I. Introduction

Hello friends, and welcome back to the Offseason Stat Series. It’s been a busy few months, but I am back and ready to continue my delightfully intricate and thorough investigation of college football phenomena. If you’ve been paying attention, you’ll have read my posts recapping TCU’s season, examining situational punting, which teams should drop down from FBS, constructing a field goal probability model, and evaluating offenses according to production theory, If you haven’t been, go catch up on those! I’ve recently spent some time reorganizing, cleaning up, and all around playing with the play by play data from the 2018 season. I’ve been thinking a lot about meaningful comparison of composite stats, and today I want to think about using central moments to describe, compare, and evaluate offenses in college football. I’ll start with some data updates and definitions of terms, then I’ll look at the college football offense landscape (and the Big 12 specifically) in terms of some common statistical measures.

II. Data and Definitions

The data come from collegefootballdata.com, as always. He has posted an API to scrape play-by-play data or has downloadable csv (spreadsheets) that are easily accessible. If you want updates or to get involved with the data, check out r/cfbanalytics on reddit. I take play-by-play data for all games in the 2018 season, from Week 0 through the national championship. These stats are descriptive, so I’m using the raw, unweighted outcomes for this analysis. I make two significant changes to my data. First, to stay consistent with the analytics literature online, I’ve updated my garbage time filter to include more plays, as Bill Connelly did with his updated S&P+ formula. Now, I filter plays when the scoring margin is 43 points in the first quarter, 37 in the second, 27 in the third, and 21 in the fourth. That’s just to allow for more data points, yet still filter out piling-on and back-up plays as we analyze the data. It’s a fine line to toe, that of “quality data points” and “small sample size”, and I haven’t done much work to validate it, but for now I trust Bill, so those are my adjustments. I think of garbage time adjustments kind of like a smoothing requirement, to remove the complete disasters and the complete routs, giving us a little bit of a “good faith” interpretation. It’s akin to your boss giving you a couple passes on being late when assessing your job performance.

Second, I for the first time in my analysis, have coded turnovers and recovered fumbles appropriately. Each turnover is coded as a 0 in yards gained, so it will appropriately drag down the total yards per play. That’s a little messy on one front - I’m intellectually amenable to an interception being a zero in yards gained, but as for fumbles, there is the issue of downfield fumbles. Parsing the text descriptions of plays to appropriately award yards before a fumble probably a project for another day. For now, we will just call it a zero. Additionally, safeties and sacks have been verified, so those negative numbers will count into the totals.

**Warning: A little math-y.

As we dive into the analysis below, I’ll focus on three “central moments” of a random variable. I’ll get through the definitions here, and try to keep it simple. The zeroeth central moment is the mean. That is the simple average: all of the yards results of a play added together, divided by the number of plays. This gives us a good baseline as to how well a team performed over the course of the season, especially for blunt comparison: Oklahoma lead the nation with 8.38 yards per play, whereas UTSA brought up the rear with 3.99. Obviously, Oklahoma’s offense was much better than UTSA’s. Thanks, zeroeth central moment!

But, we can do better. A crucial component of an offense is big plays: successful teams are the teams that can pop off a huge run or pass on top of consistently gaining enough to keep them on the field. At the extremes, comparison is obvious, but as we get more granular with teams that are more similar, the mean doesn’t provide enough information on its own. That’s where the second central moment comes in (for those of you keeping score at home, yes, I skipped the first central moment, because it’s always zero. Sometimes statisticians are real dumb about things). The second central moment is variance, which tells us how much a random variable (here, yards per play outcomes) moves around it’s mean. It’s a measure of volatility. Now, a variance centered around zero might be confusing, as a high variance could mean a large number of negative values or positive values. Fortunately for us, yards per play outcomes are effectively bounded at zero: most plays are zero, positive, or slightly negative. I ran a quick check and found that variance is effectively the same if you code all negative values at zero, which confirms the thought that variance comes from positive yards, and is generally a good thing. Troy and Minnesota both averaged 5.75 yards per play, but Troy’s variance was 108.89 whereas Minnesota’s was 78.59. Guess who had the more fun (and better) offense? Thanks, second central moment!

