# March Madness and The Stats Behind Cinderella Stories

In just a few days, some of the best teams in college basketball will compete in one of the countryâ€™s most storied annual sporting events: March Madness. From humble beginnings, the â€śBig Danceâ€ť grew from just an 8-team basketball tournament to the 68-team â€śmadnessâ€ť weâ€™ve come to love about it. Better yet, this 21-day, single-elimination tournament never fails to produce stories of nameless underdogs who overcome seemingly-insurmountable odds to achieve what no one thought was possible. This is what makes the tournament so special – no other sporting event creates Cinderella stories like March Madness does. Of course, any single elimination tournament this big is bound to produce dramatic finishes and shocking displays of luck, but how often can we expect things like this to happen? And how can we know which teams are most likely to pull off the biggest upsets?

With some investigation, itâ€™s clear the nothing has more to do with Cinderella stories than the structure of the tournament itself. If youâ€™re familiar with how any normal seeding format works, youâ€™d know that itâ€™s designed to benefit the best seeds at the detriment of the worst seeds. So, on average, we might expect that 1 seeds make it furthest in the tournament, 2 seeds make it second-furthest, and so on for seeds 3 through 16. However, when we question this simple intuition, we find something a little bit different. In Figure 1 below, each bar represents the percentage of teams since 1985 that made it to the given round of the tournament. Hopefully the first two rows make this intuitive.

Figure 1: The Proportion of Each Seed Remaining, by Round (since 1985)

After the round of 32, something strange happens. 10, 11, and 12 seeds outnumber 8 and 9 seeds in the Sweet Sixteen, 6 seeds outnumber 5 seeds in the Sweet Sixteen and Elite Eight, and 8 seeds outnumber 6 and 7 seeds in the Final Four. Is this because the selection committee systematically overvalues some teams and undervalues others? Probably not – if this were the case, we might not expect to see strictly decreasing bars in the round of 32. Rather, I reason that this is the result of the bracket itself. For convenience, see the bracket below:

Figure 2: West Regional Bracket from March Madness 2017

After the first round, the winner of the 8/9 game is (almost) certain to run into the 1 seed, who would obviously be heavily favored. Meanwhile, the 12 seed wouldnâ€™t have to play the 1 seed until the Sweet Sixteen, giving them a better chance at making it past the first two rounds.

Though, itâ€™s a little surprising to see that 10 seeds make it to the Sweet Sixteen much more frequently than 8 and 9 seeds do, given that 10 seeds usually have to play the 2 seed when they make it past the first round. This must mean that 10 seeds fare far better against 2 seeds than 8 and 9 seeds do against the 1 seeds. Indeed, they do: collectively, 8 and 9 seeds win 14.5% of their games against 1 seeds, while 10 seeds win a whopping 38.3% of their games against 2 seeds. With a p-value less than .001, we can confidently say that 10 seeds perform better against 2 seeds than 8 and 9 seeds do against 1 seeds.

Note: 10 seeds have only played 5 games against 15 seeds in the last 34 years of the tournament, and they won all 5. Theyâ€™ve played 47 games against 2 seeds. 8/9 seeds won their single game in history against a 16 seed. Meanwhile, 8/9 seeds have played 135 games against 1 seeds. All stats in this post are from 1985 through 2018, unless otherwise specified.

This difference highlights the disparity in talent between 1 seeds and 2 seeds in the tournament. Moreover, it shows that the difference between 1 seeds and 2 seeds is probably much greater than the difference between 8/9 seeds and 10 seeds. Unsurprisingly, when we look at efficiency ratings from KenPom.com, this is true: on average, the drop-off in efficiency from the 1 seed to the 2 seed is much steeper than that from the 8 seed to the 10 seed (plot below). So, if we assume that these efficiency margins accurately reflect the strength of these teams, then we would certainly expect 10 seeds to perform better against 2 seeds than 8/9 seeds do against 1 seeds. This explains why 10 seeds (and probably 11 and 12 seeds) usually make it further in the tournament than 8/9 seeds do, and why most cinderella stories come from 10-12 seeds instead of 8/9 seeds.

Figure 3: Average Efficiency Rating, by Seed (Adjusted for Pace)

Furthermore, we must remember that every 10 seed whoâ€™s ever played a 2 seed first had to get past the Round of 64. If it turns out that 10 seeds who win the 7-10 matchup in the first round are significantly better than 10 seeds who lose it, then perhaps they will put up a stronger fight against 2 seeds in the second round. Observe Figure 4 below: it shows the distributions of Â regular-season efficiency ratings for 7 and 10 seeds (represented by the shaded regions), separated into teams that won their first round game, and teams that lost.

Figure 4: Efficiency Ratings of 7 and 10 Seeds, by Winner of the First-Round Matchup. The bars in the middle represent the mean of the corresponding group, with error bars for the 95% confidence interval

Note: In case youâ€™re unfamiliar with or confused by the violin plot in Figure 4: the blue shaded regions represent the distributions of efficiency ratings for 7 and 10 seeds that made it past the first round, and the orange regions represent those for 7 and 10 seeds that got knocked out in the first round. The regions are wider where there are more points, and thinner where there are fewer (notice how they are wider near the means).

