Skill and Chance in Association Football Author(s): C. Reep and B. Benjamin Source: Journal of the Royal Statistical Society. Series A (General), Vol. 131, No. 4 (1968), pp. 581-585 Published by: Wiley for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2343726 . Accessed: 28/06/2014 19:19 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Wiley and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series A (General). http://www.jstor.org This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions 581 1968] Skill and Chance in Association Football By C. REEP and B. BENJAMIN associationfootballthe ball is passed fromplayerto playeramong the eleven membersof a side untila particularplayerloses possession of the ball eitherby interceptionor tackle on the part of a memberof the defendingteam or by an of the rulesof the game or by himselfshootingat the defendingside's infringement goal. We may definean "r-pass movement"as one in whicha playerof side A, havingjust obtainedthe ball, sets offa seriesof r successfulpasses amongmembers of his own team afterwhichthereis eithera shot at goal by the rthrecipientor an or thereis an attempted(r+ 1)th pass whichis intercepted.We note infringement that a 0-pass movementmeans that A's firstattemptedpass is interceptedor that thereis a shot at goal withouta precedingpass, as, for example,fromthe penalty thelikelihoodof a successfulrthpass: spot. Thereare a numberof factorsaffecting (1) the positionsof the playersbetweenwhom the pass is attemptedand the defendingplayerswho tryto intercept; and confidencewith (2) the relativeskills of the playersand the effectiveness whichthoseskillsare applied at thisparticularstage of the game. In evenlymatchedteamsplayingundertheconditionsnormallyobtainingin good class football(for example,in the firstthreeEnglishF.A. Divisions) the second of thesefactorsdoes not varywidelyfromone attemptedpass to anotherbut as the attackproceedstheopponentsprogressively to improvethechance disposethemselves of interception or tackle; theyare in morecompactformationand closerto theball. The probabilityof an r-passmovementP(r) = [Pl -P2 - P Pr(l Pr+1)] where Pl > P2 > P3 ***Pr > Pr+l. As to the formof the functionPr one would expectp1 to -befairlyhighthough less than unityand Pr to fall rapidlyto some low value beyondwhichthereis little further decrease;an exponentialformseemslikely. One of us (C. R.), who has an acute interestin the implicationsfor strategic has compiledcarefulrecordsof actual frequenciesagainstwhichto testthis training, togetherwithvalues Of Pr derived theory.Table 1 showsfourobserveddistributions fromthesedistributions.It will be seen that generallyPr declinesas r increases,if somewhatirregularly.At highervalues of r whennumbersare smallerthe observed ratios{P(r - 1)}/{P(r- 2)} becomeirregular and thisproducesevengreaterirregularity in p.. Investigationof the data suggestedthat P(r) would be a more convenient parameterto attemptto "smooth"in orderto producea standardmodel. In contemP(r), it seemed from platingthis "smoothing"and the shape of the distribution inspectionthatone wouldexpectP(r) to fitcloselyto a negativebinomialdistribution and thisexpectationwas justified. to fitnegativebinomialsto thedistributions Table 2 showstheresultofattempting of Table 1. Generallytheseare reasonablygood fits;the fitfor the miscellaneous IN This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions - Skill and Chancein Association REEP AND BENJAMIN Football 582 [Part4, TABLE 1 Frequencyof r-passmovesin F.A. matchesand derivedvaluesof Pr= 1-[[{P(r- 1)}/{P(r-2)}] {(l /pr,)- 1}i 42 First Division matches1957-58t r 0 1 2 3 4 5 6 7 8 9+ Total Mean Variance 