This is where it gets a little wonky. The third central moment is skewness. I can basically only explain skewness with a picture, which is more an indictment on my communicative shortcomings than it is your intelligence, dear reader. This is skewness. All you need to know in our context is that a higher skewness means a longer right tail, which means a team had lower of its plays concentrated in positive yardage. This is helpful in analyzing college football teams that have similar means and variances: which team was really more explosive? For example, Army and BYU both have yards per play about 5.6, and variances within 8 points of each other (73.5 for Army, 81.5 for BYU). Army’s skewness was 3.6 to BYU’s 2.5, more than a full point higher! So both offenses were consistent yardage gainers, but BYU had more big plays, whereas Army, running everyone’s favorite option, was ground and pound. Thanks, third central moment!

**Math-y-ness concluded.

Below, I’ll look at the leaders and losers across the country according to these three central moments, and then I’ll document the Big 12, just too keep it relevant for all us Frogs fans.

III. Mean Yards Per Play

Hello, Group of Five! There aren’t many surprises here, as most of us make our assessments of team quality based on the zeroeth central moment. Some teams are a little bit surprising, like Ohio so high and Michigan State so low (54th and 30th in Off S&P+, respectively), which highlights two facts: one, raw means are insufficient for team comparison, and adjusting stats may be important. Let’s move on and see what the second and third moments can tell us.

IV. Variance of Yards Per Play

Raise your hand if you were expecting Vanderbilt and Maryland to be at the top of any list of offensive attributes. I wasn’t. Again, I want to reinforce, the moments don’t mean much on their own. Vanderbilt and Maryland averaged 6.6 and 6.0 yards per play; Vanderbilt was 12th and Maryland 37. My theory is that the variance probably increases for teams who started hot (Maryland) or had a really strong hot streak and finished well (Vanderbilt), but the interaction of variance and mean in yards per play is still underdeveloped. Let’s explore that a bit.

That’s a pretty strong relationship; there are a few outliers, Maryland being one of them, and perhaps the most striking of the few. We see teams like WSU, Army, KSU, and Cal all below the mean, which affirms the prudence of these two measures: those teams gain consistent chunks of yardage and aren’t amazing at big plays. On the other hand, teams like Clemson, Oklahoma, Alabama, all thrive because they gain big chunks of yards and have big plays. I think there’s a lot more to parse out with this relationship, but for now, the description is important. What matters in college football isn’t just the average of yards per play, but also how volatile your offense can be when it gets a chance for a big gain.

V. Skewness of Yards Per Play

Finally, let’s look at skewness. The spread of skewness isn’t too large, about 1.8, and interpretation is messy. From top to bottom, it may not mean much, but at the margins, it could potentially be informative.

Ok, you got me. I have no idea what this means in terms of college football. This chart is one whole big shrug emoji. My thought was that a severe skew would indicate that you were distributed more around a lower value, but that doesn’t necessarily seem to add up. Let’s take a turn, though. We all agree that Oklahoma is the best offense in the nation, right? Let’s just reference every against Oklahoma, and then sort them by absolute distance from OU.

Five Best Offenses (Abs. Distance in Skew from OU):
Florida (.0001), Wisconsin (.006), Arkansas (.0196), Wake Forest (.0207), Kansas (.0211)

Five Worst Offenses (Abs. Distance in Skew from OU):
Army (1.007), TCU (.990), Duke (.938), UNLV (.908), NM (.893), Rice (.843)

Well, I feel better about the decisively worse teams? Skewness is apparently a component of style that really doesn’t mean much in terms of absolute comparison. There are good teams with heavy skewness, and bad teams with heavy skewness? So, let’s file skewness under the “TBD” category in terms of usefulness for comparing college football teams.

VI. Conclusion

In this article, I’ve begun a discussion of deeper descriptive analysis of college football offenses by exploring higher-order moments. This provides some necessary context for comparison of seemingly similar teams.

Further research will look into the defensive sides of things, and then turn to aggregation. Once we’ve solidified interpretation of these higher-order moments, reducing their dimension into a meaningful comparison number will be the next step. For now, though, you can peruse the data and let me know in the comments what you think about the usefulness of these measures.