an example. In match no 1820, played at Christchurch between New Zealand and Sri Lanka, the BQI group was 5 for both teams since Murali's Ctd-figures were 657 at 21.96 and they also had Malinga and Vaas, very good bowlers. However Murali's figures comprised of 406 home wickets at 19.65 and 251 away wickets at 25.71. Once the Ctd-Away average was applied Sri Lanka went to Bowling group 4. New Zealand firmly stayed in group 5 since Bond had done reasonably well both home and away while Martin had a 15% variation between home and career. Only Vettori had done better away but he bowled very little in the match.

Now for Recent Form. This was again very tough work. I tried doing some short cuts, in flight, as I processed the match database. However that was not effective and fool-proof. Finally I bit the bullet and took the major step of incorporating the career details, match-by-match, into the Player database. This doubled the database size at one shot since I had to provide space for 220 matches. Ha! what is this 220. Well, Tendulkar has played 184 as on date and I have estimated that it would take him over 3 years to play more than 36 Tests and THAT is very very unlikely. Well, if that happened or Kallis played another 73 Tests, I will gladly go through another re-vamp. The advantage is that I have, for the first time, the player's entire career in a single record and can come out with many different types of analysis.

The Recent form work has come out very well. It is going to be very significant also since the form swings are widely varying. There is nothing gained by looking at absolute values. A 10-Test average of 20 for Sangakkara is a disaster, 35% of his career average, while the same for Vettori is around 65% of his career average. The following tables show the extreme values in Recent forms for batsmen.

Please note that the Recent form work is for the last 10 Tests, irrespective of location. I cannot very well do it separately for Home and Away since it might take players even 5 years to play 10 away Tests and that is too long a period. However I have done a minor adjustment, based on a suggestion from Sriram. Within these 10 Tests, I have done a minor tweak of 95% for home and 105% for away performances. Also remember that the recent form work has started after the players play 10 Tests or more.

In great batting form

MtIdYearForBatsmanCareer AvgeRF-InnsRF-AvgRF-Idx%
10671987IndVengsarkar D.B42.138127.38302.3
10221985EngGatting M.W35.561197.09278.5
16032002SlkTillakaratne H.P42.886114.83267.8
09621983PakMudassar Nazar38.091093.10244.4
15721990PakImran Khan37.69891.75243.4

Probably it looks as if I should do a separate article on Recent form. Vengsarkar had the best form anyone has ever had. During a 10-Test period during 1987, he scored at 302% of his average. His scores were 37*, 126*, 33, 61, 102*, 38, 0, 22*, 164*, 57, 153, 166, 96. Gatting's golden period was during England's golden period, 1985. Tillakaratne performed at 267.8%. Incidentally during this period, both Vengsarkar and Tillakaratne performed for more than 10 Tests at only slightly lower levels. Note the wonderful batting form exhibited by Mudassar Nazar during 1983 and Imran Khan during 1990. Imran was averaging better than many a specialist batsman at his peak.

It can also be seen that it is not exactly possible for the top batsmen who average above 50 to achieve these RF-Idx % figures since they would then have to have recent 10-Tests average of over 150 or so, not very easy.

In awful batting form

MtIdYearForBatsmanCareer AvgeRF-InnsRF-AvgRF-Idx%
15212000EngNasser Hussain 37.191511.6031.2
12051992AusWaugh S.R51.061416.0031.3
14631999AusHealy I.A27.4178.6531.6
09051981EngBotham I.T33.551810.8932.5
19012008IndDravid R 53.231418.1134.0

Let us take the other end. Nasser Hussain had the worst streak anyone has ever had. Ponting can take heart. His current run is far better than Hussain's whose scores ran 15, 16, 25, 10, 21, 0, 15, 8, 10, 6*, 22, 0, 0, 7, 0 retd, 23 and 5. Steve Waugh had a similar nightmare at the early part of his career in 1992. It is amazing that Botham carried a run of 8, 35, 9, 4, 37, 7, 0, 0, 16, 26, 1, 1, 13, 16, 1, 33, 0, 0 into the Headingley Test. And how he broke this barren spell, with knocks of 50 and 149*. Everyone knows about Dravid's recent barren spell. Here it is in black and white.

Thus it can be seen that the range of RF-Idx is ten-fold, 31.2 to 302.3, that too for established batsmen who have scored 3000 or more runs.

