Overachievers and Underachievers in the FIFA World Cup, 1994-2022

Abstract

A regression model is used to predict points (countries are awarded 3 points for a win, 1 point for a draw, and 0 points for a loss) in the men’s FIFA World Cup between 1994 and 2022. Key predictor variables include population size of the 248 country-tournament observations and their GDP per capita in their year of participation. Both variables are statistically significant. Predicted point totals are then compared to actual points to compile a list of overachievers and underachievers over the last eight World Cup tournaments.

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Sommers, P.M. (2026) Overachievers and Underachievers in the FIFA World Cup, 1994-2022. Open Journal of Social Sciences, 14, 812-823. doi: 10.4236/jss.2026.148047.

1. Introduction

The Fédération Internationale de Football Association (FIFA) World Cup is arguably the most-watched single sports event in the world. Twenty-four countries entered the World Cup of 1994 hosted in the United States. Thirty-two countries entered the quadrennial men’s tournament in 1998 and every subsequent World Cup until 2026, when three countries—Canada, the United States, and Mexico—hosted forty-eight countries.

In 1994, FIFA began awarding three points for a win. Previously, teams earned two points for a win. The motivation for the rule change was to encourage more offense and discourage tactics that relied on settling for draws. Under the 32-team format used through 2022, a World Cup team could earn a maximum of 21 points (3 points each for 3 wins in the three-game round-robin Group Stage, 3 points each for the 4 wins in the single-elimination Round of 16, Quarterfinals, Semifinals, and Finals). At the conclusion of the tournament, countries might understandably want to know if they performed better or worse than expected.

The purpose of this brief research note is to help soccer fans answer the question of whether their country’s team performed better or worse than expected based not on the country’s pre-tournament seeding, but on the size of the country (by population), the country’s gross domestic product per capita, and what continent the country is located in. Apart from population size, richer countries would be expected to do better than poorer countries because they can afford better sports facilities, better coaching staffs, and talent development programs, not to mention “import … good soccer experience” (Kuper & Szymanski, 2009), that is, lure soccer players to become citizens (or nationals) of the country they will represent. At the conclusion of each tournament, which countries were overachievers? Which ones were underachievers?

2. The Data

For each of the eight FIFA World Cup tournaments between 1994 and 2022, data were collected on each country’s point total, current population and gross domestic product per capita (hereafter, GDP/N) in current U.S. dollars (at the time of that year’s World Cup). Population and GDP/N for each of the 248 country-tournament observations in our sample are from the World Bank Open Data at https://databank.worldbank.org/reports.aspx?source=2&series=SP.POP.TOTL&country=# and https://databank.worldbank.org/reports.aspx?source=2&series=NY.GDP.PCAP.CD&country=, respectively.

Several countries that played in FIFA World Cup tournaments during the study period experienced changes to their political designation or name. For example, FIFA allowed Serbia and Montenegro to compete under the name and flag of Serbia and Montenegro in the 2006 FIFA World Cup, although Montenegro had declared its independence just days before the 2006 World Cup. Serbia played as an independent nation without Montenegro at the 2010 World Cup.

Tournament point totals (3 for a win, 1 point for a draw, and 0 for a loss) increase when teams advance and play additional matches. The point totals reported in this study combine group-stage performance with knockout-stage advancement. Knockout-stage matches that end in a draw are decided by penalty shootouts. Penalty shootout victories are counted as draws, worth one point each. For example, Argentina’s 14 points in the 2022 World Cup are based on two victories in the group stage (6 points), two knockout stage wins (6 points), and two knockout stage shootout victories (one in the quarterfinals and one in the final, for an additional 2 points).

3. Methodology

To assess the effect of population size and GDP per capita on a country’s point total in each of the eight World Cups between 1994 and 2022, an ordinary least squares (OLS) regression of the following form was run:

