Three Yards and a Cloud of Dust and Rushing Performance for the CMU game

Submitted by Daniel on September 3rd, 2013 at 12:01 PM

In the aftermath of the CMU game, I’ve seen a few comments about running backs that go something like this: “If you took out X’s long run, his YPC would have only been Y, so he really wasn't that effective,” or variations thereof. This got me thinking a little about the limitations of using YPC to summarize running back performance, so I've put together a couple ways of looking at running back performance against Central.

First off, sample size concerns are rampant. Statisticians frown on many, many things, but they take particular umbrage when you do anything with a really small sample (read: less than 30). But, like our beloved coaches, we live in the real world where we have to make decisions based on incomplete information; so we continue on despite the limitations of the dataset.

Strength of competition is also suspect. We don't know for sure how good CMU will be this year, but we do know they were outscored by fifty points in the only game they've played this year. They may not be great this year.

Central Tendency

Yards per carry is calculated by summing all rushing yards for a player and dividing by number of carries, making it an average (or sample mean). A sample mean is a very useful way of summarizing data with one nagging flaw: it is particularly vulnerable to outliers. The median, on the other hand, as the most central value, can be interpreted as a more typical expectation for a dataset. One extremely high or low value will have virtually no impact on the value of the median. Here's an example: Derrick Green's YPC for the CMU game was 6.1, 2 whole yards higher than Toussaint's 4.1. But Green's median carry of 3 is an entire yard shorter than Toussaint's 4. The YPC might lead you to conclude Derrick Green was a better bet for getting yards than Toussaint, but the median says at least 50% of Toussaint's carries went for 4 or more yards in comparison with Green's 3 or more yards. Since If you needed four yards for a first down, you may want to give it to Toussaint. That's potentially valuable information not contained in the YPC. Then there's the pesky fact that TD runs have a maximum length. If we're two yards out from the end zone, that's the maximum the player can get for that carry. This artificially lowers the YPC of a player who gets the ball over the line; in particular Toussaint's YPC would probably have been higher.

The table below contains a few measures of central tendency for the players who had at least 3 carries (three is still too small, but a line had to be drawn somewhere and Rawls' touchdown seemed to merit his inclusion in this list). Rawls gets no standard deviation because three is a small number.

Player YPC StdDev Median Min Max Carries TDs
Gardner 7.4 7 6 0 22 7 2
Green 6.1 7.9 3 1 30 11 1
Toussaint 4.1 5.3 4 -3 20 14 2
Rawls 4   5 -2 9 3 1
Smith 1.7 1.8 3 -1 4 7 0
Total 5.3 7.8   -3 38 47 6

QB Devin Gardner wins the YPC sweepstakes with a blistering 7.4 YPC bolstered by a median carry of 6 yards. I would advocate getting this man some more carries, but that's a) already happening and b) potentially troublesome for our passing game. Regardless, Gardner does a good job here no matter what metric you use: no negative yardage, a great longest run and two touchdowns on only 7 carries. At least for this game, our shiny "more passing-oriented" quarterback was our most effective running back, which speaks a bit to the value of athleticism at that position.

Among the running backs, Toussaint and Green duke it out for maximal effectiveness depending on which measure you use. Green wins on YPC, longest run, and least negative minimum run. Toussaint had a higher median, most touchdowns, and most carries. Rawls has the highest median of the RB's, but since he only had three carries, sample size tells us to pay no heed.

____ Yards and a Cloud of Dust

Hearkening back to the days of Three Yards and a Cloud of Dust (TYaaCoD), I wanted to know who was more reliable if you need three yards every time you rush.  The table below contains the percent of carries the player achieved at least three yards, embodying the spirit of slightly-in-jest Schembechlerian Michigan Football.

Personally, though, I find three yards slightly lacking. If you run three yards every rushing play and you rush every play, you end up facing 4th and 1 every series. Our Fearless Leader would still go for it on fourth down every time (Heil Hoke!), but it's not an optimal situation to find yourself in. What you really want is someone who can pick up 3.5 yards or so every play, so you get a new set of downs after every three. The play-by-play is unhelpful in this regard, however, only listing integer values for yards. So I also calculated the Four Yards and a Cloud of Dust (FYaaCoD) metric, which is how the table below is sorted. If you get four yards every carry, you can go on rushing forever.