Now for the bowlers, the more important segment of Recent form analysis since these are the ones used in adjusting the BQI. The extreme values are given below.

In great bowling form

MtIdYearForBowlerCareer AvgeRF-WktsRF-AvgRF-Idx%
04601958EngLock G.A.R25.58549.96256.8
00351892EngBriggs J17.75557.84226.6
08341978EngEdmonds P.H34.183415.24224.4
14071998WinHooper C.L49.432024.30203.4
04551958EngBailey T.E29.213516.09181.6

Lock's 10-Test period saw him performing at around 2.5 times his career average. His Test performances were 3/66, 2/38, 3/86, 1/29, 11/48, 3/25, 9/29, 11/55, 8/96, 3/39. Briggs was about 225% better. However see how quickly we go below 200%. This also gives a few not-so-great bowlers like Hooper and Edmonds chances to have their golden runs and move to the top of the Recent form table. Hooper, for a period, bowled like Muralitharan and Edmonds, for a while, bowled like CTB Turner. And Bailey was giving SF Barnes a run for his money !!!

Same as for batsmen. Tough for top bowlers to have a very low % since their averages are already low. For instance Muralitharan's best period has been during 2007 when he captured 89 wickets at an average of 14.62, still giving a RF-Idx % of only around 150%, not enough to even come in the top-10.

In awful bowling form

MtIdYearForBowlerCareer AvgeRF-WktsRF-AvgRF-Idx%
11801991IndShastri R.J40.965113.836.0
5771965PakIntikhab Alam35.95996.3337.3
15932002SafNtini M28.831273.5039.2
04631958WinSobers G.St.A34.04984.5640.3
07401974EngUnderwood D.L25.841561.2742.2

On the other side, Shastri had a horror run. He performed nearly three times worse. Intikhab had a similar disaster run. Look at Ntini's barren period recently. Sobers was bowling like a millionaire and Underwood like an out-of-form Sobers.

The Recent form adjustment is briefly explained below. The adjustments are done based on averages since that is the most stable and proven of all measures.

For good form, ignore if the RF-Idx is within 20% above the career batting average. If the RF-Idx is above 120% of career bowling average, then apply the adjustment factor which will vary from 0.80 to 1.00. The Ctd-x bowling average of the bowler will be multiplied by this adjustment factor.

For Lock, the adjustment factor is likely to be around 0.82.

For poor form, ignore if the RF-Idx is within 10% below the career batting average. If the RF-Idx is below 90% of career bowling average, then apply the the adjustment factor which will vary from 1.00 to 1.20. The Ctd-x bowling average of the bowler will be multiplied by this adjustment factor.

For Shastri, the Adjustment factor is likely to be around 1.18.

Thus the RF related adjustment factor for individual players varies between about 0.8 and 1.2. Let me recapitulate that we are looking at a player's recent form. It does not matter whether he achieves this at home or away. Also remember that these are extreme values.

Let us keep in sight that these are individual bowlers' recent form indicators. When the BQI is determined, first the individual bowler's values are adjusted and then the final BQI compiled. Thus a single BQI is a composite of 4/5/6 bowlers, some in good and some in indifferent form. This leads to a lot of evening out. The team itself will carry a significant adjustment value on either side of 1.00 only if it happens that most of the bowlers are in good form or most of the bowlers are in poor form. This does not happen often. The final innings level adjustment, for the first innings, varies between 0.86 and 1.11. Second innings has a similar range. These innings, which are very illustrative and illuminating examples, are summarized below.

Two innings with extreme Recent form adjustment values

Match Id: 0834
Year: 1978
England vs Australia.
England RF-Idx for innings was 0.86.
4 out of 5 bowlers were in great form.

Match No: 0774 Year: 1976 India vs West Indies. India RF-Idx for India was 1.11. Venkataraghavan-120%, Bedi-137%, Chandrasekhar-128%, Madan Lal/Amarnath-100% (They had not even played 10 Tests). All 3 main bowlers were in poor form.

The sanctity of the word "Recent" in Recent Form will be maintained, to the extent possible. If the Recent period extends beyond 2 years, due to injuries, non-section or external factors like War, it would not be considered. However I am not able to confirm that this can be done because it is quite tricky to decide where to draw the line. The impact is likely to be infinitesimal

Now to apply all these rather complex adjustments and come out with the BQI. The revised ranges are given below. These values have changed significantly because of the various changes implemented. Hence the grouping also has changed. The BQI values are capped between 15.0 and 60.0. The distribution has a mean of 35.67 and a Standard Deviation of 7.75 which works out to a Coefficient of Variation of 0.217, indicating a very balanced distribution.

