"IB Mathematics SLInternal AssessmentIs there a correlation between the distances covered byPremier League teams and the number of pointsaccumulated?Candidate Name: Candidate Number: School Name: Teacher Name:Table of ContentsIntroduction .............................................................................................................................. 2Definitions ................................................................................................................................. 3Data ........................................................................................................................................... 4Table 1– Teams positions and distance covered............................................................... 5Least squares regression line for points (Y) on distance covered (X) ............................. 7Graph– Distance covered vs Points scored by the Teams .............................................. 9Analysis and Results .............................................................................................................. 10Conclusion .............................................................................................................................. 12Citation.................................................................................................................................... 12IntroductionEver since the bare age of 6, I?ve been a lifelong follower of the world?s mostluxurious football league, the English Premier League. The team I grew to love and supportpassionately is Manchester United Football Club. Born into a family of “Red Devils”supporters, I did not really have much of a choice. The older I got and the greater myunderstanding of football developed, I began to comprehend the reason behind my family?slove towards them. Through every game I?ve watched, I noticed that every opponent has adifferent rhythm and playing style. The further I investigated into it, I realized that everyteam?s strategy or tactic depends on the manager?s given instructions as well as thecapabilities of the available players. Moreover, I?ve noticed that during large rivalry gamesthe managers adopt a different approach to the match. This is due to the massiveresponsibility they?re tasked with to secure the 3 points and make their fans proud duringthese top tier match ups. The hate between the two rival teams? fans further adds to thepressure exerted on the manager?s and players? shoulders. The infamous tactic managersadopt in big games is to cover as much distance of the field, in order to prevent and shutdown the channels for the opposition to create a goal scoring opportunity. A prime exampleof this would be the 2015/16 Champions, Leicester City. As a result, in my Mathematicsstandard level exploration I?ve taken it upon myself to investigate whether there is an existingrelationship between the distances covered by a team with their overall performance.Through using descriptive statistics as my area of studies to prove my research question. Iwill use my boyhood club Manchester United during the 2015/16 season and compare theirresults with the champions. Leading me to the question: Is there a correlation between thedistances covered by Premier League teams and the number of points accumulated?The scope of this exploration focuses on figuring out how and why certain PremierLeague teams choose to implement a given play style. As well as whether this play styleguarantees the teams a greater aggregate of points or not. First, I will introduce a few football terms and concepts that will be used throughout the exploration. Then, I will move on toexplore ratios and correlations between the distances covered by Premier League teams.Limiting my investigation to the 2015-2016 season and focusing on two teams - my boyhoodclub, Manchester United and the season?s Champions, Leicester City. Finally, I will connectthe different mathematical sides to conclude the validity of the claim, “Greater Distancecovered results in Greater Teams.”DefinitionsFootball is a game played between two teams made up of 11 men a side. In order towin, the team has to score more goals at the end of the 90-minute match. The Premier Leagueis an English professional league for men's association football clubs. Contested by 20 clubs,it operates on a system of promotion and relegation with the English Football League alsoknown as "The English Premier League"The Premier League is a corporation in which the 20 member clubs act as contenders.Seasons run from August to May. Teams play 38 matches each (playing each team in theleague twice, home and away), totaling 760 matches in the season.A Premier League earns 3 points after