Once again, HydroTech really got me thinking.  I thought his post on Monday was excellent, and I wholeheartedly agree with his basic premise.  However, I’d like to take this line of thought a little bit further/down a different direction.

Using the Median

This post began as a comment I was going to leave on Hydro’s post, but it sort of grew wildly out of control, which brings us to today.  Specifically, I was reading through his comparisons of the run average vs. pass average of various opponents of Cal last year, and I was struck in particular by his use of our game vs. Colorado State:

Colorado State on the other hand averaged 3.7 yards per rush attempt and 10 yards per pass attempt (CSU should have passed more).

This is true, at least according to official statistics, and his conclusion is certainly accurate, but I think the statistics can sometimes be misleading.  Specifically, you’ll remember that at the end of the game, when Colorado State desperately needed to catch up, they completed a couple of long bombs for 44 and 67 yards, respectively.  That’s more than a third of the day’s passing yards right there in two passes, which certainly skews the average up.  On most passing plays, this kind of yardage is exceedingly unlikely, and I think to include them in a simple average (the mean) gives a misleadingly rosy picture of the passing game.

Instead of using the mean when trying to determine the yards per attempt, I think we should be using the median.  (For those who don’t remember their pre-algebra statistics, click on the links for basic Wikipedia articles.)  In this way, we can diminish the effect of extraordinary passing plays without discounting them entirely.  When you’re calling a specific play at some point during a game, you’re only dealing with a single, discrete event; you can certainly hope that the passing play will go for 50+ yards, but you’re more interested in what the "average" ( i.e. routine) play will go for.  In the case of Colorado State that day, the median passing play went for 6 yards, meaning that half the plays went for less than 6 yards, while half went for more.  In calling a generic passing play, CSU can reasonably expect that that play will go for about 6 yards, and they know that it’s equally likely to go for more or less than that.

Sacks Occur on Passing Plays

In the course of coming up with the median values for Colorado State, I ran into another of football’s statistical oddities that I think skews these results towards passing plays.  Namely, that sacks count against a team’s rushing yards total.  Now, while this makes a certain sort of sense (the ball was never actually passed), it really messes you up when you’re trying to predict the expected outcome of a certain play.  Virtually all sacks occur on plays that were designed for a forward pass (I can’t think of a single time where a quarterback was sacked before he had a chance to hand off, but I suppose it has happened).  Anyway, being that sacks occur on passing plays, I added them into the passing statistics when I was counting up Colorado State’s plays.

I swear I was going to pass!  I was getting to it!  I just needed a bit more time! [Photo from mybearterritory.com]

In the CSU game, Cal sacked the quarterback 5 times for a total loss of 25 yards, as well as CSU’s only fumble on the day.  Combined with interceptions and incomplete passes, sacks add to the increased risk/increased reward proposition of attempting a forward pass.  Adjusting for sacks, our new (mean) averages for running plays/passing plays are 4.7 yards per rush / 8.2 yards per passing attempt.  It still looks like Colorado State should have passed more (35 passing attempts vs. 36 rushing attempts), but not as dramatically as the statistics HydroTech cited above would suggest.  The resulting median averages remained steady, with the median rush resulting in a 3 yard gain, and the median passing attempt resulting in a 6 yard gain.

Assessing Risk/Reward on a Per-Play Basis

Of course, while I think the median gives us a better tool for predicting the outcome of a passing play, one number can only take us so far in informing play-calling.  After all, while I might get 82 yards from running 10 passing plays, I won’t get a chance to run those 10 passing plays unless I pick up this 3rd and 4 I’m facing.  And while the odds say that my median passing play nets me 6 yards, which would be enough to get the first down, what I *really* want to know is, what are the odds I pick up 4 yards or more?

Colorado State’s distribution of results following running or passing plays make for what I think is an interesting case study.  Let’s look at the results from running the ball first.  The first thing I noticed is how low-risk running the ball was for them; in handing off 36 times, CSU never turned the ball over and never lost yardage.  If all the Rams needed was a yard, they had an 86% chance of picking it up via the run.  Two yards was less automatic, but 61% is still a decent play, especially considering the negligible risk involved.  Big plays were unlikely, however; only 4 (11%) of them went for 10 yards or more.

CSU Rushing vs. Cal
Turnovers	0%
Lost Yardage	0%
No Gain	14%
1+ Yards	86%
2+ Yards	61%
5+ Yards	31%
10+ Yards	11%

Passing, on the other hand, was almost an all-or-nothing proposition for the Rams.  Lots of bad results (sacks, interceptions, incomplete passes) and lots of good ones (10 passing plays went for more than 10 yards) but very few small gains in between.  Also, notice that while CSU averaged around 10 yards per passing attempt, less than 30% of their passing plays actually resulted in gains of 10 or more yards.

CSU Passing vs. Cal
Turnovers	9%
Lost Yardage	14%
No Gain	20%
1+ Yards	57%
2+ Yards	57%
5+ Yards	51%
10+ Yards	29%

All Plays are not Created Equal

Of course, we’re still being rather simplistic in lumping all plays together into just two categories, either a run or a pass.  One could imagine (given enough data) coming up with charts like the ones above for each of the plays in a playbook.  For example, a deep pass play might result in some big gains, lots of incomplete passes, a couple interceptions and some sacks.  Conversely, a quick out to the sideline will result in far fewer sacks and interceptions (and probably incomplete passes), but most of the gains will be for much less.  Some plays, by their design, will have radically different result distributions than others.

Furthermore, I haven’t even touched on the subject of special situation play-calling, such as goal line rushing.  The same inside handoff that might gain an average of 4 yards in the middle of the field might only net 1 or 2 inside the 5 yard line, as virtually every defender is playing the run.  If we really want to get an accurate expectation of a particular play, we would have to separate out these special situations and consider them on their own.

Even if coaches don’t actually keep charts like the ones I’ve discussed for each of their plays, I’ll bet most of them have an innate sense regarding each play, analyzing its risk vs. return before calling it.  In fact, I’ll bet that’s how some playbooks are winnowed down, with the plays having poor risk/return ratios being dumped in favor of more promising schemes.

See here?  That play had a 72% chance of picking up the first down!  What happened?

The Intricacies of Play-Calling

What I’ve been discussing here is not a guide-book for play-calling.  It is merely another tool in an arsenal of which I’m only beginning to realize its complexity.  Besides the run/pass balance that HydroTech talked about yesterday, coaches will run counters to their previous plays, and even counters to their counters, as discussed here.  Coaches must consider down and distance, time left in the game, and opposing defensive tendencies, always trying to do what the defense is not prepared for.  Somehow, an offensive coordinator must reconcile all of these differing and often conflicting needs into a single play which gives the offense the best chance to succeed and, ultimately, score.  Then, once that play is over, he has to do it again.  And again.  It’s a fascinating dance, one that’s all the more intriguing the more you know about it.

In Summary

Colorado State should have passed more.  Not only were they the underdog, and so should have been more tolerant of risk, but our pass defense sucked that day.  Despite burning our secondary time and again, they still ran more than they passed, and it would have been more extreme if not for a ton of passing on their last two touchdown drives.  Given our defensive performance, Cal was actually rather lucky to escape with a win.