2343726

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 .
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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
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- 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.
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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
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584
REEP AND BENJAMIN-
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Skill and Chancein AssociationFootball
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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.
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