Poisson Distribution, coupled with historical data,
can provide a method for calculating the likely number of goals that will be
scored in a soccer match. Read on for a simple method to calculate the likely
outcome of a soccer match using Poisson Distribution.
Poisson Distribution Explained
Poisson
Distribution is a mathematical concept for translating mean averages into a
probability for variable outcomes. For example, Chelsea might average 1.7 goals
per game. Entering this information into a Poisson formula would show that this
average equates to Chelsea scoring 0 goals 18.3% of the time, 1 goal 31% of the
time, 2 goals 26.4% of the time and 3 goals 15% of the time.
How to calculate outcomes with Poisson Distribution
To
calculate the possible outcomes for a match using Poisson Distribution, we
first need to calculate a value for each team’s “Attack” and “Defence
Strength”. These numbers – calculated from last season’s performance data –
allow us to model possible outcomes for each game of the season.
Selecting
the data range is vital when using Poisson Distribution, too long and the data
wont be relevant for the teams current strength, while too short may allow
outliers to unfairly skew the data. For this analysis we are using the 38 games
played by each team last season.
Calculate the average goals scored at home and away
To get
the average number of goals scored last season, average the number of goals
scored per team, per game, per season. In mathematical terms, that’s:
 Season Goals Scored / Number
of Teams / Number of Games
In
2012/13, that was 592/20/19 at home and 471/20/19 away, equalling an average of
1.558 goals per game at home and 1.239 away. The difference from this average
is what constitutes a team’s “Attack Strength”.
 Average number of goals
scored at home: 1.558
 Average number of goals
scored away from home: 1.239
We’ll
also need the average number of goals an average team concedes. This is simply
the inverse of the above numbers (as the no. of goals a home team scores will
equal the same number that an away team concedes.):
 Average number of goals
conceded at home: 1.239
 Average number of goals
conceded away from home: 1.558
We can
now use the numbers above to calculate the Attack and Defence Strength of both
Swansea City and Manchester United for their match on August 17th, 2013.
Predicting Swansea’s Goals
Calculate
Swansea’s Attack Strength:
 a. Take the number of goals
scored at home last season by the home team (Swansea: 28) and divide by
the number of home games (28/19): 1.473
 b. Divide this value by the
season’s average home goals scored per game (1.473/1.558), to get the
“Attack Strength”: 0.946. This shows that Swansea scored 5.4% fewer goals
at home than a hypothetical “average” Premier League side.
Calculate
Man Utd’s Defence Strength:
 a. Take the number of goals
conceded away last season by the away team (Man Utd: 24) and divide by the
number of away games (24/19): 1.263.
 b. Divide this by the
season’s average goals conceded by an away team per game (1.263/1.558) to
get the “Defence Strength”: 0.81. Man Utd conceded 19% fewer goals than an
“average” Premier League side on the road.
We can
now use the following formula to calculate the likely number of goals the home
team might score:
 Swansea’s Goals = Swansea’s
Attack x Man Utd’s Defence x Average No. Goals
In this
case, that’s 0.946* 0.81 * 1.558, which equates to Swansea scoring 1.194 goals.
Predicting Man Utd’s Goals
Calculate
Man Utd’s Attack Strength:
 1. Take the number of goals
scored away last season by the away team (Man Utd: 41) and divide by the
number of away games (41/19): 2.158
 2. Divide this value by the
season’s average away goals scored per game (2.158/1.239), to get the
“Attack Strength”: 1.742. This shows that the Red Devils scored 74.2% more
away goals than a hypothetical “average” Premier League side.
Calculate
Swansea’s Defence Strength:
 1. Take the number of goals
conceded at home last season by the home team (Swansea: 26) and divide by
the number of home games (26/19): 1.368.
 2. Divide this by the
season’s average goals conceded by a home team per game (1.368/1.239) to
get the “Defence Strength”: 1.104. Swansea conceded 10.4% more goals than
an “average” Premier League side at home.
We can
now use the following formula to calculate the likely number of goals the away
team might score:
 Man Utd’s Goals = Man Utd’s
Attack x Swansea’s Defence x Average No. Goals
In this
case, that’s 1.742 * 1.104 * 1.239, which equates to Man Utd scoring 2.383
goals.
Poisson Distribution Betting – Predicting Multiple
Match Outcomes
Of
course, no game ends 1.194 vs. 2.383 – this is simply the average. Poisson
Distribution, a formula created by French mathematician Simeon Denis Poisson,
allows us to use these figures to distribute 100% of probability across a range
of goal outcomes for each side. The results are shown in the table below:
The
formula itself looks like this: P(x; μ) = (eμ) (μx) / x!, however, we can use
online tools such as this Poisson Distribution
Calculator to do most of the equation for us.
All we
need to do is enter the different goals outcomes (05) in the Random Variable
(x) category, and the likelihood of a team scoring (for instance, Man Utd at
2.383) in the average rate of success, and the calculator will output the
probability of that score.
Poisson Distribution for Swansea vs Man Utd
Goals:

0

1

2

3

4

5

Swansea

30.3%

36.18%

21.60%

8.6%

2.57%

0.61%

Man Utd

9.23%

21.99%

26.2%

20.81%

12.4%

5.9%

This
example shows that there is a 9.2% chance that Man Utd will not score, but a
22% chance they will get a single goal and a 26.2% chance they’ll score two.
Swansea,
on the other hand, are at 30.3% not to score, 36.2% to score one and 21.6% to
score two.
Hoping
for a side to score five? The probability is 5.9% if United are the scorers, or
0.61% for Swansea to do it.
As both
scores are independent (mathematicallyspeaking), you can see that the expected
score is 2 – 1 to Man Utd. If you multiply the two probabilities together,
you’ll get the probability of the 21 outcome – 0.095 or 9.5%.
Now you
know how to calculate outcomes, you should compare your result to a bookmaker’s
odds to help see how they differentiate.
For
example, taking into account all possible draw combinations (00, 11, 22,
33, 44 and 55), this method gives a probability of 0.186 or 18.6%. Pinnacle
Sports’ odds were 3.960 (a 25.3% implied probability).
Therefore
if last season’s form was a perfect indicator of this season’s results, we
should expect a 21 victory. Unfortunately it isn’t as simple as that, which is
why pure Poisson analysis has limitations.
Poisson Distribution Betting – The Limits of
Poisson Distribution
Poisson
Distribution is a simple predictive model that doesn’t allow for a lot of
factors. Situational factors – such as club circumstances, game status etc. –
and subjective evaluation of the change of each team during the transfer window
are completely ignored.
In this
case, it means the huge xfactor of Manchester United’s first Premier League
game without Sir Alex Ferguson is entirely ignored. The league is also arguably
stronger this season, so United are less likely to score as many goals as they
did last year.
Correlations
are also ignored, such as the widely recognised pitch effect that shows that
matches have some tendency to be either high or low scoring.
These are
particularly important areas in lower league games, which can give punters an
edge against bookmakers.
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