Predicting Goals Scored using the Binomial Distribution

Much is made of the use of the Poisson distribution to predict game outcomes in soccer. Much less attention is paid to the use of the binomial distribution. The reason is a matter of convenience. To predict goals using a Poisson distribution, “all” that is needed is the expected goals scored (lambda). To use the binomial distribution, you would need to both know the number of shots taken (n) and the rate at which those shots are turned into goals (p). But if you have sufficient data, it may be a better way to analyze certain tactical decisions in a match. First, let’s examine if the binomial distribution is actually dependable as a model framework.

Here is the chart that shows how frequently a certain number of shots were taken in a MLS match.

source data: AmericanSoccerAnalysis

source data: AmericanSoccerAnalysis

The chart resembles a binomial distribution with right skew with the exception of the big bite taken out of the chart starting with 14 shots. How many shots are taken in a game is a function of many things, not the least of which are tactical decisions made by the club. For example it would be difficult to take 27 shots unless the opposing team were sitting back and defending and not looking to possess the ball. Deliberate counterattacking strategies may very well result in few shots taken but the strategy is supposed to provide chances in a more open field.

Out of curiosity let’s look at the average shot location by shots taken to see if there are any clues about the influence of tactics. To estimate this I looked expected goals by each shot total. This does not have any direct influence on the binomial analysis but could come in useful when we look for applications.

source: AmericanSoccerAnalysis

source data: AmericanSoccerAnalysis

The average MLS finishing rate was just over 10 percent in 2013. You can see that, at more than 10 shots per game, the expected finishing rate stays constant right at that 10-percent rate. This indicates that above 10 shots, the location distribution of those shots is typical of MLS games. However, at fewer than 10 shots you can see that the expected goal scoring rate dips consistently below 10%. This indicates that teams that take fewer shots in a game also take those shots from worse locations on average.

The next element in the binomial distribution is the actual finishing rate by number of shots taken.

 source: AmericanSoccerAnalysis

source data: AmericanSoccerAnalysis

Here it’s plain that the number of shots taken has a dramatic impact on the accuracy rate of each shot. This speaks to the tactics and pace of play involved in taking different shot amounts. A team able to squeeze off more than 20 shots is likely facing a packed box and a defense less interested in ball possession. What’s fascinating then is that teams that take few shots in a game have a significantly higher rate of success despite the fact that they are taking shots from farther out. This indicates that those teams are taking shots with significantly less pressure. This could indicate shots taken during a counterattack where the field of play is more wide open.

Combining the finishing accuracy model curve with number of shots we can project expected goals per game based on number of shots taken.


What’s interesting here is that the expected number of goals scored plateaus at about 18 shots and begins to decline after 23 shots. This, of course, must be a function of the intensity of the defense they are facing for those shots because we know their shot location is not significantly different. This model is the basis by which I will simulate tactical decisions throughout a game in Part II of this post.

Now we have the two key pieces to see if the binomial distribution is a good predictor of goals scored using total shots taken and finishing rate by number of shots taken. As a refresher, since most of us haven’t taken a stat class in a while, the probability mass function of the binomial distribution looks like the following:

source: wikipedia


n is the number of shots

p is the probability of success in each shot

k is the number of successful shots

Below I compare the actual distribution to the binomial distribution using 13 shots (since 13 is the mode number of shots from 2013’s data set), assuming a 10.05% finishing rate.

source data: AmericanSoccerAnalysis, Finishing Rate model

source data: AmericanSoccerAnalysis, Finishing Rate model

The binomial distribution under predicts scoring 2 goals and over predicts all other options. Overall the expected goals are close (1.369 actual to 1.362 binomial). The Poisson is similar to the binomial but the average error of the binomial is 12% better than the Poisson.