Point s i,t = b 0 + b 1 Populatio n i,t + b 2 Population square d i,t + b 3 GDP/ N i,t + b 4 GDP/ N square d i,t + b 5 Europe+ b 6 Africa+ b 7 South America (1)

where Pointsi,t is country i’s point total (3 for a win, 1 for a draw, and 0 for a loss) in World Cup year t; Populationi,t is country i’s total population (in millions) as of July 1st in year t; GDP/Ni,t is country i’s GDP per capita (in thousands of current U.S. dollars) in year t; and Europe, Africa, and South America are regional dummy variables identifying the country’s location (i.e., all other countries in Asia, North America, and Oceania are excluded to avoid the so-called “dummy variable trap”). Population and GDP/N are each squared to capture a curved or inverted-U relationship between, say, Points and Population (or Points and GDP/N). Countries with large populations that have participated in the World Cup—China (2002), the United States (all years between 1994 and 2022 with the exception of 2018), and Nigeria (1994, 1998, 2002, 2014, 2018), to name a few—have struggled in the FIFA World Cup since 1994 to advance far in the knockout stages. Similarly, a linear model assumes that Points increase with GDP/N. But, rich countries that have participated in the World Cup between 1994 and 2022—Switzerland (all eight World Cups but 1998 and 2002), Norway (1994 and 1998), and Qatar (2022)—have rarely advanced to the later stages of the tournament. After Population and GDP/N are taken into account, three regional dummies capture the dominant effect countries from these continents have had on the tournament. Between 1994 and 2022, all finalists and all but three semifinalists (with the exceptions of South Korea and Türkiye in 2002 and Morocco in 2022) have come from only Europe and South America.

The resulting regression model in Equation (1) can be used to generate predicted points. The difference between actual points minus predicted points (hereafter, the residual) will be used to identify overachievers (a positive residual) and underachievers (a negative residual). Previous studies have relied on explaining variation in goal difference per game, the dependent variable, and not points (see Kuper & Szymanski, 2009: pp. 289-290) or performance in the preliminaries leading up to a World Cup (Sylla & Magel, 2016). But, in the World Cup group stage, points, not goal differential, determine which teams advance, while the difference between goals scored and allowed serves as the tiebreaker to separate teams that are tied on points.

4. The Results

Table 1 shows the regression results with and without the regional dummies. The regression results with the regional dummies show that Population and Population squared (b1 = 0.023393, p < 0.01 and b2 = −0.00002, p < 0.01) are significant at better than the 0.01 level. GDP/N and GDP/N squared (b3 = 0.09317, p = 0.032 and b4 = −0.00111, p = 0.038) are significant at better than the 0.05 level. Hawksworth et al. (2014) found that neither population nor GDP per capita explained how well a country in the World Cup performs. For both Population and GDP/N, the number of points is an inverted U-shaped relationship. That is, as population (or GDP per capita) increase, points also increase up to a certain point, after which points

decrease. The peak GDP per capita (where Points GDP/N =0 and GDP/N = b 3 / 2 b 4 )

occurs at $41,786, close to the GDP per capita of France in 2022 ($40,989). Points are significantly higher for countries in South America (b7 = 5.43548, p < 0.01) and Europe (b5 = 4.75824, p < 0.01), but not for countries in Africa (b6 = 1.53761, p = 0.106). The overall R2 for Equation (1) is 0.27.

Table 1. OLS Regression Results, FIFA World Cup Points, 1994-2022.

Dependent variable: World Cup points

Variable

With Regional Dummies

Without Regional Dummies

Population (millions)

0.023393***

0.01005*

(0.00548)

(0.0056)

Population squared

− 0.00002***

− 0.00001**

(5.06E−06)

(5.38E−06)

GDP per capita (thousands)

0. 09317**

0.14390***

(0.04308)

(0.03842)

GDP per capita squared

− 0.00111**

− 0.00169***

(0.00054)

(0.00053)

Europe

4.75824***

(0.71557)

Africa

1.53761

(0.9485)

South America

5.43548***

(0.87296)

Constant

0.37979

3.57718***

(0.80561)

(0.54751)

Observations (N)

248

248

R-squared

0.277

0.08

Adjusted R-squared

0.256

0.065

***p < 0.01, **p < 0.05, *p < 0.10, Standard errors in parentheses.

Despite differences in host status, qualification pathways, group composition, and tournament-specific competitive conditions, the estimated slope coefficients and their statistical significance are virtually the same when Equation (1) controls for tournament fixed effects. Although there are repeated observations of several of the same national teams during the study period (Argentina, Brazil, Germany, Mexico, and South Korea, and Spain appeared in all eight tournaments), country-clustered standard errors increase slightly but not enough to affect the significance tests on any of the predictor variables.

