Pythagorean Theorem for Hockey: Part D'oh

by D'ohboy on Jan 28, 2010 9:38 PM EST

After our discussion the other day, I went back to the drawing board to come up with a Pythagorean Theorem for hockey. Beware, for below the jump, there be graphs and math!

I thought about it for a while and, while I liked my PP/PK/5v5 method, several people asked for something more akin to James' original formulation that takes Runs For and Runs Against and spits out a winning percentage. However, we all quickly realized why this doesn't work for hockey - baseball has binary results (wins or losses), but hockey doesn't.

After pondering how to square this circle for a bit, I had an epiphany: what we really want to know is point totals, not winning percentage!

Armed with this realization, the NHL.com website, Excel and some patience, I devised the following method:

Figure out how many points the average team acquires in a given year (it turns out to be right around 91.3 every year since the lockout);
Figure out the average goal differential (surprise, surprise. . . it's ZERO!!!);
Figure out the average goal differential per point both positive and negative (it averages out to about 2.65). This was a bit of a kludge and deserves explanation because someone with an actual background in statistics could probably come up with a much better means of doing this. To get this number, I took the top and bottom team in the league by points in each year and figured out the absolute value of how far they deviated from the mean in both points and goals differential. Then I divided the goal differential number by the points number and averaged the top and bottom number. So for example, last year, SJS was the top team in the league with 117 points and a goal differential of 53, while NYI was the bottom team in the league with 61 points and a -78 differential. The average number of points in the NHL last year was 91.4. For SJS: 117-91.4=25.6 and 53 Diff/25.6=2.07 Gdiff/point. For NYI: 91.4-61=30.4 and 78 Diff/30.4=2.5658 GDiff/point;
Average all the data over the four full years since the lockout and do some basic algebra and...

I came up with the following:

Over a full 82-game season a team's expected points PE = 91.3 + Differential/2.644
Mid-season, you need to modify the equation like so: PE = (91.3/82)*Games Played + Differential/2.644

Again, this only works for post-lockout rules. I'm guessing that pre-lockout data would be easy enough to collect and that would allow for a much larger sample size.

I'm going to post the spreadsheets below so that others can check my work and I'll apply the theorem to this year's standings in a following post:

2005	-2006
Team	Points	GF	GA	Differential
DET	124	305	209	96
OTT	113	314	211	103
DAL	112	265	218	47
CAR	112	294	260	34
BUF	110	281	239	42
NSH	106	259	227	32
CGY	103	218	200	18
NJD	101	242	229	13
PHI	101	267	259	8
NYR	100	257	215	42
SJS	99	266	242	24
ANA	98	254	229	25
COL	95	283	257	26
EDM	95	256	251	5
MTL	93	243	247	-4
TBL	92	252	260	-8
VAN	92	256	255	1
TOR	90	257	270	-13
ATL	90	281	275	6
LAK	89	249	270	-21
FLA	85	240	257	-17
MIN	84	231	215	16
PHX	81	246	271	-25
NYI	78	230	278	-48
CBJ	74	223	279	-56
BOS	74	230	266	-36
WSH	70	237	306	-69
CHI	65	211	285	-74
PIT	58	244	316	-72
STL	57	197	292	-95
	91.36666667			0

Top	32.63333333	96	2.941777324
Bottom	34.36666667	95	2.764306499
			2.853041911

2006	-2007
Team	Points	GF	GA	Differential
BUF	113	308	242	66
DET	113	254	199	55
NSH	110	272	212	60
ANA	110	258	208	50
SJS	107	258	199	59
DAL	107	226	197	29
NJD	107	216	201	15
VAN	105	222	201	21
OTT	105	288	222	66
PIT	105	277	246	31
MIN	104	235	191	44
ATL	97	246	245	1
CGY	96	258	226	32
COL	95	272	251	21
NYR	94	242	216	26
TBL	93	253	261	-8
NYI	92	248	240	8
TOR	91	258	269	-11
MTL	90	245	256	-11
CAR	88	241	253	-12
FLA	86	247	257	-10
STL	81	214	254	-40
BOS	76	219	289	-70
CBJ	73	201	249	-48
EDM	71	195	248	-53
CHI	71	201	258	-57
WSH	70	235	286	-51
LAK	68	227	283	-56
PHX	67	216	284	-68
PHI	56	214	303	-89
	91.36666667			0

