Conventional wisdom (via a poll, sample size = 2) would think that the annual Duke game at Cameron Indoor is the single hardest game of the season for Carolina, with "hardest" defined as "least likely to win going into it". The easiest proxy for this is the closing spread going in to each game, looking at the largest margin(s) in which UNC was the underdog (which is equivalent to their lowest chance of winning). So I grabbed this data from Covers as far back as it goes (14 seasons).
That away game at Cameron was the hardest in 7 of the past 14 seasons, 50% of the time (I personally would have expected this to occur more often). The single lowest point was a spread of -15, at Duke in the 2009-10 season, which is roughly equivalent to a 7% chance of North Carolina winning. To the other extreme, Carolina's lowest win chances actually came as favorites in 2006-07 (60%) and 2008-09 (58%) - meaning they were favored to win every single game.
On average, the away game at Duke ranks as the 1.7th hardest on the schedule, and the home game in the rivalry ranks 6.9th.
Conference / non-conference splits shake out as you might expect, with the following average difficulty ranks:
- Conference Games
- Home: 24.6
- Away: 13.5
- Neutral: 14.3
- Non-Conference Games
- Home: 16.6
- Away: 8.1
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