August 12, 2020
by P.A. Jensen/CHN Reporter
Or, do teams that win more also win by more? We’ll answer both questions.
It seems obvious that teams that win more games should have larger margins of victory overall, but that needn’t be the case. Consider that the difference between a good team and a great team might be the great team’s ability to eek out close games — if that’s the case, the great team might have a lower margin of victory, even if they win more. Also, maybe great teams often play a more disciplined and lower-scoring game, which might have the same result.
Regardless, we’ll start by directly comparing average margin of victory/loss to winning percentage in college hockey. Figure 1 summarizes the results of Division-I men’s hockey regular seasons from 2013-2014 to 2019-2020. In the graph, each dot represents one school’s team for one year—your favorite school is probably represented by seven dots, one for each season in question. The x-axis is winning percentage, with better teams toward the right; the y-axis is the mean goal differential, or the average difference between that team’s score and their opponent’s score during the regular season.
Unsurprisingly, average goal differential is tightly linked to winning percentage, but the degree of the linkage is remarkable, and maybe a bit surprising. The correlation is especially strong, as shown by its tight fit around the trendline (R2 = 0.93 by linear regression, on a scale of -1 to 1), and the correlation is highly statistically significant, meaning the result is unlikely to happen by chance (p ~ 0, on a scale of 0 to 1). In addition to the tight fit and the high significance, though, the trendline crosses the 0-goal differential very near the point where a 0.500 team would sit on the graph. This means that a team that wins as much as it loses would simultaneously see the magnitudes of wins and losses cancel each other, on average. It’s exactly as one might expect.
However, and as mentioned, even if this result is predictable, it isn’t inevitable. It’s conceivable that great teams would simply win a lot of close games, flattening/bending the trendline downward on the right side of the graph. Or, maybe the poorest teams would always lose by a lot, bending the line downward on the left side of the graph, or even dragging the whole trendline down. (We may see a little bit of a downward tack in the bottom-left corner of Figure 1, but attempts to fit the data using other, non-linear trendlines still give a strikingly linear fit, with little or no increase in strength of correlation.) So, even if this result is unsurprising, it’s worth including here to highlight just how striking and straightforward the correlation is.
Of course, calculating means can hide important details. Maybe better teams have higher average goal differentials because they typically win by more goals, but maybe they’re simply more likely to win a few games by a lot of goals, bringing up their overall average. Or, maybe merely good teams win most of their games by the same margins as great teams do, but also suffer a few blowout losses, disproportionately driving their average down. Because means (and standard deviations) can gloss over these details, we’ll look more directly at the distribution of goal differentials for each caliber of team, which is shown in Figure 2.
Figure 2 condenses the approximately 400 teams in the sample (about sixty teams for seven seasons) into ten groups, or deciles, based on winning percentage: the first decile is the worst 10% of teams over the past ten years, the tenth decile is the best 10% of teams, etc. Then, it collapses all possible game results into nine categories of goal differential, which are on the vertical y-axis: 0 is for ties; wins by 1, 2, and 3 goals are listed singly (+1 – +3); and all “blowout” wins of 4 goals or more are collected into one group (“4+”). (The same scale, only negative, is used for losses.) Finally, for each decile at the bottom of the graph, the percentage of games that resulted in each goal differential is represented by color, with the lightest blue representing zero and the darkest blue representing just over 20%. In short, the vertical column of colored boxes, if added up numerically, would add to 100% of games.
So, Figure 2 shows the same general trend as Figure 1, just in more detail. For example, the worst teams over the last seven seasons (Decile 1, along the left edge of the graph) won very few games by 4+ goals (light blue, top-left corner). Many more of their games ended not only as losses generally (darker colors in the bottom half of graph), but as blowout losses (dark blue, bottom-left corner). In contrast, many of the best teams’ wins were blowout wins (dark blue, top-right corner), and when those teams in the best/tenth decile did lose games, they rarely lost by a wide margin (light blue, bottom-right corner). Again, these results are unsurprising, but they’re still noteworthy because they address so many questions that remain after simply reporting the means, as in Figure 1.
Of course, graphs like these do not directly answer the original question of whether better teams win by more goals. To do that, we need to correct for the number of wins. For example, of course better teams’ average goal differentials are going to be higher: more of their games end as wins, and fewer end as losses, meaning that their average is computed using mostly positive numbers. Similarly, most of bad teams’ games end on the negative side of the ledger, so their average will almost certainly be negative. This muddles the question of whether they win by more goals when they win, because the number of wins skews the sample.