Equation (1) with the regional dummies can be used to estimate a country’s predicted number of points. For example, Brazil 2002 (with a 2002 population, in millions, of 178.503 and a 2002 GDP/N, in thousands of U.S. dollars, of 2.8559) would have a predicted number of points equal to 9.6363. Brazil 2002’s residual (i.e., actual points minus predicted points) is 11.3637, the largest residual and hence the sample’s biggest overachiever.

Table 2 lists all 248 country-tournament observations in the World Cup between 1994 and 2022, from biggest overachiever (largest positive residual) to biggest underachiever (largest negative residual).

Table 2. Over- and Under-achievers in the FIFA World Cup, 1994-2022.

Rank

Year

Country

Points

Predicted Points

Residual

1

2002

Brazil1

21

9.6363

11.3637

2

1998

France

19

8.1042

10.8958

3

2018

Belgium

18

7.3133

10.6867

4

2010

Netherlands

18

7.3671

10.6329

5

2018

France

19

8.569

10.431

6

2014

Germany

19

8.7959

10.2041

7

2010

Spain

18

7.9946

10.0054

8

2014

Netherlands

17

7.3219

9.6781

9

1998

Croatia

15

5.7395

9.2605

10

2006

Italy

17

8.3095

8.6905

11

2002

South Korea

11

2.5144

8.4856

12

2014

Argentina

16

7.7592

8.2408

13

2022

Morocco

11

3.0732

7.9268

14

1994

Brazil

17

9.3492

7.6508

15

2014

Costa Rica

9

1.3748

7.6252

16

2018

Croatia

14

6.4051

7.5949

17

2002

Germany

16

8.5876

7.4124

18

2022

France

16

8.5898

7.4102

19

2006

Germany

16

8.8557

7.1443

20

2006

France

15

8.4634

6.5366

21

2006

Portugal

13

6.7918

6.2083

22

1994

Italy

14

7.7928

6.2072

23

2010

Germany

15

8.8689

6.1311

24

2022

Argentina

14

7.9215

6.0785

25

2002

Türkiye

13

6.9404

6.0596

26

2002

Senegal

8

2.222

5.778

27

2010

Ghana

8

2.6169

5.3831

28

1994

Sweden

12

7.0124

4.9876

29

1998

Netherlands

12

7.2332

4.7668

30

2018

Uruguay

12

7.2749

4.7251

31

2014

Belgium

12

7.3015

4.6985

32

1994

Bulgaria

10

5.4399

4.5601

33

1994

Saudi Arabia

6

1.5374

4.4626

34

2014

Colombia

12

7.5511

4.4489

35

2010

Argentina

12

7.587

4.413

36

1994

Romania

10

5.7812

4.2188

37

2010

Uruguay

11

6.8924

4.1076

38

2022

Netherlands

11

7.1577

3.8423

39

2006

Argentina

11

7.2205

3.7795

40

2002

Mexico

7

3.2301

3.7699

41

2018

Russia

8

4.2797

3.7203

42

2022

Australia

6

2.3126

3.6875

43

2002

Spain

11

7.3422

3.6578

44

2022

Senegal

6

2.4682

3.5318

45

2006

Ghana

6

2.5297

3.4703

46

2022

Croatia

10

6.5683

3.4317

47

1998

Brazil

13

9.6697

3.3303

48

2014

Mexico

7

3.8232

3.1768

49

2002

Costa Rica

4

0.8354

3.1646

50

1998

Italy

11

7.9333

3.0667

51

1998

Argentina

10

7.3312

2.6688

52

2006

England

11

8.4276

2.5724

53

1998

Jamaica

3

0.7454

2.2546

54

2022

Switzerland

6

3.7894

2.2106

55

2018

Mexico

6

3.8223

2.1777

56

1994

Mexico

5

2.8499

2.1501

57

2022

Japan

7

4.9096

2.0904

58

2002

Japan

7

4.916

2.084

59

1998

Mexico

5

2.9457

2.0543

60

2010

Japan

7

4.9932

2.0068

61

2022

Portugal

9

6.9982

2.0018

62

2006

Brazil

12

10.0264

1.9736

63

1994

Netherlands

9

7.1139

1.8861

64

2018

Sweden

9

7.1539

1.8461

65

1994

Nigeria

6

4.2875

1.7125

66

2002

Cameroon

4

2.3534

1.6466

67

2018

Senegal

4

2.4178

1.5822

68

2022

Tunisia

4

