The Blackhawks

The Blackhawks have been an interesting team this year.

First of all, there's plenty of evidence which suggests that the Blackhawks are a pretty solid team. They've accrued one of the best goal differentials in the league up to this point in the season, despite playing in the league's toughest division in the better of the two conferences. They've also soundly outshot the opposition, both at even strength and in general. In these respects, it would be difficult to argue that the Blackhawks are not one of the league's best teams.

What I find unusual, then, is that the Blackhawks expected goal differential, as calculated at hockeynumbers, is only slightly positive. The Blackhawks have allowed some 20 fewer goals than what would be predicted on the basis of shot quality, while having scored some 20 more. Given that their shot ratio is more or less in line with their goal ratio, the implication is that Chicago has been below average in both shot quality for and shot quality against. Indeed, if the expected goals numbers are translated to yield a shot quality index for each team, the Blackhawks do in fact fare quite poorly.

The fact that the Blackhawks tend to allow high quality shots against is not surprising. Firstly, shot quality is repeatable on a year-to-year basis, with Chicago having ranked 27th in the league in that regard last year.

Secondly, teams coached by Joel Quenneville tend to allow high quality shots against on average. Outside of 1999-00, for every year that Quenneville has been a head coach in the NHL, his team has ranked in the bottom half of the league in terms of save percentage, which implies that his teams were surrendering high quality shots against. While it's perhaps true that Quenneville was burdened with poorer than average goaltending during his tenure in both St. Louis and Colorado, shot quality has been directly measured from the 2002-03 season onward, with the results tending to support the argument that Quenneville-coached teams are poor in terms of shot quality against.

The Contrarian Goaltender has also found evidence that the effect of Quenneville's coaching in St.Louis was to reduce the save percentages of his own goalies, who tended to have better save percentages prior to playing under Quenneville (see, for example, the comments section of this post).

With a team save percentage of 0.912, the Blackhawks goaltending has been nothing short of superb this season. However, the fact that the team apparently allows such high quality shots against, not to mention the fact that save percentage is at least partially a product of random statistical variation, inevitably leads to the question of sustainability. Of course, It's true that both Khabibulin and Huet have proven track records, and both goaltenders are certainly better than anything Quenneville had to contend with in St. Louis and Colorado. Nonetheless, I think that it's a potential cause for concern and, at the very least, something that one ought to be mindful of in evaluating the team's prospects for the postseason.

As for the team in general, I'm inclined to think that they're still pretty good, the shot quality numbers notwithstanding. As I intimated in the above paragraph, both Huet and Khabibulin are above average netminders and the Blackhawks team save percentage is in that sense somewhat sustainable. And while it is true that the Blackhawks have exceeded their expected goals for, I'm reluctant to ascribe the difference to luck considering their plethora of offensively talented and creative players. Furthermore, even if I happen to be wrong on these points, their tendency to convincingly outshoot the other team is, if nothing else, encouraging.

Home Recording Bias: Shots on Goal

In previous posts, it was shown how some of the statistics that are recorded by the NHL are subject to a home arena bias. Home arena bias seems to be most pronounced with respect to the RTSS data, which includes statistics like hits, takeaways, giveaways, blocked shots, and so forth.

However, this bias is also observed with less subjective statistics, such as shots on goal. Below is a chart showing how the recording of shots on goal has varied on a site-by-site basis over the last 13 NHL seasons, with the more interesting information highlighted. The values contained in each cell were derived as follows:

[ (Home shots for/60 minutes played + Home shots against/60 minutes played) - (Road shots for/60 minutes played + Road shots against/60 minutes played) ]

Basically, the formula boils down to this: the total shots on goal (by both teams) in games played by a particular team at home, minus the total shots on goal in games played by that particular team on the road, with ice time controlled for. Empty net situations were not included, both in terms of shots on goal and minutes played. I should also mention that the ice time data for 1994-95 to 1997-98 is approximate.