To answer the question of whether good teams win by more goals, we have to compare good teams’ wins to poor teams’ wins, side by side, independent of how many instances there are of each. So, when they win, do better teams win by more goals than worse teams? Figure 3 shows that they do.
Figure 3 resembles Figure 1, with a team’s winning percentage along the x-axis, and its mean goal differential on the y-axis. However, in Figure 3 we see goal differentials for losses and wins separately. For example, in the left panel we see by how many goals teams lose when they lose. The trendline in the left panel shows that, as we move toward the right and teams get better, losses are by fewer goals, on average. Across the range of good and bad teams over the past seven years, the average margin of loss changes by about 1 goal: the worst teams lost by an average of about 3 goals, while the best teams lost by an average of about 2. The right panel of the graph shows the converse: when they win, the best teams win by about 3 goals, while the worst teams win by about 2. Both trends in Figure 3 are statistically significant (p ~ 0; R2 ~ 0.24 in each case).
Overall, then, worse teams not only lose more often, but they lose by more goals when they lose. Similarly, better teams not only win more often, but they win by more goals when they win. However, yet again, calculating means can hide the details. As we noted in Figure 2, the best teams seem to win a significant percentage of their games as blowouts; the reverse is true for poor teams. These blowout wins and losses might skew their mean goal differentials “upward.” In other words, maybe the worst teams lose most of their games by a typical margin, but are simply more likely to suffer a few blowout losses that drive their margin of losses up. This begs the question: do worse teams have larger margins of loss because they routinely lose by more goals, night after night, or do they just have a few more blowout losses?
A common way to address the effect of a few large numbers on calculated means is to focus on a different kind of “average,” like the median. However, because hockey is such a low-scoring game, the data are essentially ordinal—there are so few possible scores, practically speaking, that the median doesn’t change much in a large sample with a few outliers. (In our case, the median for virtually every team is about 2 goals, win or lose, good or bad.) Instead, we’ll address the effect of blowout wins and losses in two other ways: 1) we will eliminate them from the sample altogether (Figure 4), and 2) we’ll look at the overall distribution instead of calculating averages at all (Figure 5).
Figure 4 closely mimics Figure 3, except blowout wins and losses of 4 or more goals have been removed from the sample entirely. For example, the left panel of Figure 4 depicts all ~400 teams in the sample, arranged by the full winning percentage you’d see in the standings, just like in Figure 3, but the mean margin of loss is calculated for losses by 1, 2, or 3 goals only, or for “small” losses. Still, we see the same effect: even when blowout losses are removed from the sample, better teams still lose by fewer goals than other teams. The converse is true for wins in the right panel of the figure (p < 0.001 in each case). However, the strength of the correlations is relatively weak (R2 = 0.07-0.08), meaning that blowout wins and losses do, indeed, play a role in the trends we see in Figure 3.
That trend is more clearly depicted in Figure 5. Visually, Figure 5 resembles Figure 2: teams, grouped by decile according to win percentage, are on the x-axis, while goal differential is on the y-axis, and percentage of games is represented by color. However, it also resembles Figure 3, in that it examines trends in goal differential specifically when teams win and when they lose, which corrects for how often they may win or lose. The result is similar to what we’ve seen, only even more striking.
First, notice the change in scale in Figure 5. In Figure 2, dark blue represented about 20% of games resulting in a particular goal differential; in Figure 5, dark blue represents almost 50%. The darkest blue in Figure 5 is in the right panel, in the bottom-left corner: when the worst teams win, about half of those games are decided by only one goal. Contrast that with the right edge of the right panel, which shows that the best teams are about equally likely to win by 1 or 4+ goals. A similar trend exists in the left panel: when the best teams lose, they typically don’t lose by much, while the worst teams are about as likely to lose by a little or a lot. Looking at the distributions of goal differentials in this way, rather than at just the teams’ means, shows just how disparate the goal-scoring patterns are among good and bad teams.
Overall, then, winning percentage and average goal differential are linked in Division I men’s college hockey (Figures 1 and 2). Even if that linkage itself is unsurprising, the strength of the correlation is remarkable. Also, worse teams have lower average goal differentials not only because they lose more games, but also because they lose by more goals when they do lose (Figure 3). This holds true even when blowout losses are removed from the sample (Figure 4). Finally, when poor teams do win, it’s usually by only a goal or two; similarly, when the best teams lose, it’s often not by much (Figure 5).
P. A. Jensen has a crazy dream of helping college-hockey journalists with data analysis and graphing — pro bono, even. Feel free to contact him about your next project, big or small (@PrideOnIceCream).