Summary of BQI Grouping

15.0 - 27.0: Group 5 - 1002 (13.7%) Amongst the best of all time
27.0 - 32.0: Group 4 - 1657 (22.6%) A very good attack
32.0 - 38.0: Group 3 - 2085 (28.5%) Good attack
38.0 - 45.0: Group 2 - 1577 (21.5%) Below average
45.0 - 60.0: Group 1 -  997 (13.7%) A poor attack

To download/view the document containing the complete BQI / PQI tables please click/right-click here.

In the next article I will cover the Pitch Quality Index in depth.

Anantha Narayanan has written for ESPNcricinfo and CastrolCricket and worked with a number of companies on their cricket performance ratings-related systems

Comments have now been closed for this article

  • fanedlive on January 9, 2012, 5:34 GMT

    What are your opinions on home/away for sub-continental teams? Most subcontinent batsmen play similarly in subcontinental conditions (i.e. almost home for them) and differently outside? Doesn't the home/away format benefit them significantly. E.g. outside the subcontinent Tendulkar's average outside is 51.5, a good 4 points lower than the 55.7. Dravid on the other hand averages 55.8 outside the subcontinent, 1 point more than the away average reported. The averages are more skewed for Sangakkara, Jayawardena, Sehwag, Inzamam, Yousuf, etc. [[ The home/away is for bowlers. Ananth: ]]

  • fanedlive on January 5, 2012, 6:02 GMT

    Arjun, I am coming around to using the Top-10 match innings (MT10) mainly beacuse it does not ignore any innings. Relevant innings, in whichever positions these are played, will be considered. However I will not come out with a corollary article. There are too many things on the plate for me to assign two consecutive articles to the same topic. I will post a Pitch Group summary in this article and refer to it in the next (Batsman Analyaia) article. I have a few points and need your and the other readers' inputs. 1. Using the aggregate does not work out. Many matches were placed wrongly. 2. So I have to use the MT10 average. The idea is that if a batsman remained not out, to that extent the PQI value should be higher since he has not been dismissed. This results are also in line with the T7-RpW group sizes. 3. For the T7-PQI I used the BpW as an additional measure. However here there is no need to do that for reason given below. 4. The 28 innings used to determine the T7-RpW had a number of small innings, with varying balls played associations. Hence I used the BpW measure to smoothen these wide variations within a match and across matches. However in this case we would select only the top-10 innings of the match. As such I have found that the MT10 Runs have a strong correlation to the MT10 Balls played. Hence there is no need to incorporate the Balls played information, which anyhow gets determined on a pro-rata basis for a third of the matches. 5. I will post the new table, based on MT10-Avge innings in the main article itself. Will do it by tomorrow morning.

  • fanedlive on December 30, 2011, 5:21 GMT


    Regarding Gerry's claims of England attack in 2005 ashes, though the statistics shows they belong to group 3 but recalling the effect they had on powerful Aussie batting & as a whole series was astounding. In home conditions at that time Eng bowling was actually at par with any greatest attack of all time, assuming Aus batting was par excellence with all the big names. Similarly Aussie bowling in 1st MCG test yesterday might not be thick in numbers but was extraordinarily lethal given quicks bagged 19 for 370 odd runs, giving an avg of around 17 per wicket. I would say exceptional, whatever their CTD avg shows. Famed Indian batting lineup could not handle an attack which had a collective experience of about 50 test. Barring sachin (to some extent) no other batsman showed skils.

  • fanedlive on December 29, 2011, 5:43 GMT

    The other issues are a bit more complex, so let me stick to just one: In your previous version, group 5 cutoffs were 29.99. In this version, you have included pre-1900. This has brought down the group 5 upper bound a bit to 27. However, if you took a sample of only post 1920, i sense that the % of runs scored against group 5 would be only ~10% (or thereabouts), perhaps making this a bit harsh... or am i mistaken, and the upper bound would be still almost 27, even for "modern cricket"? [[ Gerry, this is a different exercise. I am finally going to form a composite of BQI and PQI and then analyze runs scored. So let us give the old one a miss. The Pitch article is due tomorrow and the combined one about 10 days afterwards. I think this would answer most comments raised at the end of the BQI article. At one stroke we would get two of the most important "opponents" into the Batting analysis. I am already very excited by this. On flat tracks, scoring against the top bowlers should get credit, and it happens likewise. Ananth: ]]

  • fanedlive on December 28, 2011, 8:26 GMT

    Ananth, with your comments, i believe of the two issued i mentioned 1) uplift and 2) delta compression, the first is getting clearer.

