I will explain how to calculate odds for football matches, using the Poisson Distribution formula.
The Basic Formula for the Poisson Distribution
The formula to find the Poisson distribution is
P(k) = (e–λ * λk)/ k!
Where
e = Euler’s constant of 2.718
λ = Expected Goals
k = Number of Goals that you are Calculating the Probability for
I will analyse a fictitious Premier League game between Crystal Palace and Chelsea.
The formula has 4 steps:
- Calculate the Attack Strength for Both Sides
- Calculate the Defence Strength of Both Sides
- Calculate the Expected Goals for Each Side
- Put the Expected Goals into a Poisson Formula
I have put a Poisson Calculator at the end of this article. You can follow this article and just punch in the expected goals into the calculator.
For current purposes, I will start by using the last 5 years of data. I will discuss how to choose the best dataset, in terms of years, in a future article.
A. Calculating Attack Strength
For attack strength, we only count the number of home goals scored for the home team and the number of away goals scored for the away team.
In other words, only home form is used for the home team and only away form is used for the away team.
There are 2 steps to calculating Attack Strength. The first is to calculate the average goals. The second step is to divide the average goals by the league average. This is an unnecessary step for Poisson. However, I will include this step in this article.
1. Attack Strength for the Home Team Formula
Step 1: Calculate Average Home Goals
Average Home Goals = Total Home Goals ÷ Total Number of Home Games
Crystal Palace have played 95 matches and scored 110 home goals in my data sample.
Therefore,
Average Home Goals = 110 ÷ 95 = 1.157
Step 2: Calculate Attack Strength
Attack Strength = Average Home Goals ÷ Home Goal League Average
We’ve already calculated Crystal Palace’s average home goals as 1.157. The home goal league average is 1.48.
Attack Strength = 1.157 ÷ 1.48 = 0.78
2. Attack Strength for the Away Team Formula
Step 1: Calculate Average Away Goals
Average Away Goals = Total Away Goals ÷ Total Number of Away Games
Chelsea have played 95 matches and scored 161 away goals in my data sample.
Therefore,
Average Away Goals = 161 ÷ 95 = 1.69
Step 2: Calculate Attack Strength
Attack Strength = Average Home Goals ÷ Home Goal League Average
We’ve already calculated Chelsea’s average away goals as 1.69.
The away goal league average is 1.27.
Attack Strength = 1.69 ÷ 1.27 = 1.33
B. Calculating Defence Strength
For Defence Strength, we only count the number of home goals conceded by the home team and the number of away goals conceded by the away team.
As with Attack Strength, only home form is used to calculate Defence Strength for the home team and only away form is used to calculate Defence Strength for the away team.
There are 2 steps to calculating Defence Strength. The first is to calculate the average goals conceded. The second step is to divide the home and away side’s average goals conceded by the home and away league average, respectively.
1. Defence Strength for the Home Team Formula
Step 1: Calculate Average Home Goals Conceded
Average Home Goals Conceded = Total Home Goals Conceded ÷ Total Number of Home Games
Crystal Palace have played 95 matches and conceded 110 home goals in my data sample.
Therefore,
Average Home Goals Conceded = 119 ÷ 95 = 1.25
Step 2: Calculate Defence Strength
Defence Strength = Average Home Goals Conceded ÷ Away Goals Conceded League Average
We’ve already calculated Crystal Palace’s average home goals conceded as 1.25. The home goals conceded league average is 1.15.
Defence Strength = 1.25 ÷ 1.15 = 1.09
2. Defence Strength for the Away Team Formula
Step 1: Calculate Average Away Goals Conceded
Average Away Goals Conceded = Total Away Goals Conceded ÷ Total Number of Away Games
Chelsea have played 95 matches and scored 116 away goals in my data sample.
Therefore,
Average Away Goals Conceded = 116 ÷ 95 = 1.22
Step 2: Calculate Defence Strength
Defence Strength = Average Away Goals Conceded ÷ Away Goals Conceded League Average
We’ve already calculated Chelsea’s average home goals as 1.22.
The away goal league average is 1.42.
Defence Strength = 1.22 ÷ 1.42 = 0.86
C. Calculate Expected Goals
Expected goals are calculated for each side individually.
1. Home Side Expected Goals
For the home side the formula for expected goals is:
Expected Goals = (Home Attack Strength x Away Defence Strength) x Home Goals Scored League Average
So, for Crystal Palace, we would calculate:
Expected Goals = (0.78 x 0.86) x 1.48 = 0.99
2 Away Side’s Expected Goals
For the away side the formula for expected goals is:
Expected Goals = (Away Attack Strength x Home Defence Strength) x Away Goals Scored League Average
So, for Chelsea, we would calculate:
Expected Goals = (1.33 x 1.09) x 1.27 = 1.84
D. Put the Expected Goals into a Poisson Formula
The easy way to obtain the results for the Poisson Distribution is to simply input the Expected Goals for Crystal Palace and Chelsea into the Poisson Calculator below.
Alternatively, Excel has a Poisson formula in which you can input the numbers.