Using a pooled t-test, the difference in average rating for 10 seeds that make it out of the first round vs. 10 seeds that donâ€™t is significant at the .05 level. This indicates that 10 seeds who win the first round are significantly stronger on average than the rest. In fact, as the plot shows, 10 seeds who win usually have higher ratings than the 7 seeds they beat. This means that theyâ€™re probably not winning by chance alone – more likely, they are actually better than the team seeded ahead of them. And just for good measure, these 10-seeded teams also have better ratings than the average of all 7 seeds combined.

Figure 4 above presents one more interesting piece of information: it shows that 7 seeds who make it past the first round are usually better than 10 seeds who make it past the first round. So why is it that 10 seeds win 38.3% of their games against 2 seeds in the second round, while 7 seeds only win 30.9%? With a p-value of .74, this is almost certainly just the result of small sample size, but still: this is what March Madness is all about.

Following the same methods, we could show similar results for 11 and 12 seeds (evidenced by their similar success in the second round). But why is there such a huge drop-off from the 12 seed to the four following seeds? Weâ€™ve explained how 10 seeds (and likely 11-12 seeds) manage to outpace 8 and 9 seeds in the Sweet Sixteen, but why does this not hold true for teams seeded 13th or worse? The surface reason is that teams seeded this low are, obviously, worse than the teams seeded above them. More than that, however, they are disproportionately worse than the teams seeded above them, as well as the teams they have to play. Figure 5 below is a barplot showing the difference in efficiency ratings for each of the 8 matchups in the first round. Notice how that difference grows exponentially large as the seeds get further and further apart.

Figure 5: Difference in average efficiency rating by first-round matchup. Error bars represent 95% confidence intervals for the true difference in mean efficiency rating

Of course, this trend is somewhat intuitive. As the seeds get further apart, you would expect bigger gaps in talent between the two teams. Though, the next plot shows us another reason why 10, 11, and 12 seeds can often reach the Sweet Sixteen, and why 8-9 seeds and 13-16 seeds almost never do. Figure 6 below is the same type of plot, but for the second round of the tournament. It shows the difference in average efficiency rating between each of the bottom 8 seeds, and each of the seeds they most often play in the second round. Here, we see that 10, 11, and 12 seeds usually have much easier matchups than the rest of the underdogs.

Figure 6: Difference in average efficiency rating by expected second-round matchup

So in addition to having relatively difficult first-round matchups, 13-16 seeds also have disproportionately tough second-round matchups. When you consider both of these factors, it becomes clear why 13-16 seeds are rarely able to reach the Sweet Sixteen, and why only one of them has ever advanced to the Elite Eight (since 1985). Meanwhile, the comparatively easy road 10-12 seeds face creates the perfect environment for thrilling upsets and exciting Cinderella stories.

In Figure 3 (Average Efficiency Rating by Seed), we noticed that the differences in strength between the teams at the top of the tournament were greater than the those of the teams in the middle of the pack. With the teams at the bottom of the tournament, we see a similar trend that helps explain the plots above: after about the 12 seed, team strength begins to fall dramatically, even dipping below 0 for the worst teams in the tournament. At first, this might come as a surprise; given the trend we see in the first 10 seeds, we could reasonably expect that the difference in talent gets smaller and smaller with each successive seed. However, we must remember that the selection committee only hands out 36 at-large bids, meaning that the other 32 teams got to the tournament just by winning their conference championships. Many of these 32 teams come from mid/low-tier conferences that canâ€™t quite compete with powerhouses like the ACC and the Big East, and so they end up stealing spots at the bottom of the bracket from more qualified teams that couldnâ€™t swing an at-large bid. The phenomenon is evidenced by Figure 7, below.

Figure 7: Percentage of Tournament Berths from Automatic Bids, by Seed (since the 2000 March Madness)

Since the 2000 March Madness, 98% of tournament berths attained by teams seeded 13-16 were automatic bids, achieved by winning their weaker conferences. Oppositely, most other teams were given at-large bids for proving that they can compete, or achieved an automatic bid for winning a tougher conference. This is why it is so difficult for 13-16 seeds to make it past the second round, even though theyâ€™re usually only playing against 5-8 seeds there.Â

If March Madness were meant to determine the best team in the nation, then itâ€™s fair to say that these teams donâ€™t belong in the conversation. But thatâ€™s not why the tournament exists, and thatâ€™s not what we get most excited about – we love watching nameless teams like 16-seeded UMBC conquer basketball juggernauts like 1-seeded Virginia, and we canâ€™t help getting attached to historic runs like 11-seeded Loyola Chicagoâ€™s Final Four push in last yearâ€™s tournament. Rare teams like these ones are why we watch this crazy 68-team dance, and they are the motivation behind the whole field of â€śbracketology.â€ť Now, we wait in anticipation to see what triumphant underdogs and inspirational narratives March Madness 2019 will bring us.

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