Microsoft Word - Math Ia - Final Final [1]

Math Internal Assessment

The Relationship Between the Total Points Earned and the Number of Yellow Cards

Received During the 2003 – 2004, 2004 – 2005 , 2005 – 2006, 2006 – 2007, 2007 –

2008 seasons of the English Premier League.

Alisha Narula

IB Math Studies

Mr. Clement

International School of Bangkok

November 27, 2009

000307 - 149

Alisha Narula

November 17, 2009

Math Studies

Math Internal Assessment

Title: The Relationship Between the Total Points Earned and the Number of

Yellow Cards Received During the 2003 – 2004, 2004 – 2005 , 2005 – 2006,

2006 – 2007, 2007 – 2008 seasons of the English Premier League.

Introduction: The managers of the premier leagues teams need to go over most of the

statistics produced during the previous seasons, in order to make certain

necessary predictions for a proper game plan for the coming seasons. Also,

buying and selling of the players becomes very interesting based on the

previous statistics of points earned, yellow cards, red cards, and other criteria.

The red cards are more of a serious matter on which the referees have been

criticized in many cases. However the yellow cards obtained more or less mild

enough to relate to the total number of points earned by each team. Comparing

and contrasting the two main criteria’s of the statistics the total points earned

and the total number of yellow cards receive should give enough of an idea to

the managers and the committee members to make logical and important

decisions, in order to succeed in the forth coming seasons.

Using the most common logical sense it shows that the teams earning

more yellow cards should be the teams not playing well enough of a fair game

ending up in less points for an overall season. In contrary to this, it is quite

evident that the better teams or rather the top ten, should be having much

less yellow cards in number, thereby depicting better players and higher

efficient strategies enforced. This is more or less, a very logical understanding

of the situation. In order to prove or see the relation, a statistical investigation

is going to be conducted where 2003 – 2008 annual season statistics are taken

and dealt in detail with various numbers of statistical data, graphs, and other

calculations with the help of which clear predictions could be seen.

In order to, move on with further investigations, I start with easier

statistics heading into the more complicated and sophisticated ones. Finding

the mean, median, mode, lower and upper quartiles, range, standard deviation,

and scatter plots if possible would further enhance for the easier parts of the

statistics. On the other hand, the complicated calculations such as the

Pearson’s Correlations Coefficient, the r value, value, the linear regression

line with the scatter plot and finally leading to the Chi Squared Value, where

some assumptions will be made and see if the hypothesis is independent or

not.

Task:

I would like to find whether or not there exists a relationship between

the total points earned and the yellow cards obtained by each tem of various

seasons of the English Premier League.

Table # 1: Season 2003 – 2004 Showing Total Points and Yellow Cards Obtained by

20 Teams of the English Premier League

Caption # 1: The table above depicts 20 various teams part of the English Premier

League, with their total points earned and total number of ye

during the 2003 – 2004 season, and also including the mean of both

Mathematical Process

Mean Number of Total Points:

2004 Showing Total Points and Yellow Cards Obtained by


: The table above depicts 20 various teams part of the English Premier

League, with their total points earned and total number of yellow cards obtained

2004 season, and also including the mean of both x and y.

Mean Number of Total Points:


: The table above depicts 20 various teams part of the English Premier

llow cards obtained

y.

=

= 51.6 ( 3 significant figures )

Using the GDC Statistical Program the following calculations are made

Minimum Value: 3

Lower Quartile: 42.5

Median: 49

Upper Quartile: 56

Max Value: 90

Mean Number of Total Number of Yellow Cards

=

= 64.6 (3 significant figures)

Using the GDC Statistical Program the following calculations are made

Minimum Value: 40

Lower Quartile: 58

Median: 63

Upper Quartile: 71.5

Max Value: 89

Dispersion of spread of the data is another very important context for which the

standard deviation can be calculated. How far away from the mean are the data spread

about would be clearly stated by the standard deviation.

The formula used for this mathematical calculation is:

Standard Deviation:

Standard Deviation Table # 2:

(x) during the Season 2003 –

Standard Deviation

=

=


Therefore, SD of (X) is: 14.8

Caption # 2: The table and calculations above depict the mathematical process of

Standard Deviation Calculation for the total points earned (

2003 – 2004 of the English Premier League.

Standard Deviation Table # 2: Standard Deviation Table for the Total Points Earned

2004 of the English Premier League

(3 significant figures)

14.8, which show the deviation from the mean on both sides.

The table and calculations above depict the mathematical process of

Standard Deviation Calculation for the total points earned (x) during the season

2004 of the English Premier League.

Standard Deviation Table for the Total Points Earned

, which show the deviation from the mean on both sides.


) during the season


Yellow Cards (y) during the Season 2003

Standard Deviation =

=

=

= 119 (3 significant figures)

Therefore, SD of (Y) is 119

Caption # 3: The table & calculations

Standard Deviation Calculation for the total number of yellow cards

during the season 2003 – 2004, English Premier League.