2.5282

1.4718

69

2018

England

10

8.5469

1.4531

70

2022

England

10

8.5484

1.4516

71

2014

France

10

8.5495

1.4505

72

1998

Yugoslavia

7

5.564

1.436

73

2010

Ivory Coast

4

2.5758

1.4242

74

2022

Costa Rica

3

1.5824

1.4176

75

1998

Nigeria

6

4.6113

1.3887

76

1994

Germany

10

8.6254

1.3746

77

2014

Switzerland

6

4.628

1.372

78

1998

Germany

10

8.644

1.356

79

2018

Iran

4

2.6721

1.3279

80

2022

Cameroon

4

2.6958

1.3042

81

1998

Morocco

4

2.7013

1.2987

82

2010

Australia

4

2.709

1.291

83

2006

Australia

4

2.7676

1.2324

84

1998

Romania

7

5.8236

1.1764

85

2006

Spain

9

7.8863

1.1137

86

2006

Switzerland

8

6.8925

1.1075

87

2002

China

0

−1.0375

1.0375

88

1998

Iran

3

1.9652

1.0348

89

2006

South Korea

4

3.0046

0.9954

90

1994

Spain

8

7.0819

0.9181

91

2010

South Korea

4

3.0887

0.9113

92

2002

South Africa

4

3.2389

0.7611

93

2014

Algeria

4

3.3315

0.6685

94

2010

New Zealand

3

2.3561

0.6439

95

2006

Ukraine

7

6.4154

0.5846

96

2022

South Korea

4

3.43

0.57

97

2006

Mexico

4

3.4534

0.5466

98

2006

Ivory Coast

3

2.5029

0.4971

99

2018

Tunisia

3

2.5134

0.4866

100

2010

Mexico

4

3.5909

0.4091

101

2014

Ivory Coast

3

2.6607

0.3393

102

2014

Brazil

11

10.703

0.297

103

2022

Iran

3

2.7352

0.2648

104

2010

Honduras

1

0.7466

0.2534

105

2010

South Africa

4

3.7613

0.2387

106

2018

Saudi Arabia

3

2.8432

0.1568

107

2022

Ghana

3

2.874

0.126

108

2022

Qatar

0

−0.0654

0.0654

109

2022

Mexico

4

3.9886

0.0114

110

2018

Switzerland

5

5.0031

−0.0031

111

2022

Saudi Arabia

3

3.0473

−0.0473

112

2002

Denmark

7

7.1288

−0.1288

113

1998

Denmark

7

7.1289

−0.1289

114

2002

England

8

8.2552

−0.2552

115

1994

South Korea

2

2.2575

−0.2575

116

1998

Cameroon

2

2.3182

−0.3182

117

2014

Chile

7

7.3455

−0.3455

118

2010

Paraguay

6

6.3664

−0.3664

119

2006

Ecuador

6

6.4315

−0.4315

120

2018

South Korea

3

3.4362

−0.4362

121

2006

Netherlands

7

7.4496

−0.4496

122

2002

United States

7

7.4506

−0.4506

123

2018

Costa Rica

1

1.4936

−0.4936

124

2018

Colombia

7

7.4993

−0.4993

125

1998

Saudi Arabia

1

1.5332

−0.5332

126

2010

Brazil

10

10.5437

−0.5437

127

2002

Russia

3

3.5888

−0.5888

128

2018

Brazil

10

10.5913

−0.5913

129

2006

Angola

2

2.6414

−0.6414

130

2022

Brazil

10

10.6546

−0.6546

131

1994

Russia

3

3.6688

−0.6688

132

2014

Honduras

0

0.788

−0.788

133

2018

Denmark

6

6.794

−0.794

134

2006

Costa Rica

0

0.9395

−0.9395

135

1994

Belgium

6

6.9749

−0.9749

136

2018

Japan

4

4.9828

−0.9828

137

2010

The Democratic People's Republic of Korea

0

1.0054

−1.0054

138

2002

Republic of Ireland

6

7.0845

−1.0845

139

1998

Paraguay

5

6.1015

−1.1015

140

1998

South Korea

1

2.1391

−1.1391

141

2006

Saudi Arabia

1

2.1655

−1.1655

142

1998

England

7

8.1874

−1.1874

143

2010

Chile

6

7.2107

−1.2107

144

2014

Uruguay

6

7.2265

−1.2265

145

1994

Argentina

6

7.2333

−1.2333

146

1998

South Africa

2

3.2539

−1.2539

147

1994

Cameroon

1

2.2762

−1.2762

148

2006

Iran

1

2.3159

−1.3159

149

1998

Tunisia

1

2.3459

−1.3459

150

2002

Tunisia

1

2.359

−1.359

151

2006

Tunisia

1

2.4546

−1.4546

152

