One of the more interesting dynamics in hockey is the relationship between Goals For and Goals Against, and how that relationship correlates to a team’s overall record.

Obviously, a team needs to score more than its opponent to win, but the question of exactly how the differential between goal-production and goal-prevention relates to a team’s overall record — and more importantly, whether it can be used as a predictive or projection-based tool — requires a bit of mathematical analysis.

Back in 2010, Chemmy at Pension Plan Puppets put together an intriguing analysis looking at the effects of save percentage on a team’s record. The analysis was based in part on one of the key concepts put forward by baseball stats guru Bill James.  James’ analysis, known as Pythagorean Expectation, sought to establish a formula by which a team’s expected percentage of wins (e.g. the percentage of games they “should have” won) could be predicted based upon the difference between a team’s runs scored and runs allowed.  Now, adapting this formula to hockey is nothing new; a number of different writers have in the past presented a hockey equivalent of the formula as an expected results percentage based on goals for and against.

The problem encountered is, James’ formula is geared toward projecting win percentages. This works fine for hockey analysis if wins are all were are interested in analyzing. If we wish to look at points — e.g. how many points “should” the team have earned based on goal-scoring and goal-prevention — we run into the issue of overtime/shootout losses.

Without getting too heavy into the mathematical theory behind this type of analysis (you can read more about that here), essentially the hockey version of Pythagorean Expectation works out to:

Expected Points Percentage = CORR * ((GF^2) / (GF^2) + (GA^2))

… where CORR is a correction factor to account for the difference between a team’s points percentage and win percentage. This is calculated as (points percentage) / (win percentage), or to be more precise (Points/(GP*2)) / (Wins/GP).

To begin, I recreated Chemmy’s analysis from 2010 based upon this season’s results to date.  In this analysis, I used each team’s individual correction factor for purposes of precision (as opposed to the Mean correction factor, which may be substituted to create a universal formula).

Here are the results:

Expected Points Percentage: Eastern Conference
TEAM GP WIN % PTS % GF GA EXP WIN % CORR EXP PTS %
NYR 68 .632 .684 188 148 0.617 1.081 0.668
BOS 68 .588 .610 222 164 0.647 1.038 0.671
FLA 68 .471 .566 166 191 0.430 1.203 0.518
PIT 68 .618 .654 219 173 0.616 1.060 0.652
PHI 68 .574 .625 220 197 0.555 1.090 0.605
NJD 69 .580 .616 195 179 0.543 1.063 0.577
OTT 70 .514 .579 216 206 0.524 1.125 0.589
WSH 69 .507 .551 184 193 0.476 1.086 0.517
WPG 69 .464 .522 181 195 0.463 1.125 0.521
BUF 69 .464 .522 171 194 0.437 1.125 0.492
TBL 68 .456 .507 191 233 0.402 1.113 0.447
TOR 69 .435 .493 200 212 0.471 1.133 0.534
CAR 69 .377 .486 181 207 0.433 1.288 0.558
NYI 69 .406 .478 160 206 0.376 1.179 0.433
MTL 69 .391 .464 183 193 0.473 1.185 0.561

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Expected Points Percentage: Western Conference
TEAM GP WIN % PTS % GF GA EXP WIN % CORR EXP PTS %
STL 70 .643 .693 183 135 0.648 1.078 0.698
VAN 69 .609 .667 215 172 0.610 1.095 0.668
DAL 69 .551 .587 185 183 0.505 1.066 0.539
DET 69 .638 .659 217 162 0.642 1.034 0.664
NSH 68 .588 .640 195 175 0.554 1.088 0.602
CHI 70 .529 .586 209 206 0.507 1.108 0.562
PHX 69 .493 .565 178 173 0.514 1.147 0.590
CGY 69 .464 .551 173 191 0.451 1.188 0.535
LAK 69 .464 .551 154 152 0.507 1.188 0.602
COL 70 .514 .543 183 187 0.489 1.056 0.516
SJS 67 .493 .560 184 173 0.531 1.136 0.603
ANA 69 .420 .493 171 193 0.440 1.172 0.516
MIN 69 .420 .493 150 193 0.377 1.172 0.442
EDM 68 .382 .434 180 206 0.433 1.135 0.491
CLB 69 .319 .370 161 223 0.343 1.159 0.397

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What these numbers provide us is another way to look at the results to date thus far, namely, how a team “should” have fared over a particular span.  Indeed, a quick scan of Expected Points Percentage as compared to Actual Points Percentage allows us to see who is over-performing/under-performing in terms of the relationship between their goal-scoring and goal-prevention trends.