securing a win over the opponent. Whendrawing level (both teams scoring the same amount of goals), the points are split evenly witheach team claiming one point. When a team is defeated, 0 points are allocated to the losingside. After the 38 games in the season, the team with greatest number of accumulated pointsis crowned as champions. The distance covered by Premier League teams is total amount of ground runthroughout a football game by the players. In order to calculate the amount of groundcovered, we have taken into account the distance each individual player has covered andfound the mean or the total average of the collective team.Counter attacking Football involves a team withdrawing players into their own halfbut ensuring that one or two players are committed to the attack. The idea is to try to getstrikers isolated from the defenders by lurking on the halfway line, and to create space for aquick break. Possession football is a style of play in football characterized by short passing andmovement, working the ball through various channels, and maintaining possession of theball.DataThe first set of data collected will be the amount of points each team accumulated aswell as their respective positions for the 2015-2016 season. The average distance covered byeach Premier league team is taken as Y and the average points collected by each Premierleague team is taken as X.The following table is made to understand the correlation between X & Y.Table 1– Teams positions and distance covered.Pos Teamitio 2 2 n X(Points) Y (Km) X Y XY1Leicester 81 10.97738984 6561 120.5030877 889.1685772Arsenal 71 11.17815526 5041 124.951155 793.64902353Spurs 70 11.81045816 4900 139.4869219 826.73207124Man City 66 10.69947778 4356 114.4788248 706.16553355Man Utd 66 10.6954338 4356 114.3923042 705.89863086Southampton 63 11.26083443 3969 126.8063921 709.43256917West Ham 62 10.91748636 3844 119.1915084 676.88415438Liverpool 60 11.88441589 3600 141.239341 713.06495349Stoke 51 10.7928232 2601 116.4850326 550.433983210Chelsea 50 11.18355948 2500 125.0720026 559.17797411Everton 47 10.99432853 2209 120.8752598 516.73344091247 11.40034564 2209 129.9678807 535.8162451Swansea13Watford 45 10.78675824 2025 116.3541533 485.404120814West Brom 43 11.50069067 1849 132.2658859 494.52969881542 10.63022947 1764 113.0017786 446.4696377Crystal Palace16Bournemouth 42 11.98207606 1764 143.5701467 503.247194517Sunderland 39 10.70410396 1521 114.5778416 417.460054418Newcastle 37 10.81605308 1369 116.9870042 400.19396419Norwich City 34 10.92825156 1156 119.4266822 371.56055320Aston Villa 17 10.66664896 289 113.7774 181.3330323? ? = ? ? ? =1033 ? = 221.8095204 ? =57883 ? =2463.410603 11483.35541 Using the data from the table, we find out the Pearson?s correlation coefficient (r) which isthe degree of linearity between two random variables X and Y, given „n? ordered pairs(x ,y ), (x ,y ), (x , y )………(x ,y ) of data is1 1 2 2 3 3 n n ? ? ? ? =? ? ? ? Where „r? is called Pearson?s correlation coefficient?? S is called the covariance of X and Y, Sxy = ? ? - xy ?S is the standard deviation of X, x ? ( ? )? ? = ? -? ? S is the standard deviation of Y,y SolvingS S and S xy, x ywe get ,? ( ? )? ? = ? -? ?? ? Sxy = ? ? - ? 1033 221.80952 ? = 11483.3554 -? ? 20 ? = ? ? .? ? ? ? ? ?? ? ? ( ? )? ? = ? -? ? 21033 ? = 577883 -? 20 ? = ? ? .? ? ? ? ? ?? ? ( ? )? ? = ? -? ?2221.80952 ? = 2463.410608 -? 20 ? = ? .? ? ? ? ? ? ? ? ?? Finally Substituting the values of S S and S in the equationxy, x y ? ? ? ? = ? ? ? ?we get , ? ? .? ? ? ? ? ? = ? ? .? ? ? ? ? ?? .? ? ? ? ? ? ? ? ?? = ? .? ? ? ? ? ? ? ? ? ?Least squares regression line for points (Y) on distance covered (X) A statistical method used to determine a line of best fit by minimizing the sum ofsquares created by a mathematical function. A "square" is determined by squaring thedistance between a data point and the regression lineThe regression line equation is given by : ? - ? = ? ? - ? ? ?To plot the graph using technology In the MS Excel spreadsheet, enter the data in columns A and B as shown below,Select the column A and B, click on SCATTER option in INSERT toolbar, which resultsin giving a graph as shown below:Graph – Distance covered vs Points scored by the Teamsy = 0.005x + 10.78 Distance covered vs Points R² = 0.046 12.2 12 11.8 11.6 11.4 11.2 11 10.8 10.6 10.4 0 10 20 30 40 50 60 70 80 90 Points? X axis To calculate the slope „m?, we use the relation,? ? ? ? =? (? ) ? S is called the covariance of X and Y xy S is the standard deviation of Xx ( ? )( ? ) ? = ? ? -? ? ?1033221.80952? = 11483.3554 -? ? 20 ? = ? ? .? ? ? ? ? ?? ? ? ( ? ) ? ?? = ? -? ? 21033 2? = 57883 -? 20 2? = 4528.55? ? ? .? ? ? ? ? ? ? =? ? ? ? .? ? ? = ? .? ? ? ? ? ?Distance ?y axis"