    Firstly i had taken 30.2 as the batting avg, whereas the bowling avg in 2024 matches is 31.7 (due to non-bowler dismissals and extras). So 31.7 is indeed the correct measure. The second argument is the ratio of aggregates v/s avg of averages. Between these two the bulk of the uplift should get explained. The rest would be adjustments, justified.

    2) delta compression of home / away. It is 4.3% if one takes a simple avg of avgs in your XL sheet. But it is ~10% if one takes cricinfo ratio of home aggregates v/s away aggregates. The uplift due to taking ratio of aggregates v/s avg of averages should be similar in home/away, so the home away delta should be similar in cricinfo & your XL file.

    So are the adjustments you are making contributing to the home/away delta compression - that is the central question since this whole article is about home/away. [[ Probably only partly. It is only the question of computing at Test level against the comparing value taken across 2024 Tests. Incidentally I have taken away a few minor adjustments, such as limiting the lower end of bowling avge to 15. I have re-done the Excel sheet and uploaded. I have also done the Home-Away myself. the numbers are 34.9 and 36.3. If you have the time, do the analysis for one specific country, say India, which performs much better at home. My feeling is also that there are many, many bowling spells by bowlers in their early 50s, confirmed by the following numbers. Career wkts summary Greater than 86 : 189/34706. Between 50-86 : 149/9596. Below 50 (1-49): xxx/12500. Total wkts: 59000 app That means the lower level adjustment to 40 is done for quite a number of wickets captured by bowlers who are below 87 wkts. For a substantial part of wickets of those 149 who are in between 50 and 87 and for all the wickets captured by the hundreds of bowlers who have captured between 0 and 49 (12500 or so). If you look at this carefully you will see that the adjustment has been primarily done for those lower level bowlers who have captured below 87 wkts and that too only below 50 wkts. Ananth: ]]

  • fanedlive on December 28, 2011, 4:16 GMT

    Dear Ananth, trying to squeeze out a few minutes intermittently during work each day (unfortunately a very busy week). 1) I had asked if the upper bound in Group 5 would change materially if pre-1900 were to be excluded 2) Are we by any chance making RF adjustment to first 50 wickets / 87 wickets 3) The two cut-offs at 50 wickets and 87 wickets seem to be attracting most comments, so while i study your other comments, i would venture to comment that if this work has to find wider readership, as it undoubtedly deserves to, perhaps you can simplify here somewhat, by using a single number. This is not to suggest oversimplification, however. [[ 1. Yes, more or less. The first few Tests have been 1. Then from about 1988, about 20 consecutive innings have BG 5.There is a profusion of 4s/5s until 1898 when the picture changes. 2. The RF adjustments are done only after a bowler crosses 50 wkts and 10 Tests. 3. There is really one cut-off. At 50 wkts. It is only that the following formula, expressed in English language, is applied.

    If (Ctd wkts less than 50)
       If (Career-wkts greater than or equal to 87)
          If (Ctd-avge less than 40)  then take this as 40.
          Else take the Ctd-avge.    
         Take the Ctx-Avge.
    I think nothing could be clearer. I think the lack of comments has to do with the Readers' lack of interest in Bowling-related analyseis and the absence during XMas-New Year time. When I come to the third article the interest will come through since there the batsmen come in, even though irt wouyld be a bowler analysis.. Ananth: ]]

  • fanedlive on December 27, 2011, 11:41 GMT

    Gerry, I hve given the following info to bolster the argument given in my last response. I have just about completed the PQI work. The batting average of the top-7 batsmen, taken across all matches, is 35.91 (this matches exactly Crcinfo figure). The mean of the Top-7 RpW match values is 37.39. This is the same data with absolutely no adjustments whatsoever. You can see there is a 4% variation. In other words Sum(T7Runs) / Sum(T7Wkts) for 2024 population entries taken as a single sum, varies by 4% to Sum(T7Runs/T7Wkts) / 2024.