While the home recording bias for shots on goal is not large, it is nonetheless clear that not all NHL arenas record shots equally. The recorders in Vancouver have over the years been very conservative in their shot counting, although the effect appears to have been moderated in the last couple seasons. There were more shots/60 in Colorado and Anaheim home games than there were in road games played by those two teams for every single season analyzed. Shot recording in Nashville has been generous ever since their inaugural season (although I'm not sure what happened in 2003-04), whereas the reverse has been true in Minnesota. In both New Jersey and Dallas, shots have been harder to come by since around the turn of the millennium. Finally, a bias towards overcounting seems to have materialized in Sunrise over the last couple years.

Of course, the above values are not necessarily demonstrative of a bias; they are merely suggestive of it. They ought to be supplemented with data on shooting percentage in order to allow for a more confident interpretation of the effect.

Why shooting percentage? Well, if an arena does in fact undercount or overcount shots on goal, the bias should concern saves rather than goals. The reasoning here is not difficult. Each shot on goal that results in a goal is necessarily a shot on goal -- there is no room for the exercise of discretion on the part of the shot recorder. However, in the case of a shot on goal that does not result in a goal, the shot recorder is permitted a modicum of discretion, and what constitutes a shot for some may not constitute a shot for others. Undercounting shots should have a positive effect on shooting percentage, whereas overcounting would be expected to have a positive effect on save percentage.

Therefore, it can be seen how accompanying data on shooting percentage will shed light on the extent to which a true bias is present. For the teams for which a bias is suspected -- Florida, Dallas, New Jersey, Nashville and Minnesota, I've included information below on the shooting percentage in games played by those teams, broken down into road and home situations. The data in the left column is the shooting percentage in road games played by the team indicated in the upper left hand corner. The data in the right column is the shooting percentage in that team's home games. It is important to stress that these figures do not merely represent the road and home shooting percentages of the team in question. Rather, the figures represent the overall shooting percentage (that is, both for the team in question as well as their opponents) in road or home games played by that team during the season indicated.

In terms of New Jersey, it seems that the tendency for undercounting shots at Continental Airlines arena began during the 2001-02 season. In every subsequent year, the shooting percentage in New Jersey home games has been higher than in New Jersey road games. Indeed, this bias has had some negative effect on the home save percentage of Devils goaltenders during the period in question (keeping in mind that the average home save percentage for NHL teams tends to be 0.003 to 0.007 higher than the average road save percentage).

The Devils led the league in shot quality against from 2002-03 to 2006-07. They also surrendered the fewest powerplays against during each of these seasons. While the Devils' road save percentage during the period in question reflects this fact, their home save percentage does not. At least part of the discrepancy can be accounted for by recording bias.

The shooting percentage data for Dallas games suggests that the bias emerged during the 1998-99 season. The effect appears to be large.

The data for Minnesota is less clear. The difference is in the predicted direction for four of the seasons (2001-02, 2002-03, 2003-04, 2005-06), the opposite direction for two of them (2006-07, 2007-08), with there being no difference in 2000-01. It's possible that:

a. The Wild simply play more conservatively at home.
b. The bias has lessened over time.
c. The results can be explained through some combination of the above factors.

The data on Nashville reveals that the bias is genuine, or at least was genuine prior to 2006-07 and 2007-08. The results for the last two seasons, taken together, indicate that the bias may no longer persist.

Finally, the data on Florida implies that there exists no recording bias at the BankAtlantic Center. The shooting percentage in Panther road games is largely indistinguishable from the shooting percentage in Panther home games. The fact that Florida goaltenders (Luongo, Anderson, Vokoun) have placed among the league leaders in save percentage in each of past several seasons (outside of 2006-07) have led some to conclude that their must be something amiss, given that the Panthers do not employ any type of defensive system and have not been an otherwise successful hockey team during that period. However, the most probable explanation is that the Florida has merely benefited from having a series of good goaltenders -- indeed, both Luongo and Vokoun have posted very impressive numbers elsewhere (granted, the data on Nashville suggests that the latter's save percentage may have been somewhat inflated by recording bias during his stay in the Music City).