2022

United States

5

6.5351

−1.5351

153

2018

Australia

1

2.6263

−1.6263

154

2014

Iran

1

2.6515

−1.6515

155

2002

Saudi Arabia

0

1.654

−1.654

156

2018

Panama

0

1.691

−1.691

157

2014

Ghana

1

2.7357

−1.7357

158

2010

Switzerland

4

5.808

−1.808

159

2014

Greece

5

6.8773

−1.8773

160

2014

Nigeria

4

5.8798

−1.8798

161

2010

Portugal

5

6.9175

−1.9175

162

2018

Portugal

5

6.9521

−1.9521

163

2018

Spain

6

7.9976

−1.9976

164

2002

Belgium

5

7.0105

−2.0105

165

2018

Morocco

1

3.0437

−2.0437

166

2002

Paraguay

4

6.0632

−2.0632

167

2006

Togo

0

2.1023

−2.1023

168

1998

Norway

5

7.1291

−2.1291

169

2002

Sweden

5

7.136

−2.136

170

2010

Algeria

1

3.1696

−2.1696

171

2006

Sweden

5

7.2705

−2.2705

172

2014

South Korea

1

3.3263

−2.3263

173

2014

Australia

0

2.372

−2.372

174

1994

Republic of Ireland

4

6.4215

−2.4215

175

2014

Russia

2

4.4439

−2.4439

176

2010

Cameroon

0

2.4982

−2.4982

177

2010

Slovakia

4

6.5198

−2.5198

178

2014

Cameroon

0

2.5748

−2.5748

179

1994

Morocco

0

2.6372

−2.6372

180

2010

United States

5

7.6737

−2.6737

181

2014

Bosnia and Herzegovina

3

5.676

−2.676

182

2014

Ecuador

4

6.7366

−2.7366

183

2010

Slovenia

4

6.7528

−2.7528

184

2002

Croatia

3

5.7747

−2.7747

185

2022

Ecuador

4

6.7859

−2.7859

186

2010

Serbia

3

5.8242

−2.8242

187

2014

Portugal

4

6.8948

−2.8948

188

2002

Argentina

4

6.9092

−2.9092

189

1994

United States

4

6.9312

−2.9312

190

2018

Serbia

3

5.9411

−2.9411

191

2022

Canada

0

2.9673

−2.9673

192

1994

Norway

4

7.0072

−3.0072

193

2022

Spain

5

8.0122

−3.0122

194

2018

Nigeria

3

6.0922

−3.0922

195

2006

Paraguay

3

6.1641

−3.1641

196

1998

Spain

4

7.2174

−3.2174

197

1994

Switzerland

4

7.2431

−3.2431

198

2022

Belgium

4

7.268

−3.268

199

2002

Ecuador

3

6.3064

−3.3064

200

2014

Croatia

3

6.3334

−3.3334

201

2022

Poland

4

7.3357

−3.3357

202

2022

Uruguay

4

7.3598

−3.3598

203

2002

Portugal

3

6.3985

−3.3985

204

2002

Poland

3

6.4606

−3.4606

205

2010

England

5

8.4729

−3.4729

206

2006

Czech Republic

3

6.5377

−3.5377

207

1998

Chile

3

6.637

−3.637

208

2014

United States

4

7.6388

−3.6388

209

2006

Poland

3

6.7556

−3.7556

210

2018

Argentina

4

7.7627

−3.7627

211

2002

Nigeria

1

4.7635

−3.7635

212

1994

Colombia

3

6.824

−3.824

213

1998

Colombia

3

6.9071

−3.9071

214

2010

Denmark

3

6.917

−3.917

C215

2002

Italy

4

7.9409

−3.9409

216

2006

Japan

1

4.9707

−3.9707

217

2014

Japan

1

4.9853

−3.9853

218

1998

Belgium

3

7.0198

−4.0198

219

2010

Greece

3

7.0872

−4.0872

220

2018

Peru

3

7.1379

−4.1379

221

2006

Croatia

2

6.1627

−4.1627

222

2018

Poland

3

7.1843

−4.1843

223

2002

Uruguay

2

6.2592

−4.2592

224

2018

Egypt

0

4.3998

−4.3998

225

2010

Nigeria

1

5.4823

−4.4823

226

1998

Bulgaria

1

5.4958

−4.4958

227

2022

Germany

4

8.8129

−4.8129

228

2018

Iceland

1

5.8157

−4.8157

229

1998

Japan

0

4.8873

−4.8873

230

2014

Spain

3

7.9675

−4.9675

231

1998

Austria

2

7.0333

−5.0333

232

1994

Bolivia

1

6.0663

−5.0663

233

2022

Serbia

1

6.1151

−5.1151

234

2014

Italy

3

8.389

−5.389

235

2022

Denmark

1

6.4688

−5.4688

236

