The Effect of Drawing Penalties

As Leafs fans, we've become rather accustomed to seeing Nazem Kadri dive embellish an infraction to get a call. In fact, he's one of the best in the league at doing just that. While the sportsmanship of it can be debated, this ability is absolutely a skill with real value.

This naturally leads to the question of whether we can quantify the impact of this skill on a team.

Methodology, and some simplifying assumptions

The math for evaluating the advantage of being put on the power play versus even strength is very simple. That said, we're going to make a few assumptions in order to facilitate comparisons across teams and to make some calculations easy.

1. We assume all players are placed on a hypothetical team that is completely average at even strength, completely average in terms of power play goals for, and completely average in power play goals against. This is to facilitate comparisons across different teams.
2. We assume all penalties drawn are minor penalties, and that all penalties take us from 5 on 5 to 5 on 4. We also assume that there are no other penalties until this power play expires. Effectively, power plays either end by the penalty expiring or the advantaged team scoring. This simplifies our work.
3. We assume goals scored on the power play are uniformly distributed from zero to two minutes. Naturally, the expected time of goal, given that a goal has been scored will be one minute into the power play. This is another simplifying assumption, and while it doesn't totally reflect reality, it's reasonable enough. Perhaps a better method would be to assume they are uniformly distributed from 5 seconds to two minutes, to account for the initial faceoff on a power play.

So, lets get into the math. We're looking to evaluate the expected goals added to a team being on the power play for two minutes as opposed to even strength for those two minutes.

The first parenthesis represents the value of being on the power play, and we subtract the second, as we are replacing even strength play. However, thanks to assumption 1, this term disappears, as an average team will score exactly as much as they concede at even strength. Therefore, our revised formula is much more simple:

EMG = (Expected Goals For on PP - Expected Goals Against on PP). Note these are per two minutes.

So, let's get to the calculations. Data was taken from stats.hockeyanalysis.com from 2011-2015. Expected Goals For on PP: Thanks to assumption 2, this is relatively straight forward.

Power plays end when the advantaged team scores. We know that the probability that the attacking team scores is represented by PP%. Over the time span I mentioned above, the average PP% was approximately 17.9975% (source). Since the maximum the advantaged team can score on the power play is one, Expected Goals For on PP = 0.179975.

Expected Goals Against on PP: This is a little different. Since there is no cap on the amount of goals the shorthanded team can score while shorthanded, this effectively is a Poisson Process (in case you want to learn more about it, here's the Wikipedia page). In order to calculate this, we have the following formula:

Expected Goals Against on PP = Average Goals Against Rate on PP * Average Length of PP

From HockeyAnalysis, we have that Average Goals Against Rate on PP is 0.7687 per 60 minutes (we can adjust this easily, so I'm omitting that). We can use the PP% above, along with assumption 3 to find the average length of power plays. Since we assume the average length of power play given that the advantaged team scored is 1 minute, the average length of a power play is:

1 minute * 0.17995 + 2 minutes * (1-0.17995) = 1.82205 minutes, or 1 minute, 49 seconds.

Put this all together, and we get that the Expected Goals Against on PP is 0.02334528.

This leaves us with EMG being 0.15460742.

So, how much value does this provide?

I've calculated individual totals for 2013/2014 and 2014/2015. Unsurprisingly, Nazem Kadri tops the list in 2014/2015, as his penalty drawing ability puts his teams in situations where they can expect to get 3.25 more goals per season, compared to a player who takes and draws penalties in equal measure.

He also ranks very highly in 2013/2014, at 11th in the league with 2.32 goals per season from this skill. So it seems like on the high end, this skill can add (approximately) 2-3 goals to a team's value.

On the low end of forwards, pluggers and grinders such as Chris Neil and David Clarkson take away about 1.5-2 goals from their team. So it seems that for forwards, the advantage of this skill is notable, but not hugely significant.

Defenders, by the nature of their position, will often take more penalties, so they are disproportionately near the bottom of the list. The most penalized defensemen tend to lose 3 goals through this, and the best defensemen tend to add 1-1.5 goals through this.

I haven't combed through the data much, but there might be some interesting insights there. Here's my spreadsheet, and as always, if there are any issues or suggestions, let me know.