## Wednesday, April 4, 2018

### Some Thoughts on Rangers' Defense

Rangers have conceded an average of 1.18 goals per game so far this season. Last season through 32 games, their average was 1.06 goals against per game. That's an increase of 11%. While that seems like a big difference, we have to remember that conversion rates are highly influenced by luck in small samples, and 32 games is a small sample.

In this post, I take a look at whether or not the difference between 1.18 and 1.06 is statistically significant. Specifically, I will assess how likely it would be to concede 1.18 goals per game in the short term (over 32 games) if the long term average was 1.06. To do this I will run a simulation experiment in R based on the Poisson distribution. All data were downloaded from football-data.co.uk.

It has previously been shown that goal scoring generally follows a Poisson distribution. For example, the charts below compare the actual distributions of goals scored per game in the SPFL Premiership since 2013-14 (blue bars) with distributions of values randomly selected from a Poisson distribution with the same average (gray bars). As you can see, the observed distributions match very closely the randomly generated Poisson distributions.

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Because goals per game follows a Poisson distribution, we can use the rpois() function in R to sample repeatedly from a Poisson distribution with an average  of 1.06. This allows us to replicate a large number of 32 game samples and plot the distribution of average goals per game for each replicate to see the range of variation due to small sample size.

As you can see from the chart below, when we repeatedly sample from a Poisson distribution and get the mean, we end up with a nice symmetrical distribution of average goals conceded per game centering on 1.06. The dashed line represents Rangers' current 1.18 average, which is approximately the 75th percentile. In other words, roughly 25% of the distribution contains mean values that are equal to or greater than 1.18 despite being drawn from a Poisson distribution with mean = 1.06. This relatively wide range of variation is due to the small sample size of 32 in each replicate.
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Thus, given this evidence, I think it's fair to say that Rangers' goals concede per match this season is not significantly different from last season despite conceding 4 more goals through 32 games this season. This relatively minor difference is easily explained by small sample size.

Moreover, goals conceded is generally a very poor indicator of a team's defensive quality since it conflates goalkeeper performance with outfield defensive performance. A better metric to analyze a team's defense is shots on target conceded per game.

The chart below shows shots on target conceded per game for SPFL Premiership teams this season versus last season. As can you can see, Rangers are among the most improved defenses with regard to limiting opposition shots on target, along with Motherwell and Kilmarnock. So there is actually some evidence to suggest Rangers have improved defensively this season.

The real problem with Rangers' defense might be the goalkeeper. Their current save rate (defined as 1-[goals conceded/shots on target conceded]) is 0.64, which is 8.6% lower than the 0.70 save rate expected for a typical keeper of average quality. Rangers ended last season with a save rate of 0.74, which is 13.5% higher than the current save rate. This kind of variability is not unusual for save rate, which is one of most volatile and unpredictable metrics in football analytics.

So I think it's safe to say Rangers' defense has not gotten worse this season, and in some respects it may have improved. However, regardless of any of this evidence, many fans may feel it's still not good enough given Rangers' resources, and that's probably fair.