Team Rankings and Playoff Probabilities

The first chart shows how each team in the league has fared thus far in terms of adjusted winning percentage. Adjusted winning percentage is essentially each team’s Pythagorean Expectation, with the exception that, instead of goals for and goals against, I use adjusted goals for and adjusted goals against. In computing each team’s adjusted GF and adjusted GA, I simply take each team’s actual GF and GA, subtract shootout goals and empty netters, and then make a second order correction for schedule difficulty. In determining schedule difficulty, oppositional strength is determined through the goal differential of the opponent, the location of the game (i.e. whether it’s a home or away game), and the circumstances of the game – namely, whether or not it’s the second half of a back-to-back for the road team.

If you compare these rankings to the actual standings, most teams are similarly positioned. There is, however, one notable outlier.

The Rangers are currently 9^th in the league in points per game, yet 26^th by this metric. Not surprisingly, they’ve had a ton of success in the shootout so far (record: 9-4), which is basically equivalent to sheer luck. While some may point to the Rangers shot differential, especially at EV, as evidence of them being not that bad of a team, I’m inclined to disagree. Reason being: they're in the red in terms of expected goals, which suggests that they’ve been below average in terms of shot quality for, shot quality against, or both.

Of course, there are a few teams who can be labeled as either lucky or unlucky in general – notwithstanding the fact that that these rankings aren’t too different from the standings. In other words, teams who are either better or worse than these rankings would suggest.

In terms of teams that probably aren’t as good as their adjusted winning percentage would indicate, I’m thinking of BOS, FLA, and PHI. These teams have all been greatly aided by the percentages this year. I think that the success that each of these teams has experienced thus far is unlikely to continue during the remainder of the regular season and the playoffs. Granted, the Flyers outperformed their underlying numbers last season as well. As the sample size in games played increases, it becomes increasingly difficult for one to point to randomness in an attempt to account for success with the percentages. On the other hand, I find it very difficult to look at a team that’s scored 15 shorthanded goals and conceded none and say that they haven’t been at least somewhat fortunate. I just don’t think that they're an inherently good hockey team.

And for teams in which the opposite is true, I’m thinking of OTT, LAK, COL, and TOR. These teams have all been – for lack of a better term – utterly screwed by the percentages this season, to the point where none of them have a realistic shot at making the playoffs. This is unfortunate in the sense that, if you were to compare this group of teams with the three listed above, I don’t think that there’s much to choose between them. Hell, I think that one could make a reasonable argument for the Kings being the best team of the seven – at least, looking at it in terms of which team is most likely to experience success from this point forward.

Anyway, here the playoff probabilities for all 30 teams (updated on 02/19/09). The left hand column contains seeds 1-15 in each conference, with the corresponding column for each team showing the probability of finishing the season in that position, expressed as a percentage. So, for example, the Blackhawks have an (approximately) 1% chance of finishing in 1st place in the West. The final two rows contains each team's probability of making the playoffs (in the second last row) and each team's probability of winning the division (in the last row). Future game probabilities are based on the respective adjusted winning percentages of the involved teams, game location, and whether or not the game is the second half of a back-to-back for the road team.

Coming off a Win/Loss: The effect of Prior Results

I’ve often wondered if the outcome of a team’s previous game has any affect on the result of that team’s subsequent game. Intuitively, I wouldn’t expect there to be much of an effect. The outcome of any given game is determined by many different factors, some of which are known to have a large effect.

While the result of the previous game could conceivably be one of these factors, it would probably rank pretty far down the list in terms of importance. In other words, if there is such an effect, I would expect it’s magnitude to be small.

That said, I’ve heard it argued before that the previous game does in fact have an effect on a team’s performance in the following game, so it’s something worth examining, I think.