12 First Division matches1961-62+ 36 Miscellaneous matches1965-66 1I World Cup matches1966? No. of moves Pr No. of moves Pr No. of moves Pr No. of moves Pr 10,580 6,923 3,611 1,592 608 280 107 33 9 11 23,754 1.00 1150 *445 *555 *475 *424 400 *427 *381 *380 497 2,331 1,629 1,035 565 304 154 60 31 15 6 6,130 1.30 2K11 *380 *620 *571 *523 *502 *466 *420 *463 400 3,407 2,320 1,435 767 422 215 130 62 24 20 8,802 1-35 2-52 *387 *613 570 *534 *533 *518 *526 *456 *431 1,862 1,220 785 518 316 153 78 65 27 39 5,063 1-56 3-29 *368 *632 '619 *604 *567 *534 *577 *626 503 t All involvingSheffield Wednesday. + All involving Arsenal. ? One matchwentto extratime. TABLE 2 Frequencyof r-passmovesin F.A. matches-negativebinomialtests r 42 First Division matches1957-58t 12 First Division matches 1961-62+ 36 Miscellaneous matches1965-66 Ex- A-E pected A-E Actual Expected Actual Ex- A-E pected Actual 0 10,580 10,542 + 38 1 6,923 7,075 -152 2 3,611 3,542 +69 3 1,592 1,572 +20 4 608 653 -45 5 280 260 +20 6 107 101 +6 7 33 38 -5 8 9 14 -5 11 8 9andover +3 Total 23,754 23,805 Mean 1P00 Variance 1-495 p(x 2) > 10 2,331 1,629 1,035 565 304 154 60 31 15 6 6,130 1a303 2-109 2,224 1,788 1,061 556 271 126 57 25 11 10 6,129 +107 -159 -26 +9 +33 +28 +3 +6 +4 -4 <*0001 3,407 2,320 1,435 767 422 215 130 62 24 20 8,802 1-348 2-519 3,341 2,407 1,428 786 415 215 109 55 27 22 8,805 +66 -87 +7 -19 +7 0 +21 +7 -3 -2 11 World Cup matches1966? Actual Ex- A-E pected 1,862 1,777 +85 1,220 1,313 -93 785 829 -44 518 503 +15 316 293 +23 153 165 -12 93 -15 78 65 52 +13 27 29 -2 31 +8 39 5,063 5,085 1P558 3-286 > 30 < 01 t InvolvingSheffield Wednesdayon each occasion. I InvolvingArsenalon each occasion. ? One matchwentto extratime. This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions 1968] - Skill and Chancein Association REEP AND BENJAMIN Football 583 seriesis veryclose whilethatforthe Arsenalseriesis bad. But theseare onlyfour exampleschosenarbitrarily fromthe extensiverecordsshownin Table 3. TABLE 3 Passingmovedistributions 1953-67 No. of Frequency Detail matches 1953-54Wolverhampton W. 1955-56Miscellaneous 1955-56Sheffield W. 1956-57Sheffield W. 1956-57Miscellaneous 12 15 42 42 18 *417 395 *433 *443 *446 1957-58Sheffield W.t 42 445 *291 152 067 1957-58Miscellaneous 1958 WorldCup 1958-59Miscellaneous 1959-60Miscellaneous 1960-61Miscellaneous 1960-61Tottenham H. 1961-62TottenhamH. 1961-62Arsenalt 1961-62Miscellaneous 1961-62Burnley 1962-63TottenhamH. 1962-63Miscellaneous 1962-63Coventry 1962 WorldCup 1963-64Miscellaneous 1964-65Miscellaneous 1965-66Miscellaneoust 1966 WorldCupt 1966-67Miscellaneous 12 11 33 41 38 12 12 12 42 8 11 42 14 18 19 17 36 11 18 414 *384 *398 395 *389 395 *378 *380 *378 *384 400 *396 *434 330 *391 *366 *387 *368 *369 *158 *161 *170 *159 *160 *156 *157 *169 *166 *175 *160 *160 *151 *170 *153 *167 *163 *155 *163 All matches-Actual Expected+ 578 A-E 1 0 *299 *295 *297 297 *296 *284 *262 *290 *296 *266 *259 *260 *266 *262 *275 *251 *268 *294 *245 *268 *263 *263 *241 *231 394 *275 *398 *275 -004 - 2 *169 *167 *158 *152 *145 3 I 071 081 *068 *064 *062 076 090 077 *080 090 *092 094 *092 *092 *083 *092 *086 *067 *110 *091 093 *087 *103 *102 *162 *084 *158 *084 + 004 - 4 *029 *036 027 *027 *029 5 009 *017 *011 *010 *013 *025 013 *036 *051 *036 039 049 049 055 050 *051 040 050 045 *032 *066 -047 *056 *048 *062 057 018 027 *017 *017 *025 *028 *031 *025 *028 *022 *026 *024 *014 *036 *027 *025 *025 030 033 044 *022 043 *022 +*001 - 6 7+ 004 005 004 005 *006 *002 004 *002 *002 003 005 *002 007 012 *008 007 *012 *012 *012 *010 *012 *012 *012 *012 005 *020 *013 *014 *015 *015 *023 007 *013 004 007 009 009 *013 *008 *011 009 009 009 003 *023 *010 *016 *012 *026 *022 *010 009 *011 009 -*001 t Shownin Tables 1 and 2. of a negativebinomial. + On thehypothesis In thistablethefrequency distributions havebeenstandardizedto a totalfrequency of unity. Theirgeneralsimilarity is striking.Taking all matchestogether, thefitto the negativebinomialdistribution is almostexact,and thisis the patternadheredto by mostof the seriesfrom1958 to 1967. Priorto 1958,thenegativebinomialis still representative thoughthe parametersare those of the firstdistributionshown in Table 2. Generaltimechangesin the styleof play would be expectedto affectthe basic parametersof the distribution but not its mathematicalcharacter. There are otherregularities to be observedin associationfootball,and some of theseare shownin Table 4. If thepitchis dividedintofourquarterslaterallythenthe This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions 584 REEP AND BENJAMIN- *~ *j, [Part4, Skill and Chancein AssociationFootball z X o.s.Sz tn N 00 c cs m- t- tS 00 r s 00 "o 'I N -. S? Z: .Sk .ON ,j ci~ ~ t~~~~~O n t wo X . . . . tn m tntn t0 . . . . z~~~~~~~~~~~~~~~~ . t m DWO n? 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N. .q cq. .q cq. .q WI0 0 (i2 cq cq m CD0q\ . m WO . oo \000 o000 O~~~~ S r N . m~~~~~~~~~~~~~~~~~~~~~e ON_>> < t to ^ E 4~~~~~~~~~~~~~~~~O *? s ~~~~~. C Z toC a~~~~~~~ i t- m > N N -. ) . . u I n" . . . d d u d . -4q .t . m Q \0 o \1..O ~.O\0 o \ 1 I 4N tn tn tn tn \.I \_."C ".I ".I 1-- " It - . t I" V- _ 111 "- 11 -- This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions o 1968] REEP AND BENJAMIN- Skill and Chancein AssociationFootball 585 twoquarterscontainingthegoals are referred to as the"shootingareas". As mightbe expectedmoregoals come frompassingmovesbeginningin thatarea; theproportion can be seen fromcolumn(4) to be consistently just about 50 per cent. Column(10) shows that such shootingarea originattacks are in the ratio of 1: 3 to attacks reachingthe shootingarea, and fromcolumn(11) we see thatof theseshootingarea originattacksslightlymore than 50 per cent are regainedpossessions,that is, the defencefailsto get the ball clearlyaway fromthe shootingarea. The size of these proportionsare not surprising but theirnearconstancyis remarkable.Again column (6) showsthat30 per centof regainedpossessionslead to shotsat goal. It can be seenfromcolumn(8) that15 percentofall attacksreachingtheshooting area lead to shots at goal, but fromcolumn (7) that of attacksoriginatingin the shootingarea 22 per centor so lead to shotsat goal. Column (9) refersto "own half" breakdown,that is, to passingmoves which breakdown in the attackers'own halfof thepitch. Of all goals scoredagainstthem 50 per centcome fromsuchfailuresto movetheball intothedefenders'half. Finally,column(5) showsthatwithrareexceptions(forexample,the 1966World Cup series)it takes 10 shotsto score 1 goal. The observationthatthereis a stochasticelementin the numberof goals arising froma particularnumberof shotsin one match(as wellas a near-constant proportion overa largerseriesof matches)is easyfora statistician to accept; indeedhe wouldbe surprisedif it wereotherwise.It indicates,of course,thatan excessof shotsby one team does not mean that,by chance,the otherside will not getmoregoals and thus win the match. All this is so far removedfromcurrentsoccer beliefsand tactics thatgeneralacceptanceof therandomelementhas been inhibited(thoughone of us, C. R., has shown that a successfulstyleof play can be built upon it). It seems, however,thatchancedoes dominatethegame and probablymostsimilarball games. This content downloaded from 193.142.30.220 on Sat, 28 Jun 2014 19:19:49 PM All use subject to JSTOR Terms and Conditions