If we take the average of these distributions between 8 and 13 shots (where the sample size is greater than 40) the bumps smooth out.

source data: AmericanSoccerAnalysis, Finishing Rate model

source data: AmericanSoccerAnalysis, Finishing Rate model

The binomial distribution seems to do well to project the actual number of goals scored in a game, and the average binomial error is 23% lower than with the Poisson. When individually looking at shots taken 7 to 16 the binomial has 19% lower error if we just observe goal outcomes 0 and 1. But so what? Isn’t it near impossible to predict the number of shots a team will take in the game? It is. But there may be tactical decisions like counterattacking where we can look at shots taken and determine if the strategy was correct or not. And a model where the final stage of estimation is governed by the binomial distribution appears to be a compelling model for that analysis. In part II I will explore some possible applications of the model.

Jared Young writes for Brotherly Game, SB Nation’s Philadelphia Union blog. This is his first post for American Soccer Analysis, and we’re excited to have him!


Game Of The Week Review: Montreal Impact Visit Sporting Kansas City

I know I shouldn’t be surprised by the Impact stealing a match on the road, especially considering Sporting’s lack of strength at home as of their recent string of outcomes. Though, with all the statistical pointers, it’s quiet uncanny that they came up with even a point, let alone all three.


It’s hard to look at the tackles, interceptions and clearances and not think that it’s a by product of the Impact largely being on their heels for the majority of the match. That in large part is due to the style which the Montreal Impact implements. The team as a whole has functioned with 48% possession through 12 matches and even less possession (44%) in away games. It’s not a bad thing, but it naturally produces more defensive events.

Much of our discussion during the podcasts has dealt with shots and their predictive nature. Montreal has been at the forefront of the discussion, with amazing results despite being outshot on both total attempts at goal (12 to 15 per game) and actual shots on target (4.9 to 5.2) Montreal Impact is currently now sporting 26 points with a goal differential of +7. Not to mention they are boasting the highest conversion rate in the league of 15.3%. Better than the next highest (FC Dallas, 13.9%) by nearly a whole point and a half.

Matthias, Drew and I have discussed whether or not Montreal can continue to maintain such a high finishing rate. It’s a legitimate question considering the construct of the situation but, as pointed out by Ravi Ramineni in a discussion this morning on twitter, ‏the problem with making such assertions is that we’re looking purely at the shot totals rather than looking at the qualitative state of the shot.

However, while it’s interesting enough to question whether or not the Impact are going to stick around and continue to score goals at their current rate, I’m going to leave that for another day. It’s even more interesting that Kansas City came up with twice the amount of attempts on goal and the only scored once. That one goal was on a foul that was made right on the line of the 18 yard box. Had the linesman not been on his game, that call could have easily been a free kick.

The question that I really have is more of why was Sporting unable to build upon their chances. Looking at the amount of clearances that the Impact had  I kind of wondered if the fact was that they just couldn’t maintain the needed pressure upon Troy Perkins goal.

Kansas City Attempts Name Minutes
Miss Joseph Peterson 6′
Attempted blocked Paulo Nagamura 19′
Miss Claudio Bieler 25′
Miss Claudio Bieler 42′
Goal Claudio Bieler 49′
Miss Seth Sinovic 49′
Miss Claudio Bieler 56′
Miss Kei Kamara 60′
Attempted Save Benny Feilhaber 65′
Miss Aurélien Collin 69′
Attempted Save Paulo Nagamura 70′
Miss Paulo Nagamura 71′
Attempted blocked Claudio Bieler 76′
Miss C.J.Sapong 78′
Attempted Save Joseph Peterson 82′
Attempted Save Aurélien Collin 85′
Miss Joseph Peterson 90′
Attempted Save Claudio Bieler 92′
Miss Kei Kamara 94′


Looking at this you can see three real bunches. First at the 69th-71st minute, Again with the 76th and 78th minute and then in the final moments game a solid run of 90 to 94, ending with Kei Kamra’s shot that just drifted wide.

Ultimately, I’m more inclined to believe that Sporting did just as much to not earn a result as the Impact did to really earn one. But while most people would be willing to chalk this game up to luck, I just think it’s the largest example of what the Impact do well, and that’s disrupting opposing teams while allowing the Impact to sit in their own defensive third. I’m still not inclined, as I’m sure Matty isn’t either, to give the Impact the full rights of being a team that is “for real”. But they certainly continue to prove their case week in and week out.