  • fanedlive on December 27, 2011, 4:36 GMT

    In your example of Jones, should not his average when he crosses 50 wickets be used as CTD for all the 18 matches before. Why should 40 be used instead of his average when he crosses 50? [[ Raghav, this is what I have responded. It is clear that after Jones crossed 50 wkts, his actual average kicks in. It is only in the pre-50 phase that he does not get the protection since he finished his career at 59 wickets. The protection is only for bowlers who have taken the cut-off wkts (87). If you want the protection come down to 50 wkts, it will mean almost all bowlers, including Hirwani. I am not sure whether it is correct. "" Jones 17 @ 31.93 (Total 44 wkts and career is only 59 wkts, so taken as 40.00) Incidentaly for Test 1762, Jones move to 27 at 29.75 and a total wickets of 54 so his average is taken as 29.75. And the England attack moves to 29.0 and Group 4. "" Ananth: ]]

  • fanedlive on December 26, 2011, 16:28 GMT

    Extremely well-written. No doubts about the methodology.

    I always wondered how the legends of past like Bradman, Hutton, and Hobbs would compare with modern great players.

    Ananth, I have only a small query-cum-suggestion. Are you, by any chance, going to add a factor for (i) the protective equipment and (ii) the uncovered pitches of the past? [[ These are totally subjective factors, Vivek. How can I make allowance for these and justify that. Ananth: ]]

    (i) We have seen Gavaskar playing bare-headed, and heard of Bradman facing bodyline with pads and gloves. Today's batsmen look like astronauts. They have no fear that the ball would hit them fatally. I wonder how our Sehwag-likes would rate against a bodyline attack or 80's Windies attack with no protectives, no restrictions on bouncers, and no rule against front-foot dragging.

    (ii) The pitches used to be uncovered in the past. Today's modern equipment known as pitch cover protects the pitch to a great extent. I hope you are going to take care of the cover-factor. [[ The covered/uncovered nature of the pitches will come through in the Pitch Type calcultions. Ananth: ]]

    Waiting for the next part.

  • fanedlive on December 26, 2011, 13:32 GMT

    Ananth, just trying to understand. Not in an acceptance or non-acceptance frame of mind. If I don't understand how a global average of 30.17 from cricinfo for all 7300 odd innings scales up to 35.67, how am I going to appreciate your work? So first I must understand where the uplift is coming from.

    Taking a simple average of BQI for home and away in your XL gives me a home/away delta of 4.3%. Whereas unadjusted cricinfo stats show 10%. So the home/away distinction in your BQI is smaller than the raw stats. So to fully understand your work, it is necessary to understand why this compression of delta is happening. The delta of home/away is the whole reason for this article anyway. [[ Gerry, I presume your figure has been calculated in the following manner. = Sum(Bowler runs) / Sum(Bowler wkts). It is the purest amd macroest of all calculations. A single sum of probably 6000+ figures divided by another single sum of 6000+ figures. Okay, home and away separately. Mine, on the other hand, is a very low leval computation in a very low micro level manner. The overall formula is the Arjun suggested Reciprocal method. Briefly BQI for each match = Sum(Balls) / (Sum(Balls/CTx-Avge)). The Ctx-Avge is adjusted by quite a few factors. This BQI is further limited to specific values. There are limits fixed at various levels. How can you compare the two totally independent figures and say these should be close to each other. I will give you a comparison. Philander's Bowling Average = 297/24 = 12.38, determined at the macro level. What if, for some valid reason, I did the averaging at the micro level, as given below. Analysis Avge Ctd 3 for 63 = 21.00 0.0 5 for 15 = 3.00 21.00 1 for 47 = 47.00 9.75 5 for 70 = 14.00 13.89 5 for 53 = 10.60 13.93 5 for 49 = 9.80 13.05 Avge = 21.08 Note the very significant variation, even though both are in a way bowling averages. Mainly because of the 47.0. And the Ctd average has varied like a yo-yo. However it would settle after a few Tests are played. Until then the multiplying factor will be this varying figure. There is no comparison between the two cases. However I have given this example to indicate how macro level and multiple micro level calculations will produce great variations in seemingly similar values. You have to stop thinking of BQI as equivalent to Bowling Average. It is a Bowling measure derived from Bowling average and 5 other factors. I am not going to anything more to convince you. I will only check my work thoroughly once more. That is all. Ananth: ]]

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