2006

Serbia and Montenegro

0

5.7065

−5.7065

237

2018

Germany

3

8.8355

−5.8355

238

1998

Scotland

1

6.882

−5.882

239

2006

Trinidad and Tobago

1

6.941

−5.941

240

2022

Wales

1

7.0718

−6.0718

241

2002

Slovenia

0

6.1181

−6.1181

242

1994

Greece

0

6.2675

−6.2675

243

2010

Italy

2

8.3762

−6.3762

244

2006

United States

1

7.5745

−6.5745

245

2002

France

1

8.1104

−7.1104

246

1998

United States

0

7.2296

−7.2296

247

2014

England

1

8.4766

−7.4766

248

2010

France

1

8.5234

−7.5234

1Italicized names are winners of the men’s FIFA World Cup.

The three biggest overachievers are Brazil 2002, France 1998, and Belgium 2018. Brazil did exceptionally well in the 2002 World Cup, winning all seven of their matches (without needing extra time or penalty kicks) and that year’s championship. France hosted the tournament in 1998 and defeated Brazil 3-0 in the final to claim their first-ever world title. Belgium did incredibly well in the 2018 World Cup. They finished in third place, which was the country’s best-ever result in the tournament’s history.

The three biggest underachievers are the United States 1998, England 2014, and France 2010. The United States in the 1998 World Cup lost all three group stage games and finished in last place in their group. England in the 2014 FIFA World Cup hosted by Brazil finished at the bottom of their group, losing games to Italy and Uruguay and recording a draw with Costa Rica. England finished with just one point, their worst-ever performance in the tournament’s history. The biggest underachiever was France in 2010 which was eliminated in the first round after finishing last in their group.

5. Concluding Remarks

A regression model using a country’s population and GDP per capita at the time of its participation in the FIFA World Cup is estimated to predict points earned by countries between 1994 and 2022. A comparison between actual points earned and predicted points enables one to easily identify overachievers (actual points earned exceed predicted points) as well as underachievers (predicted points exceed actual points).

Future researchers might estimate a regression model to predict points in the Women’s FIFA World Cup which also operates on a four-year cycle. Although 32 countries will participate in the FIFA Women’s FIFA World Cup in 2027 hosted by Brazil, an extension of the work presented here for the men’s FIFA World Cup to include the results of the 2026 World Cup must recognize that the 48 countries in 2026 could earn more points. That is, the 2026 tournament added a new Round of 32 knockout phase. Spain and Argentina will have played a total of eight (not seven) games each by the conclusion of the 2026 World Cup final.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1] Hawksworth, J., Zimmern, W., & Broadfield, D. (2014). The PwC World Cup Index: What Can the Dismal Science Tell Us about the Beautiful Game?
https://www.pwc.com/im/en/assets/document/pwc_world_cup_index_-_june_2014.pdf
[2] Kuper, S. & Szymanski, S. (2009). Soccernomics.
https://www.scribd.com/document/980987303/Soccernomics-PDF
[3] Sylla, M. & Magel, R. (2016). Predicting the Winner of Games in World Cup Soccer Matches. Journal of Advance Research in Mathematics and Statistics, 3, 10-21. [Google Scholar] [CrossRef]

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