On the one hand, some have suggested that the momentum of winning the previous game carries over to the next game, thus enhancing a team’s chance of success. According to this line of reasoning, the average team should do slightly better when coming off a win than when coming off a loss.

Conversely, others have suggested that winning breeds complacency, with losing having the opposite effect. This approach predicts that teams should do better when coming off a loss, on average.

I don’t think that either of these arguments have much merit. Both are based on the idea that psychological factors have a measurable effect on game outcomes, a premise with which I personally disagree. While casual fans often resort to folk psychology when discussing success and failure at the NHL level, its relevance has never, to my knowledge, been demonstrated through actual evidence.

In any event, I attempted to determine if the preceding game has any effect on following game results. My methodology was pretty straightforward. The sample included all regular season games played during the seasons of 2005-06, 2006-07 and 2007-08. Each game played was classified as a win, a loss, or a tie for both of the involved teams. For the sake of simplicity, any game that went past regulation was considered to be a tie. I then looked at whether that team won, lost or tied in its next game. Here are the results for 2007-08. The teams that had a better record when coming off a win compared to coming off a loss are shaded green. Teams for which the opposite was true are shaded orange.

Below is a chart of the average winning percentages of all 30 teams in each situation (coming off a win, coming off a loss, and coming off a tie) for all three seasons. The left hand column shows the average winning percentage for all 30 teams in games played after a win. The middle and right hand columns do the same, only for games where the team was coming off a loss and tie, respectively. It’s necessary to look at the average winning percentages rather than the aggregate winning percentages for one simple reason: better teams, by virtue of winning more games, tend to play a higher percentage of their games when coming off a win. For example, the Thrashers played a mere 18 games coming off a win last season; Detroit played 46. It needn’t be explained as to how this could confound the results.

Also included is a chart that breaks down the number of teams that had a better record after winning vis-à-vis their record after losing, and vice-versa.

The results are pretty consistent with my expectation in that the effect of the preceding game appears to be fairly small. In the 90 ‘team-seasons’ analyzed, 41 teams had a better record after winning, whereas the other 49 had a better record after losing. The average winning percentage for teams coming off a win was slightly less than 0.49. For teams coming off a loss, that figure was approximately 0.505. Therefore, it can be said that teams have, since the lockout, done slightly better after losing their previous game than they have when coming off a win. Of course, the margin is quite small and well within the potential range of random variance. Even supposing that the results are statistically significant, the influence of a team’s preceding game upon the outcome of its following game appears to be limited.

Even Strength Shooting Percentage

To what extent is team-to-team variation in even strength shooting percentage the product of random variation? I'm not sure what the answer is, but I suspect that the contribution is substantial. I've included several graphs below in order to illustrate this. The table below the first graph contains the data upon which each distribution is based.



The first graph. The yellow line is the actual spread in EV ( note: 5 on 5 only) shooting percentage that exists among NHL teams at this point in the 2008-09 NHL season.

The X-axis contains the percentage 'categories' in which the figure listed is the midpoint value of the category.

They Y-axis is the relative frequency of each individual percentage 'category'.

As an example, 6 teams in the NHL this year currently have an EV shooting percentage that is between 0.08 and 0.085. As there are 30 teams in the league, the relative frequency is 0.2 ( as 6/30 = 0.2). The midpoint value for this category is 0.0825. Therefore, the relative frequency of the '0.0825' category is 0.2.

The pink line shows the predicted spread in EV shooting percentage if each team had the exact same underlying shooting percentage at ~0.085 ( i.e. the league average 5-on-5 shooting percentage). This was determined through the following.

1000 "seasons" were simulated.
For each "season", each team has an artificial shooting percentage.
This percentage is the number of goals that a team scores over x number of trials.
The number of trials is equivalent to the number of EV shots that the team has taken through this point in the season.
The probability of "scoring" in each individual trial is the same for every team at 0.085.
Therefore, any team-to-team variation will be the product of randomness.

A specific example will hopefully make this clear.