Standard Deviation Table # 3: Standard Deviation Table for the Total Number of

) during the Season 2003 – 2004 of the English Premier League



The table & calculations above depict the mathematical process of

Standard Deviation Calculation for the total number of yellow cards (y) obtained

2004, English Premier League.

for the Total Number of



the mathematical process of

obtained






Standard Deviation Table # 5

(x) during the Season 2004



The table above depicts 20 various teams part of the English Premier




during the Season 2004 – 2005 of the English Premier League




y.




=

=



Caption # 5: The table and calculations above depict

Standard Deviation Calculation for the total earned points

– 2005, English Premier League.




=

=


Therefore, SD of (Y) is 81.4, which show the deviation from the mean on both sides.



and calculations above depict the mathematical process of

Standard Deviation Calculation for the total earned points (x) during the season 2004






mean on both sides.

the mathematical process of

during the season 2004

Standard Deviation Table for the Total Number of



Caption # 6: The table & calculations above depict the mathematical process of

Standard Deviation Calculation for the total number of yellow cards (

during the 2004 – 2005 season, in the



Caption # 7: The table above depicts 20 various teams part of the English Premier League, with

their total points earned and total number of yellow cards obtained

also including the mean of both x and

Standard Deviation Table #

(x) during the Season 2005– 2006 of the English Premier League.



2005 season, in the English Premier League.



The table above depicts 20 various teams part of the English Premier League, with

points earned and total number of yellow cards obtained during the 2005 – 2006 season, and

and y.


2006 of the English Premier League.


) obtained


The table above depicts 20 various teams part of the English Premier League, with

season, and



=

=




Standard Deviation Calculation for the total earned points (





=

=







, English Premier League.







the season 2005

Number of

of the English Premier League




during the 2005 – 2006 season, in the English Premier League.

Table # 10: Season 2006 – 2007 Showing Total Points and Yellow Cards Obtained

by 20 Teams of the English Premier League


League, with their total points earned and total number of ye




season, in the English Premier League.

2007 Showing Total Points and Yellow Cards Obtained






ined



llow cards obtained

y.


Earned (x) during the Season 2006


=

=






Standard Deviation Table # 11: Standard Deviation Table for the Total Points

during the Season 2006– 2007 of the English Premier League.





, English Premier League.

Standard Deviation Table for the Total Points



) during the season 2006




=

=





during the 2006 – 2007








the Total Number of

of the English Premier League



ined

Table # 13: Season 2007 – 2008 Showing Total Points and Yellow Cards Obtained



League, with their total points earned and total number









of yellow cards obtained

y.


Earned (x) during the Season 2007


=

=






Standard Deviation Table # 14: Standard Deviation Table for the Total Points

during the Season 2007– 2008 of the English Premier League.






Standard Deviation Table for the Total Points



) during the season 2007




=

=





during the 2007 – 2008 season, in the English Premier League.








e Total Number of




) obtained

MEASURING CORRERELATION

Dealing with the linear association, a concept called correlation is used to

measure the strength and direction. The correlation coefficient lies between

The r-value of 0 shows no linear association at all,

and positive correlation respectively. The positive correlation shows an increase in

one variable resulting in an increase in the other. The negative correlation shows an

increase in one variable resulting in a decrease in the other. Using the data

points earned) and y (the total number of yellow cards), Pearson’s’ Correlation

Coefficient is calculated showing the degree of linearity between the two variables

and y. In order to do that, a table of values showing

a particular formula which is given below is used to calculate the

Certainly a revision of the results will be corrected by enforcing the GDC function.

Formula for Pearson’s Correlation Coefficient

Table # 16: The Table of Values depicting the values from the 2003

the English Premier League

r- value :

MEASURING CORRERELATION


measure the strength and direction. The correlation coefficient lies between –

of 0 shows no linear association at all, -1 and 1 shows perfect negative



one variable resulting in a decrease in the other. Using the data x,

(the total number of yellow cards), Pearson’s’ Correlation

Coefficient is calculated showing the degree of linearity between the two variables

In order to do that, a table of values showing x, y, xy, , and is created and

a particular formula which is given below is used to calculate the r-value manually.


ula for Pearson’s Correlation Coefficient (r-value):

The Table of Values depicting the values from the 2003 – 2004 season of


– 1 and 1.

1 and 1 shows perfect negative



x, (the total

(the total number of yellow cards), Pearson’s’ Correlation

Coefficient is calculated showing the degree of linearity between the two variables x,

is created and

manually.


2004 season of

r = - 0.592 = 0.350 = 35.0% The -0.592 r – value shows a moderate negative relationship between the total points

earned, x, and the total number of yellow cards obtained, y. Furthermore, there is a

35.0% correlation evident.