At this point, you might be saying “Great, you updated Chemmy’s 2010 results to 2012. So what?”  Typically, this sort of analysis has been used retrospectively; that is to say, where could we expect the team to be, points-wise, had their save percentage been higher?  This is easy enough to do; simply plug the to-date data into the Expected Points Percentage equation above, and then adjust GF and/or GA accordingly to create those retrospective scenarios.   However, my interest lies not in what this data can tell us from the perspective of hindsight, but rather to what extent these numbers can be used to generate projections for games yet to be played.

A Simple Application of Expected Points Percentage to Games Remaining

In that light, the first question to ask is, in the event that GF/gm and GA/gm trends established over the (appx) 85% of the season to date were to hold steady over the course of the remaining (appx) 15% of the season, what would be the expected effect on the final standings?

Adjusting the formula for established per-game GF and GA trends (which will be applied to games remaining), the formula is written as follows:

EXP PTS PCT = CORR * ( (GF/gm*GR)^2 / ( ((GF/gm*GR)^2)+((GA/gm*GR)^2) ) )

In a hypothetical scenario where GF and GA trends remain similar over the remaining course of the season, applying the Expected Points Percentage formula to the maximum possible points remaining for each team produces the following results:

Projecting Expected Final Points: Eastern Conference
Team PTS EXP PTS % GR EXP PTS GR EXP FINAL PTS
NYR 93 0.668 14 19 112
BOS 83 0.671 14 19 102
FLA 77 0.518 14 14 91
PIT 89 0.652 14 18 107
PHI 85 0.605 14 17 102
NJD 85 0.577 13 15 100
OTT 81 0.589 12 14 95
WSH 76 0.517 13 13 89
WPG 72 0.521 13 14 86
BUF 72 0.492 13 13 85
TBL 69 0.447 14 13 82
TOR 68 0.534 13 14 82
CAR 67 0.558 13 15 82
NYI 66 0.443 13 12 78
MTL 64 0.561 13 15 79

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Projecting Expected Final Points: Western Conference
Team PTS EXP PTS % GR EXP PTS GR EXP FINAL PTS
STL 97 0.698 12 17 114
VAN 92 0.668 13 17 109
DAL 81 0.539 13 14 95
DET 91 0.664 13 17 108
NSH 87 0.602 14 17 104
CHI 82 0.562 12 13 95
PHX 78 0.590 13 15 93
CGY 76 0.535 13 14 90
LAK 76 0.602 13 16 92
COL 76 0.516 12 12 88
SJS 75 0.603 15 18 93
ANA 68 0.516 13 13 81
MIN 68 0.442 13 11 79
EDM 59 0.491 14 14 73
CLB 51 0.397 13 10 61

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Such a projection reveals a few interesting possibilities. The analysis suggests that if current goal-scoring and goal-prevention trends were to maintain the same rate over the span of the remainder of the season, one could expect:

(a) A minimum 89 points required to secure the final playoff spot in the East;
(b) A minimum 93 points required to secure the final playoff spot in the West;
(c) San Jose to sneak into the final playoff spot in the West;
(d) Toronto to miss the playoffs, miss out on the draft lottery and be looking at the 7th – 9th pick in the draft (depending on tiebreakers).

It is important to note that the application of the formula in this regard poses a significant problem: that being, we cannot predict the total number of goals for and against that will be scored in games yet to be played. But what we can do is use the formula to predict the goal-production and goal-prevention rates to reach a set number of points in a particular span of games … including those yet to be played. I’ll get to that in a moment.