Philadelphia has taken 984 shots at EV at this point in the 2008-09 season. Therefore, Philadelphia has 984 trials. The probability of scoring in each individual trial for Philadelphia is the league average EV shooting percentage at ~0.085. In Philadelphia's first "season", they scored 107 times. As 107 / 984= ~0.109, Philadelphia's EV shooting percentage for their 1st "season" is 0.109. I then did this for every team and repeated the process 100 times (i.e. simulated 100 seasons). Here's how the first 48 or so shaped out:




Even though the probability of a goal on any given "shot" is 0.085, the artificial shooting percentage will necessarily differ from 0.085 due to insufficient sample size. While it goes without saying, as the sample size (number of trials) increases, any given team's artificial shooting percentage will more closely approximate 0.085. Therefore, for teams that have taken more shots through this point in the 2008-09 season will have more "trials". The spread in shooting percentage for these teams will be lower due to them having a greater number of trials. For example, the standard deviation for Detroit's 100 seasons is ~0.007. By comparison, the same value for Pittsburgh is ~0.009.

The same rules regarding the x and y axes that apply to the yellow (actual) distribution also apply to the pink (random) distribution. The relative frequency for the pink distribution is the proportional representation of each artificial shooting percentage category. As an example, as there were 100 "seasons" and 30 teams, the entire sample consisted of 3000 artificial shooting percentages. 601 artificial percentages fell between 0.08 and 0.085. The relative frequency for the '0.0825' category is therefore ~0.2, as 601/3000 = ~0.2.


As many will note, the spread between the worst ( NYI at 0.069) and best ( BOS at 0.108) teams appears to be sizable, as is indicated by the breadth of the yellow distribution.

However, the pink distribution is itself fairly broad. In fact, it very closely resembles the yellow distribution. As would be anticipated, the yellow distribution is slightly broader than than its counterpart, but the difference is not large. This suggests that much of the inter-team variation in EV shooting percentage is the result of randomness.


The second graph, shown above, contains a 'smoothed' version of the actual distribution, which is represented by the dark line. The average shooting percentage in the league is currently ~0.085, as has been mentioned. The standard deviation is currently ~0.01. The dark graph is simply a normal distribution (bell curve) with a mean of 0.085 and standard deviation of 0.01.

The light line is merely the pink distribution reproduced. Again, the two distributions are very similar to one another.

The fact that the actual distribution is somewhat broader than the expected distribution shows that teams do indeed differ in their underlying shooting percentage at EV. Nonetheless, this variation is only very slightly larger than what would be predicted by chance alone. The underlying differences appear to be minimal.

Vic Ferrari has done a lot of excellent, excellent work over at his site that is similar to this. Much of his work has examined the ability of individual players to influence shooting and save percentage while on the ice. His findings are comparable in that the vast majority of inter-individual variation seems to be due to random variation.

EDIT: I've included some supplementary data tables for the purposes of clarity.

I should mention that the data I used for this post was obtained at behindthenet -- an awesome site that I highly recommend. Without it, this post wouldn't have been possible.

The Bruins

The Bruins have been one of the surprise teams this year, what with them having the best record in the league at the halfway point (few would have predicted this to be so). They also have the best goal differential, so it's not as if they've been lucky in the conventional sense by winning a lot of close games. However, just because a team's record is proportional to its goal differential doesn't necessarily mean that it hasn't been lucky.


This is a chart showing how the Bruins have fared in various game situations so far this season. The numbers are as of 01/07/08. A couple things can be said about these numbers:

1. The Bruins success appears to largely be a product of the percentages. They have the best shooting percentage in the league, as well as the best save percentage. This also holds true at even strength.

2. For a team with such a good record and goal differential, the Bruins are anomalous in that they're pretty average with respect to shot differential. In fact, they get outshot on average.

The Percentages

The problem for Boston is that there isn't a great deal of repeatability in terms of the percentages, particularly at even strength. This post by Tyler at mc79hockey demonstrates how the sum of a team's even strength shooting percentage and its even strength save percentage tends to regress to 100 as the season progresses. The Bruins currently sit at ~105. If I was a betting man, I'd place money on that figure significantly decreasing by April.