Likewise the “r” value for other years is calculated way the statistical mode on the

GDC.

Seasons “r values” “ values”

2003 – 2004 - 0.592 0.350 = 35.0%

2004 – 2005 - 8.32 6.92 = 692%

2005 – 2006 0.0378 0.0015 = 0.15%

2006 – 2007 -0.148 0.0218 = 2.18%

2007 – 2008 -.0.428 0.183 = 18.3%

Caption # 16: The tables and calculations above depict the Pearson’s Correlation

Coefficient also known as the r – value, which is – 0.592.

LEAST SQUARES REGRESSION

The next part of the investigation consists of finding the least squares

regression line or, also known as the line of best fit. This along with the scatter plot

would at a glimpse depict a clear picture of the relationship between the variables.

Not only does it serve this purpose, but it also aids in calculating an equation, which

holds a great importance in further calculating the values outside the graph drawn,

that is extrapolation. Interpolation is not at all a problem, as that can be directly seen

from the graph. Once again, a similar table of values is used to find the equation of

least square regression.

Table # 17: The Table of Values depicting the values from the 2003 – 2004 season of

the English Premier League.

Caption # 17: Using this chart

regression.

Linear Regression Formula:

The details to work out the equation:

Using this chart, values will be used in order to calculate the line of

Linear Regression Formula:

The details to work out the equation:

values will be used in order to calculate the line of


Putting all the details in the main equation:

The scatter plot showing the line of best fit is shown as follows:

Graph # 1: Scatter Plot of Data from Season 2003

Line & it’s Equation

Caption # 1: As seen above, the scatter plot depicts a negative and weak correlation,

having a linear regression line of

0

20

40

60

80

100

0 20

Nu

mb

er

of

Ye

llo

w C

ard

s

Scatter Plot of Yellow Cards vs. Goal Points Earned


Putting all the details in the main equation:

The scatter plot showing the line of best fit is shown as follows:

Scatter Plot of Data from Season 2003 – 2004 with a Linear Regression

As seen above, the scatter plot depicts a negative and weak correlation,

having a linear regression line of y= -0.437x + 87.16

y = -0.4374x + 87.167

40 60 80 100Goal Points

Scatter Plot of Yellow Cards vs. Goal Points Earned

for 2003-2004

2004 with a Linear Regression


Furthermore, using the GDC Function and the

the linear regression line formula. The line

the other seasons. Specifically seasons 2004

2008 during the English Premiere League.

Season 2004 – 2005: Table # 18: The Table of Values depicting the values from the 2004

of the English Premier League.

Caption #18: Using this chart, values will be used in order to calculate the line of

regression. (On the calculator)





0

10

20

30

40

50

60

70

80

0 20

Ye

llo

w C

ard

s

Scatter Plot for Yellow Card vs. Goal Points Earned

Furthermore, using the GDC Function and the x, y, xy, , and table of values, and

the linear regression line formula. The linear regression line equation was found for

the other seasons. Specifically seasons 2004- 2005, 2005-2006, 2006-2007, and 2007

English Premiere League.

The Table of Values depicting the values from the 2004 – 2005 season




Scatter Plot of Data from Season 2004– 2005 with a Linear Regression


having a linear regression line of y= 0.002x + 51.45.

y = 0.0022x + 51.454

40 60 80 100

Goal Points Earned

Scatter Plot for Yellow Card vs. Goal Points Earned

for 2004-2005

table of values, and

ar regression line equation was found for

2007, and 2007-

2005 season


2005 with a Linear Regression


Season 2005 – 2006

Table # 19: The Table of Values depicting the values from the 2005






Caption #3: As seen above, the scatter plot depicts a negative and weak correlation,


0

10

20

30

40

50

60

70

80

0 20

Ye

llo

w C

ar

ds

Scatter plot of Yellow Cards vs. Goal Points Earnerd

for season 2005

The Table of Values depicting the values from the 2005 – 2006 seaso




er Plot of Data from Season 2005– 2006 with a Linear Regression



y = -0.0118x + 59.182

40 60 80 100Goal Points Earned

Scatter plot of Yellow Cards vs. Goal Points Earnerd

for season 2005 - 2006

2006 season


with a Linear Regression


Season 2006 – 2007

Table # 20: The Table of Values depicting the value








0

20

40

60

80

100

0 20

Ye

llo

w C

ard

s

Scatter Plot of Yellow Cards vs.

Goal Points Earned for 2006

The Table of Values depicting the values from the 2006 – 2007




er Plot of Data from Season 2006– 2007 with a Linear Regression


having a linear regression line of y= -0.103x + 66.61.

y = -0.1034x + 66.618

40 60 80 100Goal Points

Scatter Plot of Yellow Cards vs.

