Expected goals models attach a probabilistic estimate of the outcome of every attempt on goal using a variety of variables, notably shot location and type.
Comparing a striker's actual scoring output against the "average player" outcome predicted from the model provides a simple benchmark for over or under performance and the same can also be done from the perspective of the keeper.
Often such results are condensed down to a single figure.
For example, a keeper who allows ten goals, when the cumulative total of the expected goals from all the on target attempts he has faced comes to eleven may be considered to have over performed by allowing 9% fewer goals than an average keeper might have.
While this approach conveys some information, it does lose a lot of the granular nature of the initial shot data.
In addition the distribution of expected goals over a range of shots having widely differing individual goal expectations is not always wholly represented by the cumulative total and sample size is also omitted in the catch all figure.
Running simulations of all the shots faced by a keeper maintains some input for both the quality and the quantity of the dataset.
Liverpool's Simon Mignolet had to deal with 112 attempts ranging from 1 in a 1,000 long shots to 9 out of ten near certainties. The cumulative expected goals total for all chances he faced came to just over 36 goals.
If we simulate every attempt faced by Mignolet using the output from a shot model based on prior seasons, we can see how likely it is an average Premier League keeper would concede 42 goals or worse. This was the record actually achieved by Mignolet in 2015/16.
That value comes to 9%. So, nearly 91% of the average keeper iterations result in fewer than the 42 goals Mignolet actually conceded. Therefore, Klopp's search for another keeper in time for 2016/17 is perhaps understandable.
By contrast, Fraser Forster faced 67 attempts, saving 50 and conceding 17 goals against an expectation of nearly 20. So an above average season long performance.
This can be quantified in a similar manner to Mignolet's under performance. Just as an average keeper would likely perform as bad or worse than Mignolet had in 2015/16 only 9% of the time, such a keeper would only perform as well or better than Forster in 22% of the iterations.
Models cannot capture every aspect of a chance, keepers may mature and decline with age and injury related fitness. But a probabilistic approach such as this can at least demonstrate that Mignolet's season was likely to have been a disappointing one and Forster's above par. Although based on this single season alone, there may be a 22% chance that he was just an average keeper buoyed by luck.
These plots can be summarised in spreadsheet format for all keepers from 2015/16.
The above heat maps illustrate the most likely number of goals conceded by an average keeper for attempts faced by the four keepers, The actual number of goals allowed by each keeper is highlighted in red.
Mignolet's poor season is shown by the small percentage of iterations that fall below his highlighted % figure.
Schmeichel's season was slightly above average, as his highlighted figure lies slightly above the most likely outcome of 40 goals conceded.
However, we perhaps shouldn't be too confident that his slight over performance is entirely down to talent. There is a greater than 30% possibility that an average keeper would achieve a results as good or better than Schmeichel did simply through randomness.
Lloris and Adrian also recorded above average shot stopping qualities, but again not by a large enough margin that we might confidently conclude that they are certainly superior to our average goal keeping benchmark.
The remainder of the keeping achievements from the 2015/16 season are summarised below.
The higher the red actual figure is above the darkest green formatting denoting the performance of an average keeper, the more likely the keeper is to have put in an above average performance.
And the lower a player's actual figure is below the benchmark average, the more likely he is to have exhibited below average levels of talent.
Brad Guzan's numbers were so bad he has broken the system, but as his colleague, Mark Bunn also scrapes the bottom of the barrel, it does seem possible that their car obsessed defence may have contributed levels of incompetence that was largely absent among other Premier League teams.
The above heat maps illustrate the most likely number of goals conceded by an average keeper for attempts faced by the four keepers, The actual number of goals allowed by each keeper is highlighted in red.
Mignolet's poor season is shown by the small percentage of iterations that fall below his highlighted % figure.
Schmeichel's season was slightly above average, as his highlighted figure lies slightly above the most likely outcome of 40 goals conceded.
However, we perhaps shouldn't be too confident that his slight over performance is entirely down to talent. There is a greater than 30% possibility that an average keeper would achieve a results as good or better than Schmeichel did simply through randomness.
Lloris and Adrian also recorded above average shot stopping qualities, but again not by a large enough margin that we might confidently conclude that they are certainly superior to our average goal keeping benchmark.
The remainder of the keeping achievements from the 2015/16 season are summarised below.
The higher the red actual figure is above the darkest green formatting denoting the performance of an average keeper, the more likely the keeper is to have put in an above average performance.
And the lower a player's actual figure is below the benchmark average, the more likely he is to have exhibited below average levels of talent.
Brad Guzan's numbers were so bad he has broken the system, but as his colleague, Mark Bunn also scrapes the bottom of the barrel, it does seem possible that their car obsessed defence may have contributed levels of incompetence that was largely absent among other Premier League teams.
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