*(This text in a sense builds on parts of this 2014 article by Danny Page and also this recent one by Ferdia O'Hanrahan and there are probably a lot more people who did similar work already. However, I wanted to get some intuition about one or two things that I haven't seen being discussed deeply enough to really understand what's going on. Hence, this article exists.)*

Based on a number of chances with xG values assigned to them, the number of goals a team scores during a football match can be modeled as a so called poisson binomial distribution. This distribution is generated by doing binary success/failure experiments repeatedly with varying probabilities of success. In this case, the experiments are shots (plus maybe other dangerous situations) and the probability of success is the xG value assigned to them. The probability distribution assigns each number of goals a probability that this number of goals is scored from the chances (respectively their xG values) it was generated from.

The xG sum is also the

*mean*of this distribution.* If you take a look at the example, 2.2 and 0.3 lie pretty much in the center of the distributions. If you would simulate these shots 10,000 times, add all the goals up, and divide them by 10,000 you would get about 2.2 goals for Team Red and 0.3 goals for Team Blue.

Now, the problem is that the poisson binomial distribution can look very differently depending on which xG values you aggregate (and that part of this effect on shape is hard to quantify). Blue's distribution above is highly asymmetrical, sharp and not far spread out, whereas Red's distribution looks more symmetrical, more spread out and flatter.

Without considering the shape of the underlying distribution, the mean (i.e. the xG sum) is prone to misinterpretation.

Here are three insights I gained from playing around with this stuff:

#### 1. The (rounded) xG sum is not necessarily equal to the most probable number of goals

Look at this example:

Both teams have an xG sum of 1.8. Team Blue achieved this with many low-xG-chances and Team Red with few high-xG-chances. The mode (= the most probable number of goals = the peak of the probability distribution ≠ the mean) of Team Red lies clearly at 2 goals, the number nearest to 1.8. However, for Team Blue, scoring two goals and scoring one goal are pretty much equally likely (actually, scoring two goals is even slightly

*less*likely than scoring one goal!).

So, if Team Blue scores one goal from this game, you actually shouldn't be more surprised than if they score two. The notion that Team Blue "underperformed" is however still valid. The probability that they score

*two or more*goals from these chances is higher than scoring one goal or none.

The second insight is related:

#### 2. More chances lead to more variance in the result.

This one is intuitively clear. The more chances you have, the more different amounts of goals you can score and (on average) the lower the chance of scoring a particular number of goals. This is especially the case if you have many chances, the outcome of which is hard to predict, i.e. chances valued around 0.5 xG (which correspond to pretty high quality in the low scoring game football). If you have twenty chances worth 0.02 xG, you will most likely score zero or one goal. The distribution will have low variance. Yet, any additional chance inscreases the variance of the number of goals scored. The closer they are to 0.5 xG, the more they do so.**

Some consequences of this are:

- A high deviation of a team's actual goals from its expected goals should be less surprising if they had a lot of chances. See the example above.
- When the expected scoreline is close, a draw can still be unlikely if there were many chances in the game.
- While more shots and higher chance quality obviously increase your chances of winning (as they increase your xG sum), they almost inevitably make your goal output more erratic.

#### 3. You can be less likely to win a game despite having a higher xG sum.

When you multiply the probability of team A scoring 2 goals with the probability of team B scoring 0 goals you get the probability of the game ending with a 2:0. Adding up these "result proabilities" you can compute a probability of team A winning, team B winning and a draw.

What's interesting is, that even when the xG sum of team A and team B are equal, the winning probabilities of these teams are almost always unequal. Starting from an extreme instance of this you can even construct cases where the team that leads by xG actually has a lower chance of winning the game. For example:

You can compute that Team Red has a winning chance of 37.1% and Team Blue has a winning chance of 33.7%

**–**a difference of 3.4 percent points. Projected onto a season of 34 games this corresponds to a difference of about three and a half points.

From the example (and as Mark Taylor already suspected in this 2014 article) it seems like this effect could be a many chances vs few big chances thing. If you have one or two high-qualitiy chances, you are very unlikely to score none of them and that way you may decrease your probability of losing by an exceptionally high amount.

To make more precise statements, you would have to analyze what affects the shape of the distribution and its conversion to goal difference and winning percentages, which is a lot more complicated than looking at the plain mean and variance.

Still, it would be interesting to see how this asymmetry works, what inuitively causes it and how relevant it is.

________________

* The mean of a poisson binomial distribution is simply the sum of the individual "success probabilities" of the single experiments. See wikipedia.

** The variance of a poisson binomial distribution is the sum of terms of the form (p_i * (1 - p_i)) ranging over all experiments, where p_i is the success probability of a single experiment. If you plot this term over p_i, you will see a peak at 0.5 and a value of zero at 0 and 1. Again, see wikipedia for the formula.

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