Top	21.63333333	66	3.050847458
Bottom	35.36666667	89	2.516493874
			2.783670666

2007	-2008
Team	Points	GF	GA	Differential
DET	115	257	184	73
SJS	108	222	193	29
MTL	104	262	222	40
PIT	102	247	216	31
ANA	102	205	191	14
NJD	99	206	197	9
MIN	98	223	218	5
DAL	97	242	207	35
NYR	97	213	199	14
COL	95	231	219	12
PHI	95	248	233	15
WSH	94	242	231	11
OTT	94	261	247	14
CGY	94	229	227	2
BOS	94	212	222	-10
CAR	92	252	249	3
NSH	91	230	229	1
BUF	90	255	242	13
EDM	88	235	251	-16
CHI	88	239	235	4
VAN	88	213	215	-2
FLA	85	216	226	-10
PHX	83	214	231	-17
TOR	83	231	260	-29
CBJ	80	193	218	-25
NYI	79	194	243	-49
STL	79	205	237	-32
ATL	76	216	272	-56
LAK	71	231	266	-35
TBL	71	223	267	-44
	91.06666667			0

Top	23.93333333	73	3.050139276
Bottom	20.06666667	44	2.19269103
			2.621415153

2008	-2009
Team	Points	GF	GA	Differential
SJS	117	257	204	53
BOS	116	274	196	78
DET	112	295	244	51
WSH	108	272	245	27
NJD	106	244	209	35
CHI	104	264	216	48
VAN	100	246	220	26
PIT*	99	264	239	25
PHI*	99	264	238	26
CGY	98	254	248	6
CAR	97	239	226	13
NYR	95	210	218	-8.00
MTL	93	249	247	2
FLA	93	234	231	3
STL	92	233	233	0
CBJ	92	226	230	-4
ANA	91	245	238	7
BUF	91	250	234	16
MIN	89	219	200	19
NSH	88	213	233	-20
EDM	85	234	248	-14
DAL	83	230	257	-27
OTT	83	217	237	-20
TOR	81	250	293	-43
PHX	79	208	252	-44
LAK	79	207	234	-27
ATL	76	257	280	-23
COL	69	199	257	-58
TBL	66	210	279	-69
NYI	61	201	279	-78
	91.4			0

Top	25.6	53	2.0703125
Bottom	30.4	78	2.565789474
			2.318050987

Four Year Average:

Avg. Pts	Avg. GDiff/Pt
91.3	2.644044679

If this FanPost is written by someone other than one of the blog's editors, the opinions expressed in it do not necessarily reflect those of this blog or SB Nation.

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Comments

Reading this, I’m realizing just how much I hate math. Credit to you for doing this legwork. Holy crap.

Familiar Rapports: Bald Pollack, F&B, Gould Old Days.

Lobbies: Osala, Perreault, Erskine, Pothier, Neuvirth, Flash.

Fan of: Mean Lars Backstrom, Line Mashing, Cake.

by Whiter Mage on Jan 28, 2010 10:21 PM EST reply actions

Reading this, I’m realizing how much I LOVE math. Haha :) Great job D’ohboy – very interesting post.

"No Brooks Laich, no win. Know Brooks Laich, know win."

by kellobellow on Jan 28, 2010 11:04 PM EST up reply actions

What’s weird is that I used to hate math. And now I’m doing stuff like this in my spare time. I was so absorbed in this I forgot to eat dinner. :)

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:24 PM EST up reply actions

It happens – a lot of times, teachers, middle/high school math classes, etc. can discourage people from liking math. (Part of why I do what I do!) It takes finding something you are truly interested in to discover what math can really do.

"No Brooks Laich, no win. Know Brooks Laich, know win."

by kellobellow on Jan 29, 2010 8:39 AM EST up reply actions

I think for me it was one part frustration/youthful impatience, and one part rebellion against my mother, who was a physicist.

This is not a game of who the f*ck are you...

by D'ohboy on Jan 29, 2010 9:28 AM EST up reply actions

Waiting for PEs…….

by fnralch on Jan 28, 2010 10:32 PM EST reply actions

They’re up now.

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:24 PM EST up reply actions

I had an epiphany: what we really want to know is point totals, not winning percentage!

Please tell me you lurked my most recent run in with MarioD.

Great work and great effort. I dislike math enough that I would never venture to do this. My reservation is that you only picked the top and bottom team to determine the goal differential per point. It seems like you should be using the data for each team but I’m not exactly sure how you would do it.

Now let's say you and I go toe to toe on bird law and see who comes out the victor.

by Rob Parker on Jan 28, 2010 10:43 PM EST reply actions

No, what happened?

I’m also not happy with the kludge, but it was a fast and dirty way of establishing a good deviation from the mean. If I were someone with actual statistician, I’d figure out how to do an actual measurement of deviation, then I’d use that as the multiplier. If you read the following post, though, you’ll see that it actually works quite well. Honestly, I’m actually a little shocked that it worked as well as it did.

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:23 PM EST up reply actions

Points percentage is win percentage. Idiot.

Now let's say you and I go toe to toe on bird law and see who comes out the victor.

by Rob Parker on Jan 28, 2010 11:31 PM EST up reply actions

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:36 PM EST up reply actions

I’m guessing you’re talking about MarioD, since I was using points totals. . .