Using Expected Points Percentage for In-Season Comparisons

Before proceeding, I should note there are further applications to this type of analysis than simply projecting expected points. We can use this system to add an additional level of evaluation to a team’s performance over a given span of games relative to the results over the course of the season to date.  For example, looking at the Leafs‘ last 16 games (in which they sport a 2-12-2 record) as compared to the games prior, we get the following:

Toronto Maple Leafs“>

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What results is a very stark look at not only just how precipitously the team’s GF/gm fell and GA/gm rose over that span, but the effect of the relationship between goal production and prevention — illustrated by the sharp change in the differential — on the team’s expected points percentage. Of particular note is the “PERF” column, used to denote actual performance relative to expected performance. What this indicates is, in the first 53 games the Leafs pretty much played to their level of expectation (62 actual points to 64 expected).  While we are all aware of how sharply the Leafs‘ play declined, the PERF for the previous 16 games gives a very good indication of just how far below expectation their play has fallen; even with a .282 expected points percentage, the team is managing to produce points at a rate of just 67% of that figure. Ouch.

Expected Points Percentage as a Predictive Tool

Returning to the predictive element of the equation, one might look at the above in-season comparison and be inclined to ask “well, what of the remaining 13 games?”  As we saw in the initial analysis, the Leafs are expected to earn 14 points in their last 13 games — if their GF and GA production over that span remains the same as it has been over the first 69 games.  But as we obviously have no way of actually knowing if those numbers will indeed hold, can such a projection still be made?

The short answer is yes, and it leads us to perhaps the most fascinating aspect of this analysis: using the Expected Points Percentage formula to project a variety of scenarios over the remaining games on the schedule. We’ve already illustrated expected final points using season-to-date trends, but how can we use this same logic to predict the type of production required might be required to attain a set number of points over the remainder of the season?

Looking at the Expected Points charts generated earlier, one can project the expected cutoff for the post-season to be approximately 89 points. The Maple Leafs’ season-to-date Expected Points Percentage of .534, if it were to be maintained, would result in the team earning 14 of the remaining 26 possible points in their season. By comparison, in order to attain the 21 points necessary to meet the post-season cutoff, the Leafs would need to play at a .789 clip the rest of the way (21/26 = .789).

The question, therefore, becomes: If we set .789 as our expected points percentage for the remaining games, at what rate would the Maple Leafs need to produce – and prevent – goals in order to meet that figure?

To illustrate the practical aspects of Expected Points Percentage in that regard, I’ve mapped out five hypothetical scenarios below. The first looks at what could be expected to happen if the Leafs maintain the GF/gm and GA/gm trends established during their recent 16-game skid. The second looks at what happens if the Leafs return to the GF/gm and GA/gm trends over the course of the season. Obviously, they have no hope of coming close to .789 under either scenario.

The third scenario looks at the change in GF/gm necessary to reach a .789 Expected Points Percentage should the team match their GA/gm over the season to date. The fourth scenario looks at the change in GA/gm necessary to reach that number should the team match their GF/gm over the season to date.  Finally, the fifth scenario takes into account the loss of Joffrey Lupul’s production (which equates to roughly 5 goals over the remaining 13 games).

The results are … well … not pretty:

Using EXP PTS PCT to Predict Performance
TREND GR GF/Gm GA/Gm DIFF. EXP PTS % EXP PTS GR
Recent Trends 13 2.06 3.56 -1.50 0.282 7
Season To Date 13 2.90 3.07 -0.17 0.534 14
To attain a .789 Expected Points Percentage …
GA (season rate) 13 4.65 3.07 1.58 0.789 21
GF (season rate) 13 2.90 1.91 0.99 0.789 21
Injury Factor 13 2.52 1.67 0.87 0.789 21

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(1) Recent
Should the Leafs maintain the GF/Gm and GA/Gm established over their previous 16 games, they would be expected to earn 7 points in the remaining 26 games, resulting in a probable lottery pick. This is undoubtedly the scenario Leafs fans would like to see most … but the likelihood of the team maintaining a recent level of performance so far below that throughout the majority of the season is questionable at best.