Goal Points Earned for 2006 -

2007

2007 season




Season 2007 – 2008:

Table # 21: The Table of Values depicting the value





Line & its Equation

Caption # 5 As seen above, the scatter plot depicts a negative and weak correlation,


0

10

20

30

40

50

60

70

80

90

0 20 40

The Table of Values depicting the values from the 2007 – 2008




er Plot of Data from Season 2007- 2008 with a Linear Regression



y = -0.2304x + 72.481

60 80 100

Series1

Linear (Series1)

season of




Table #22: This table depicts the least square regression for annuals seasons

beginning from 2003 – up until 2008.

Seasons Least Square Regression

2003 – 2004

2004 – 2005 y= -0.002x + 51.45.

2005 – 2006 y= -0.011x + 59.18

2006 – 2007 y= -0.103x + 66.61

2007 – 2008 y= -0.230x + 72.48

Caption # 22: From the table above it is noted, that all the least square regression

equations for the following seasons from 2003 – 2008 have had a negative and rather

weak correlation.

Test of Independence

This test is done to show whether the data are independent of each other. Should one

set affect the other is the question that has to be answered. Usually the test finds

the difference between the observed and expected value by using the formula:

Test of Independence Formula:

Where is an observed frequency, and is an expected frequency.

Table # 23: This is the Contingency Table of the Observed Values for annual seasons

beginning from 2003 – 2008 in the English Premier League.

Observed Values Table

TOTAL

20

30

50

32

18

50

TOTAL

52

48

100

Caption # 23: The Contingency table above depicts observed values.

Yellow

Points

Total

Points

Table # 24: This is the Contingency Table of the Expected Values and the Process for

finding Expected Values for annual seasons beginning from 2003 – 2008 in the

English Premier League.

Expected Values Process & Formula Table

Yellow Cards

Total Points

TOTAL

Expected value =

= = 26

Expected value =

= =24

b = 50

Expected value =

= = 26

Expected value =

= =24

d =50

TOTAL a = 52 c =48 100

Caption # 24: Through the table above the processes for finding expected values is

clearly depicted.

Title # 25: This is the Contingency Table of the Expected Values for annual seasons

beginning from 2003 – 2008 in the English Premier League.

Expected Values Table

Yellow Cards

Total Points

TOTAL

26

24

50

26

24

50

TOTAL 52 48 100

Caption # 25: The contingency table above depicts expected values.

The calculation of the contingency table proceeds to the calculation of the

manually. For the calculation, the table is created as follows:

Table # 26: The following table is created to manually calculate the value.

20 26 (20 – 26) 36 1.385

30 24 (30 - 24) 36 1.5

32 26 (32 - 26) 36 1.385

18 24 (18 - 24) 36 1.5

Total: 5.77

Null Hypothesis :

The total points earned and the number of yellow cards obtained are independent.

Alternative Hypothesis :

The two related matters are not independent.

Degree of Freedom

Formula :

(r – 1) (c – 1) = (2 – 1) (2 – 1)

= 1 x 1

= 1

Through investigation it is clear that that total number of points earned and the

number of yellow cards are very well related which would be an important factor for

all the managers to evaluate the decisions of buying players and confirming their

quality of play. Thus according the number of yellow cards that certainly leads to red

cards.

Validity:

Working in the beginning of the project, once the scatter plot and line of linear

regression were drawn, they showed an evidence of almost no relationship between

the total number of points earned and the yellow cards. Also, the Pearson’s

Correlation Coefficient was quite weak in most cases. However, working through the

test of independence, it is noted that there lies a relationship between the two

factors, which holds an immense importance to all English Premier League Managers.

Furthermore, the probability value to

Conclusion:

To conclude, this investigation hasn’t only aided my in expanding my

understanding of various mathematical processes, such as the , least square

regression, and other mathematical processes.

Firstly, the conclusions made for the least square regression equation was that

these two factors being, yellow cards and the total earned points, had no correlation as

the equations constantly showed a pattern amongst equations that depict a negative

and weak correlation such as, , y= -0.002x + 51.45. ,y= -0.011x +

59.18 , y= -0.103x + 66.61, y= -0.230x + 72.48. Therefore, after having completed

the least squares regression conclusions were made that the data had basically no

correlation. However, after having completed the table, it is clearly noted that the

total number of points and the total earned points during the seams, are in reality very

well related. Furthermore, this is further supported as the null hypothesis is rejected

and the alternative hypothesis is accepted, as the is greater than the critical

value. In addition, the probability value is 0.0163, which is less that 0.05. That is a

further evidence of the fact that the null hypothesis is rejected.

More importantly, this investigation not only demonstrated that there is an

evident correlation between the yellow cards and the total points earned. Also, these

results are highly valuable for managers as it they can make sure to include various

techniques that prevent players from obtaining more yellow cards as an evident

correlation is seen.

Documents

Microsoft Word - Math Ia - Final Final [1]