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:37 PM EST up reply actions

Yeah, he spent a lot of time and spewed a lot of venom because we ignorant Caps fans could not understand the simple concept that points are imaginary and meant to be a proxy for win percentage, and that NHL standings in fact are determined by win percentage.

Now let's say you and I go toe to toe on bird law and see who comes out the victor.

by Rob Parker on Jan 29, 2010 12:11 AM EST up reply actions

Then he’s not just a moron, he’s an ignorant moron. Where was this? I’d love to read it.

This is not a game of who the f*ck are you...

by D'ohboy on Jan 29, 2010 12:44 AM EST up reply actions

start here and scroll down at your own risk.

by Natty Bumppo on Jan 29, 2010 10:58 AM EST up reply actions

Great work. It’s pretty impressive how consistent those numbers are, so it would definitely be interesting to see what it’s like for the whole league.

You should be able to calculate the standard deviation with an excel formula. The longer but still easy way to do it would just be to take the sum of all the deviations from the mean and divide it by the number of teams (30).

Of all our iniquities ignorance may be the worst

by Killer_Carlson on Jan 29, 2010 2:04 AM EST up reply actions

I’m guessing that the formula is something like =SDEV, right?

I’ll try it out and see what I get. s

This is not a game of who the f*ck are you...

by D'ohboy on Jan 29, 2010 8:10 AM EST up reply actions

I don’t usually use it, but it is something like that. You should be able to find it easily from the pull down menus, or if all else fails do a help search for standard deviation.

Of all our iniquities ignorance may be the worst

by Killer_Carlson on Jan 29, 2010 3:46 PM EST up reply actions

Here’s how you do it: plot all differentials, make a linear fit, and the slope will give you what you’re looking for.

by fnralch on Jan 28, 2010 11:37 PM EST up reply actions

By: “make a linear fit” do you mean “just eyeball it”?

This is not a game of who the f*ck are you...

by D'ohboy on Jan 28, 2010 11:58 PM EST up reply actions

No, there’s free soft (e.g., xmgrace for linux) that can do any kind of fit. The result, however, would be just marginally different from your average.

by fnralch on Jan 29, 2010 12:07 AM EST up reply actions

Yeah. I was guessing that, by getting the two “endpoints” of the line, the average would more or less be the slope.

This is not a game of who the f*ck are you...

by D'ohboy on Jan 29, 2010 12:43 AM EST up reply actions

It might be close, but there’s a statistical way to find the actual line of best fit… I don’t know how to on any computer software, but I know how on a TI-83, haha :)

"No Brooks Laich, no win. Know Brooks Laich, know win."

by kellobellow on Jan 29, 2010 8:38 AM EST up reply actions

B=(X’X)^(-1)X’y in my opinion. That’d be a the simple linear regression, but I’m not sure there shouldn’t be some weighting for other factors, like sv%, but that’s not getting added without a third dimension.

Not computationally efficient, but we’re not dealing with large variable sets, either.

Only YOU can prevent idiots from commenting!

by Knee high to a duck on Jan 29, 2010 6:09 PM EST up reply actions

Its probably close, but we can always help you to get just a bit closer.

Of course, at some level, the precision starts to matter less and less since a) you get into fractional points and b) its a hockey game, so bad bounces, bad calls, and fluke plays are going to undermine the value of additional precision at some point.

Still fun to ponder… maybe I’ll dig a bit deeper Sunday and finally get some use from the half dozen math textbooks I still have from college.

by iwearstripes on Jan 29, 2010 5:39 PM EST up reply actions

sorry, this comment is off the subject.
Can you please tell me how to include excel tables in fanposts? I’ve been working on one for some time but just can’t figure out how to get my tables in it. I thought it was impossible, and your post proved me otherwise.
I tried to send a private message, but for the life of me, couldn’t find how to do that either…
thanks for any help.

by benjik on Jan 30, 2010 5:33 PM EST reply actions

I just copied the table and posted it into the text of the post….

…and right about now you’re like “f*$& you a$$hole I tried that already and it didn’t work!!!”

I’m guessing the problem you had was that your table showed up with all sorts of formatting gobbledygook above and below the table. To solve this problem, you need to preview your post and find out exactly where the formatting text is within the post. Then go back to the editing page, switch over to HTML view and find the offending formatting text and delete it (it’s usually bracketed by something like “start fragment” and “end fragment”). Then just keep going back and forth from preview to HTML editor view until it’s all gone. It takes a couple of minutes per table.

If you’re having problems copying and pasting the table itself, I recommend copying it into a MS Word doc, then using the “Paste from MS Word” function in the editor page.

Good luck!

This is not a game of who the f*ck are you...

by D'ohboy on Jan 30, 2010 11:57 PM EST up reply actions