(2) Season to Date
As projected by our earlier analysis, should trends established over the course of the season to date be matched in the final 13 games, the Leafs can be expected to earn 14 of the possible remaining 26 points.

(3) If Goal Prevention Matches Season To Date Trends
Assuming the Leafs’ GA/gm remains at its current 3.07 for the balance of the season, the Leafs would need to score at a rate of 4.65 GF/gm to gain 21 points.

(4) If Goal Production Matches Season To Date Trends
Assuming the Leafs’ GF/gm remains at its current 2.90 for the balance of the season, the Leafs would need to post a 1.91 GA/gm rate the rest of the way to gain 21 points.

(5) Injury Factor
Losing a top-line winger (in this case, Joffrey Lupul) is bound to have an adverse effect on a team’s goal-production. This scenario factors the loss of Lupul’s production into the season to date GF/Gm rate, taking it from 2.90 to 2.52 in a 13 game span. In this circumstance, the Leafs would require a sterling 1.67 GA/Gm over the final 13 to have a chance at gaining 21 points.

What we can establish from this is a practical use for projection-based analysis beyond simply attempting to ascertain a team’s place in the final standings. By setting an expected points percentage required to meet a certain number of points (in this case, .789 to meet 21 points over 13 games), we are able to then fluctuate per-game goals-for and goals-against rates required to satisfy the expected points percentage. In effect, this gives us a useful projection as to the expected level of performance required to attain a certain number of points within a given span of the season, be it an application to a part of the season already played (as was done in Chemmy’s analysis) or applied to a portion of the season yet to be played.

One Last Scenario: The “New Coach” Factor

Although Randy Carlyle’s tenure has been short, and as such does not provide much of a sample size to work with, few would argue the Leafs have looked noticeably better from a goal-prevention standpoint (even as the offense seems to have dried up of late). In 5 games, Carlyle’s Leafs have produced at a rate of 1.80 GF/Gm and 2.40 GA/Gm. With the team only beginning to adapt to Carlyle’s methodology and strategies, it’s worth taking a look at how the trends established under his watch thus far may conceivably affect the Leafs’ final 13 games.

Leafs under Randy Carlyle
Scenario GR GF/Gm GA/Gm DIFF. EXP PTS % EXP PTS GR
Current Trends 13 1.80 2.40 -0.60 0.408 11
GF Increase 13 2.35 2.40 -0.05 0.554 14

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If the Leafs were to maintain the pace they’ve shown under Carlyle through the final 13 games, our equation suggests they could be expected to earn 11 points in that stretch, thus finishing with 79 on the season and on the edge of qualifying for a lottery pick.

Although the Leafs may not be able to reach their season to date average of 2.90 GF/gm over the final 13 games (especially in the wake of the injury to Lupul), it is conceivable they will not remain at a lowly 1.80 GF/Gm for the duration either. What I’ve done is set a hypothetical increase in the team’s GF/Gm to midway between what they have produced under Carlyle (1.80) and what they have accomplished on the season as a whole (2.90). In short, if the Leafs are able to maintain the level of goal-prevention (2.40 GA/Gm) they have produced under Carlyle, our equation suggests a goal-production rate of 2.35 GF/Gm the rest of the way would result in an expected 14 points.

At this point you might be inclined to ask how the team can have an expected points percentage of above .500 if they are allowing more goals per game than they are scoring. Such is life in a league which rewards overtime losses with a single point. It is imperative to remember the percentage is an indicator of expected points, not necessarily wins.

A Final Note

Of course, it must be acknowledged these calculations do not take into account the possibility of other teams deviating from current trends; indeed, we have been witness to exactly that with the Leafs’ February/March decline.  In that light, these numbers should not be viewed as a prediction of the final result so much as a projection of an expected result based on established goal-production and goal-prevention trends over a set span of games played.

Looking forward to your thoughts as always,

garrettbauman@www.mapleleafshotstove.com