Are Boston's percentages at all sustainable?

We know from past posts that, while fluctuations in the percentages do indeed have little sustain in the future, a team is able to reliably influence its shooting/save through shot quality. Shot quality is moderately correlated with the percentages and is substantially reliable. Thus, over a sufficiently large sample of games, there would still likely be team-to-team variation in the percentages, with this effect being mediated by shot quality.

In past seasons, the team that leads the league in shot quality for typically has a shot quality index of roughly 1.1. That is, that team takes shots that, on average, are 10% more likely to result in a goal than the average team.

Conversely, the team that leads in the league in shot quality against typically has a shot quality index of roughly 0.9. That is, that team allows shots that, on average, are 10% less likely to result in a goal against than the average team.

If we make the very conservative assumption that Boston currently leads the league in both shot quality for and shot quality against, then we can estimate the Bruins' expected shooting percentage based on these adjustments.

Expected shooting percentage = shot quality for index * league average shooting percentage
Expected save percentage = 1- ( shot quality against index * league average shooting percentage)

League average shooting%: 0.0917

Boston's expected shooting percentage: 1.1*0.0917 = 0.10
Boston's expected save percentage: 1-(0.9*0.0917)= 0.917

Boston's actual shooting percentage: 0.118
Boston's actual save percentage: 0.93

Therefore, even if we assume that Boston currently leads the league in both shot quality for and shot quality against, the Bruins' have still outperformed their expected shooting percentage and expected save percentage. While far from constituting definitive proof of good luck, it is suggestive of it.

In actuality, the Bruins have not been leading the league in either shot quality for or in shot quality against. Hockeynumbers tabulates data on shot quality that is periodically updated throughout the season. While the data is only available for specific game situations (EV, PP, SH), figures for overall shot quality can be obtained by dividing each team's expected goals for/goals against by their corresponding shots for/shots against total, and then expressing the resulting figure relative to the league average.

In addition to having a negative shot differential, the Bruins are below average in both shot quality for and shot quality against, thus making it even less likely that they'll replicate their impressive shooting/save percentage in second half. Indeed, Boston is in the red in terms of its expected goal differential, as is nicely illustrated here

Boston's 'true' even strength shooting/save percentage

As displayed in the table at the beginning of the post, the fact that Boston has managed to lead to the league in both shooting and save percentage is largely tied to even strength play -- that is, the overall percentages are largely being driven by the exceptional even strength percentages. Therefore, the sustainability of Boston's overall percentages is critically contingent upon sustaining its high percentages at even strength. Boston's shooting/save percentage at even strength will likely fall to something more reasonable by the time the season has ended. At the same time, however, it's unlikely that its EV shooting/save percentage is merely average.

To illustrate this, assume that Boston's true underlying even strength shooting percentage is exactly league average (~0.084), with the same holding true for its even strength save percentage (~0.916). The Bruins have taken 890 shots at even strength so far this season, while allowing 920. If a team with a true EV shooting percentage of 0.084 takes 890 shots, the probability of shooting 0.109 or better by chance alone is remote (about 4 times per thousand). Likewise, if a team with a true EV save percentage of 0.916 has 920 shots against, the probability of having a save percentage better than or equal to 0.939 by chance is equally minuscule (about 5-6 times per thousand). Thus, Boston's underlying EV shooting/save percentage -- while almost certainly lower than what they've attained thus far -- is probably above average. Therefore, a complete regression to the mean is unlikely.

The First Goal

In hockey, scoring the first goal is important. Last season, every single team in the league had a better record in games where they scored first compared to games where they did not.

