A STATISTICAL STUDY OF

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A STATISTICAL STUDY OF MANCHESTER UNITED IN
THE PREMIER LEAGUE
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CONTENTS

• Perception of Statistics
• Why Manchester United and the Premier League?
• Summary Statistics
• Hypothesis Testing
• Chi Square Test
• Summary

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Perception of Statistics
MEANINGLESS
MIND-NUMBING
UNEXCITING
BORING
POINTLESS
TEDIOUS
STATISTICS
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Perception of Statistics
Statistics are like bikinis.  What they reveal
is suggestive, but what they conceal is vital.
Aaron Levenstein
Lies, damned lies, and statistics Mark Twain
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Perception of Statistics

• Problem
• Boring Data Sets.
• How many pupils are interested in the duration of
  light bulbs?
• Solution
• Something adolescents can relate to.
• Something fun.
• So what did I choose to study?

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MANCHESTER UNITED
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Why Manchester United and the Premiership?

• Football
• Tangible and interesting.
• Manchester United
• Reigning champions of the Premiership.
• Fan base of 333 million (5 of the world
  population).
• Most popular football team in the world?
• Premiership
• Makes for nice statistical analysis.

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Premier League

• Contests Englands top 20 teams.
• Each team plays all others twice, once at home
  and once away.
• 38 matches per season (August-May).
• 3 points Win, 1 point Draw, 0 Lose.
• Most points wins.
• Bottom 3 teams relegated to the Football League
  Championship.

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Summary Statistics

• Three types of average
• Mean
• Median
• Mode
• Three types of spread
• Range
• Interquartile Range (IQR)
• Standard Deviation

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Summary Statistics Average

• For a discrete data set x1, x2,,xn arranged in
  ascending order the averages have the following
  properties

Mode
Median
Mean
?
Denoted
Mode

Method
Value that occurs most frequently in the data
set.
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Summary Statistics Spread

• For a discrete data set x1, x2,,xn arranged in
  ascending order the spreads are as follows

Interquartile Range
Standard Deviation
Range

s
Denoted
Range
IQR
Method
IQR Q3 Q1
Range xmax - xmin
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Summary Statistics Quartiles

• Three quartiles
• Q1 (Lower Quartile) 25th percentile.
• Q2 (Median) 50th percentile.
• Q3 (Upper Quartile) 75th percentile.
• Calculating Quartiles
• Q2 median
• Then for
• n even split data set into lower and upper
  halves.
• n odd discard the median before splitting into
  halves.
• Q1 median of the lower half of the data set.
• Q3 median of the upper half of the data set.

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Summary Statistics Standard Deviation Example

• Find
• the total number in the sample n
• the mean of the sample
• the deviation
• the deviation squared
• the sum of all , i1,,n
• the Standard Deviation using the formula

GOALS SCORED AT HOME AGAINST CHELSEA (8 years)
xi
i
0.25
-0.5
1.5
1
1
0.25
-0.5
1.5
1
2
0.25
-0.5
1.5
1
3
0.25
-0.5
1.5
1
4
0.25
0.5
1.5
2
5
2.25
-1.5
1.5
0
6
2.25
1.5
1.5
3
7
2.25
1.5
1.5
3
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Summary Statistics Examples

• We can calculate the averages for the number of
  goals scored and conceded over the past four
  seasons by Manchester United in the Premier
  League. This takes into account 152 games!
• Likewise, we can calculate the spreads for such
  data

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Summary Statistics Examples

• Goals Scored Against Top Performing Teams (past 8
  seasons)
• Goals Conceded Against Top Performing Teams (past
  8 seasons)

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Summary Statistics Boxplot

• Quick analysis of the distribution of data
• Q1, Q2, Q3
• Whiskers
• Outliers
• Whiskers end on the last given value which is
  within or on the upper and lower boundaries.
• Lower Boundary 1.5 (IQR)
• Upper Boundary 1.5 (IQR)
• Any value which lies outside these boundaries is
  extreme and is called an Outlier.

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• Goals Scored per Game (4
  seasons)

• Shots per Game (4 seasons)

Outlier

• Fouls per Game (4 seasons)

• Yellow Cards per Game (4 seasons)

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• Goals Scored per Game against Top Performing
  Teams (8 seasons)

• Goals Conceded per Game against Top Performing
  Teams (8 seasons)

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• Goals Scored and Conceded per Match per Season

• Fouls and Yellow Cards per Match per Season

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Summary Statistics Histogram

• Graphical display of the frequency of disjoint
  categories of the data.
• Area of a bar represents number of observations
  in the category.
• Mathematically
• a mapping mi that counts the number of
  observations (n) falling into k categories such
  that

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• Goals Scored per Match at Home and Away (4
  seasons)

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• Shots Taken and Shots on Target per Match (4
  seasons)

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Hypothesis Testing

• Test based on statistical data where you choose
  between two hypotheses
• H0 vs HA
• H0 must be stated and exists solely to be
  falsified by the sample.
• Set a value for a, the significance level. This
  is the probability for rejecting the null
  hypothesis when it is true.
• If sample has a smaller probability than a,
  reject H0.

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Chi-Square Test

• When data can be split into two categorical
  variables such as games with yellow cards dealt
  and those without and win/no win, we call this
  categorical data.
• The chi-squared test (?2 test) finds whether
  there is significant evidence of a relationship
  between two such variables.
• Test for independence.

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Chi-Square Test Method

• Observed Values 2 x 2 contingency table (rc2)

Sample size

• nij number in row i, column j.
• ni. sum of row i.
• n.j sum of column j.

• pij probability of being in row i, column
  j.
• pi. probability in row i.
• p.j probability in column j.

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Chi-Square Test
Rows and Columns independent P(A n B) P(A) x
P(B)

• H0
• Estimate each
• Under H0
• Cell (i,j)
• Test Statistic

Binomial Oij B(n,pij) E(Oij) npij
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Chi-Square Test

• Expected Table

• ?2 Table

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Chi-Square Test

• Large values of ?2 provide evidence against H0
  (suggests there may be a relationship).
• For large n, given H0, the sampling distribution
  is chi-square.
• Degrees of Freedom
• d.f. (r 1)(c 1)
• 1 (for a 2 x 2 table)

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Chi-Square Test Example

• Two categorical variables
• Aim To find whether there is a significant
  relationship between yellow cards being given and
  outcome of the match when played at home.
• H0 independence of rows and columns.
• Significance Level a 0.05.

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Chi-Square Test Example

• Observed

• ?2

• Expected

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Chi-Square Test Example

• ?2 Table
• .
• 0.919 1.767 1.875 3.605
• 8.166
• p-value (from tables for 1 d.f.) 0.004269
• p-value is evidence that outcome of the match
  and yellow cards given are
  related.

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Summary

• Statistics should be interesting and fun.
• One of many data sets.
• Pupils could pick their own
• Favourite football team.
• Formula1.
• Tennis etc.
• Perform hypothesis tests for paired data
• T-test
• Sign test
• Wilcoxon Rank Sum Test

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THANK YOU