Puckerings archive: Harmonic Points (08 Apr 2002)

What follows is a post from my old hockey analysis site puckerings.com (later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on April 8, 2002.

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Harmonic Points
Copyright Iain Fyffe, 2002

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The way it is now, assists are more important than goals in determining scoring championships. Why do I say this? Because for every goal, there are 1.7 assists awarded. Therefore, playmakers have an advantage over goal-scorers, because there are more assists for them to get a piece of. This is not fair. There is absolutely no evidence that playmaking is more important than goal-scoring in terms of scoring goals.

Total Hockey's Adjusted Scoring stats account for this somewhat, by using historic assist rates, which are lower than current rates. But it does not go far enough. Since there is no evidence to indicate which of goal-scoring and playmaking is more important, it is only fair to assume that they are equally important. Thus, when determining a "scoring champion", we should adjust the number of assists to equal the number of goals, on a league-wide basis.

More to the point, I believe we can further refine how we decide who is a "champion" scorer. For instance, say we have three players, all of whom have 80 adjusted scoring points. Player A has 25 goals and 55 assists, Player B has 40 goals and 40 assist, and Player C has 55 goals and 25 assists. I contend that Player B is the superior scorer. Why? Because he is less reliant on other players to produce goals. Player A is a playmaker; if he has no one of talent to pass to, his scoring will suffer. Player C is a goal-scorer; he needs a playmaker to maximize his value. Player B is a more complete player; he is less reliant on teammates, and is therefore a superior individual player.

I do, of course, realize that hockey is a team game, and it takes an entire team to win. But when we are assessing individual players, we should remove the effect of his teammates as much as possible. In this case, we do this with the Harmonic Points system (HP).

HP is based on the mathematical concept of the harmonic mean. The harmonic mean of two numbers is a middle number such that by whatever part of the first term the middle term exceeds the first term, the middle terms exceeds the second term by the same part of the second term. Whew! In other words, if the harmonic mean is 20% (of the lesser term) greater than the lesser term, it will be 20% (of the greater term) lower than the greater term. Still confused? Maybe a numerical example will help.

Take two numbers: 100 and 200. The harmonic mean of these numbers is 133. 133 is 33% (of 100) greater than 100, and 33% (of 200) less than 200. 

I won't keep you in suspense any longer. Here's how to compute HP (which is simply the harmonic mean of goals and assists, times two):

HP = 2 x {(2 x G x A) / (G + A)}

Where HP is Harmonic Points, G is goals, and A is assists. The formula is multiplied by two to retain the "look" of the number of points, since we're taking an average of goals and assists. A player who has an equal number of adjusted goals and adjusted assists will have HP equal to his adjusted points.

In applying HP, I have used Total Hockey's Adjusted Scoring statistics. This is to eliminate much of the bias created by a player's time and place, allowing us to compare players from different eras. In addition, I will be indicating Adjusted Games Played (games played divided by length of schedule times 82), which are not disclosed in Total Hockey, but should be.

But using the idea that playmaking and goal-scoring are equal in importance, we cannot use Adjusted Scoring stats as they are. Adjusted Assists are based on historic assist rates, which, of course, are higher than historic goal rate. So I have adjusted Adjusted Assists to use the same base figure as goals.

Here are the single-season NHL leaders in HP per 82 Adjusted Games Played (minimum 20 AGP), from 1917/18 to 2000/01. There have been 34 100-HP pace seasons in NHL history:

Rank	Name	Club	Year	AGP	HP	Per 82
1.	Howie Morenz	Montreal	1927/28	80	145	149
2.	Mario Lemieux	Pittsburgh	1992/93	59	103	143
	Mario Lemieux	Pittsburgh	1995/96	70	122	143
4.	Wayne Gretzky	Edmonton	1983/84	76	128	138
	Mario Lemieux	Pittsburgh	1988/89	78	131	138
6.	Wayne Gretzky	Edmonton	1981/82	82	129	129
7.	Wayne Gretzky	Edmonton	1984/85	82	127	127
8.	Wayne Gretzky	Edmonton	1982/83	82	122	122
	Mario Lemieux	Pittsburgh	2000/01	43	64	122
10.	Howie Morenz	Montreal	1930/31	73	108	121
11.	Wayne Gretzky	Edmonton	1986/87	81	117	118
12.	Phil Esposito	Boston	1970/71	82	115	115
13.	Mario Lemieux	Pittsburgh	1987/88	79	110	114
14.	Ralph Weiland	Boston	1929/30	82	113	113
15.	Irvin Bailey	Toronto	1928/29	82	112	112
	Jaromir Jagr	Pittsburgh	1995/96	82	112	112
17.	Jaromir Jagr	Pittsburgh	1998/99	81	110	111
18.	Wayne Gretzky	Edmonton	1985/86	82	110	110
19.	Mario Lemieux	Pittsburgh	1989/90	60	80	109
20.	Phil Esposito	Boston	1973/74	82	108	108
	Mario Lemieux	Pittsburgh	1991/92	66	87	108
22.	Mario Lemieux	Pittsburgh	1996/97	76	99	107
23.	Phil Esposito	Boston	1971/72	80	103	106
	Wayne Gretzky	Los Angeles	1988/89	80	103	106
25.	Phil Esposito	Boston	1968/69	80	102	105
26.	Teemu Selanne	Anaheim	1998/99	75	95	104
27.	Aurel Joliat	Montreal	1927/28	82	103	103
28.	Ebbie Goodfellow	Detroit	1930/31	82	102	102
	Gordie Howe	Detroit	1952/53	82	102	102
	Jaromir Jagr	Pittsburgh	2000/01	81	101	102
31.	Mario Lemieux	Pittsburgh	1993/94	22	27	101
	Eric Lindros	Philadelphia	1996/97	52	64	101
33.	Wayne Gretzky	Los Angeles	1990/91	80	98	100
	Steve Yzerman	Detroit	1988/89	82	100	100

It's clear, by this analysis, that Mario Lemieux is the greatest offensive player in NHL history, bar none. His competition is, of course, Wayne Gretzky. Lemieux is on this list nine times to Gretzky's eight, but Lemieux also dominates the top of the list, appearing three times in the top five (to Gretzky's once), and seven times in the top 20 (to Gretzky's six). Lemieux is the only player with multiple 140-HP pace seasons (Gretzky never had one), and the only player with multiple 130-HP pace seasons (three, to Gretzky's one).

In terms of career HP per 82 AGP, there are four distinct classes of players: (1) Mario Lemieux, (2) Wayne Gretzky, (3) current stars in their prime, and (4) everyone else. Lemieux, through the 2000/01 season, has 1077 HP in 799 AGP, for a per-82 game figure of 111. No one else is even remotely close. Gretzky is second with an average of 96 (1805 HPP in 1543 AGP). Following these two are a bunch of players in the 80's, all current players in their prime: Eric Lindros, Jaromir Jagr, Teemu Selanne and Paul Kariya. Their averages will most likely drop over time to put them in the final group. The "everyone else" group is headed by Mike Bossy (75 average), Howie Morenz (73) and Phil Esposito (73). Other high averages belong to Gordie Howe, Jean Beliveau, Steve Yzerman, Joe Sakic, Marcel Dionne, and Bobby Hull.

The degree of separation between these classes of players serve to demonstrate how truly impressive Mario Lemieux's (and, to a lesser extent, Wayne Gretzky's) scoring exploits really are. These are the complete scorers, players who can carry a team's offence on their backs, all by themselves.