However, one has to wonder: is scoring the first goal as important it's made out to be? For example, hockey media types love to harp on the importance of scoring first, invariably citing team A's record when managing to do, or how team B's losing streak is explicable through its tendency to surrender the lead early in the game. Not only does this emphasis conflate cause and effect, but it's insufferably repetitive and trite. One would intuitively expect the team that scores first to have a higher probability of winning, and it's fairly obvious that such a relationship exists. In fact, I suspect that the probability of winning when scoring first is not significantly different than what would otherwise be expected on a mathematical basis.

Moreover, scoring in general is important, whether it be the first goal of the game or the last one. Any given goal is more or less significant and potentially determinative of the game's outcome. To make the distinction between the first goal and any other goal scored during the game just smacks of arbitrariness. Scoring first is probably more correlated with winning than, say, scoring second, but I'd be surprised if the difference was large, and shocked if it was large enough to warrant the special attention.

Thus, this post seeks to answer two questions:

1. When a team scores scores first,what percentage of the time does it win the game? Is this value any different from what probability theory predicts it to be?

2. How much more important is the first goal than the second goal?

The first question can be answered through application of the poisson distribution. As Alan Ryder explains in this paper, goal scoring in hockey is essentially a poisson process. Ryder has determined that a team's probability of winning by z goals in regulation at any particular point during the game can be found through application of the following formula in Microsoft excel.

Pr(Win by z) = EXP(-(mt+vt)) * (mt/vt)^(z/2) * BESSELI(2*SQRT(mt*vt),ABS(z))

where:

m = that team's average goals for per game*
v = that team's average goals against per game*
t = the time remaining in regulation divided by 60
z = the margin of victory

* - Adjusted goals ought to be used here as they provide the best measure of a team's true ability to score and prevent goals.

Through use of this formula, the theoretical probability of winning in regulation for a team that scores first can be determined.

In the 2007-08, there were 1222 games that had at least one goal scored in regulation. On average, the first goal was scored just prior to the 12 minute mark of the 1st period. Thus, our value for t is 0.803.

The m and v values are, for the purposes of the formula, 2.639 and 2.545. These figures are adjusted to reflect the following:

1. That the team that scores first is, on average, slightly better than the average team.
2. That the team that gives up the first goal is, on average, slightly worse than the average team.
3. That the home team is more likely to score the first goal.

For determining the probability of winning, the z value ranges from 0 to sufficiently high n (~10), as the team that scores first must only maintain the existing margin -- or increase it -- in order to win. For the probability of losing, z is -2 to sufficiently low n (~-10), as the trailing team must outscore the opposition by 2 or more goals in order to win. For the probability of a tie, z is -1, as this will restore the original margin of zero.

Thus, all of the necessary input variables having been determined, the theoretical probability of winning when scoring first can be computed. Below is a comparison of the theoretical probability against the actual probability.



As can be seen, the actual and theoretical probabilities more or less mirror one another, save for the fact that probability theory predicts ties to occur less frequently than they actually do. This reflects the fact that teams do, to some degree, play to the score, particularly as the end of regulation nears. The upshot of there being more ties in reality is that the first goal is somewhat more valuable than what the values expressed in the chart would otherwise indicate. For example, while the actual probability of winning is nominally lower than its theoretical counterpart (0.599 vs 0.615), if one examines only those games resolved in regulation, the actual probability of winning is slightly higher than the theoretical probability (0.764 vs 0.744). Nonetheless, the important part is that the theoretical and actual values are essentially equivalent to one another. If the actual probability of winning was substantially higher than the theoretical probability, the large amount of emphasis placed on scoring first may be justifiable. However,the fact that they are virtually the same means that the advantage conferred by scoring first is neither surprising nor contrary to expectation, thus making it unworthy of mention.

What about the importance of the second goal vis-a-vis the first goal?



Scoring second is very nearly as highly correlated with winning as scoring first. And yet, it is the latter that -- rather unfairly -- receives all of the attention. In this sense, the emphasis that's placed on scoring first seems more than a little arbitrary. It's simply not very accurate to accord the first goal special status when, in actual fact,the vast majority of goals scored throughout the course of a hockey game are significant.