I did make a slight modification to the success rates of both metrics: I counted a touchdown as a success regardless of how many yards the play was because there is no further to go.

Row Labels Total Yds Carries TYaaCoD FYaaCoD
Gardner 52 7 71% 71%
Rawls 12 3 67% 67%
Toussaint 57 14 57% 57%
Green 67 11 73% 55%
Smith 12 7 57% 29%

For TYaaCoD, you would want the following players rushing in order: 1. Green 2. Gardner 3. Rawls 4. Toussaint 5. Smith 6. Johnson. All players are between 50% and 75% successful at getting 3 yards against CMU, which is heartening. Moving to  FYaaCoD, you would want 1. Gardner. 2. Rawls 3. Toussaint 4. Green, 5. Johnson 6. Smith.

There's some shuffling when you move to FYaaCoD: Derrick Green drops from first to fourth, and Smith falls to sixth at a slightly disappointing 29% success rate. Rawls still has only three carries, but two of them pass the FYaaCoD test, so he has a terrific success rate of 67%. Almost as good as Devin Gardner, who had over twice as many carries. Devin's ability to scramble is probably for real. Toussaint's actual strength as a running back comes through a bit more on the FYaaCoD metric. On his 14 carries, he hit 4+ yards 57% of the time, and he often surpassed four. That increases the chance of success for future plays, as the distance to the first down marker is smaller.

I thought about running the same analysis with passing yards, but it didn't feel right since yards per catch vary widely based on the play. Your wideout running the deep route will end up with more yards per target than the slot ninja you toss the bubble screens to. That is more schematic than based on individual skill. It is true that running plays are also not all created equal. But every running play starts behind the line of scrimmage and heads as far as possible into enemy space, making comparison a reasonable exercise.


Any statistical summary is just that: a summary. We lose information when we look at average, median, min, max, total yds, TYaaCoD, FYaaCoD, etc. that is available to us in the actual dataset. Our lizard brains just can't process significant amounts of data in numerical form in any reasonably quick fashion. But there is one thing we are great at: reading charts. So I've assembled the information from each rushing effort for everyone with 3+ rushes in order from least yards gained to most. I've colored the touchdowns Highlighter Yellow™ so you can include/exclude them from your mental calculations as needed.

For recent time's sake, Drake Johnson. Fare thee well, 2013 Drake. We hardly knew ye.

Conclusions? Inferences?

A. We were completely misguided to push for Devin-Gardner-to-wide-receiver last year when his natural position is clearly running back. The fact that QB's get an extra blocker has no bearing on this.

B. At this exact moment in time, the staff's decision to go 1. Toussaint 2. Green 3. The Field. is pretty justified. We saw flashes of brilliance from both of them—maybe even more from Green—but Toussaint overall had a better day. If Green sheds a few pounds and picks up just a hair more speed in the process, though—and I think we all expect that to happen— he could become the clear #1 even by mid-October. De'Veon Smith is not yet ready for world-beating, but he did display that vaunted balance. Hold off on judgment on him at this point.

C. Charts are indeed fun to look at.

D. Norfleet had one rushing effort for 38 yds, which I didn't include in this analysis because dividing by zero is difficult and because his YPC would make Brian cry.



September 3rd, 2013 at 1:25 PM ^

Standard deviation is another meaningful statistic to characterize a running back. Of Gardner, Toussaint and Green, Toussaint has the lowest SD (5.5) so clearly he is the most consistent one.


September 3rd, 2013 at 2:09 PM ^

Standard Dev would only tell you about all consistency, when you really only care about consistency with bad plays. Take two players:

PlayerA: 5, 5, 5, 40

PlayerB: 4, 3, 4, 3

Player B will be more "consistent" as defined by standard deviation because there is less variation in those runs, but (1) they are all worse, and (2) you're essentially penalizing Player A for having a great run.

I think you were suggesting only looking at standard dev, but looking at it in combination with average, but it still is going to give a very cloudy picture (I'd be a little surprised, for one, if runs really do resemble a normal curve, so the calculations that standard deviation allows would need some work). I think Daniel's original suggestion of median, or percentage of runs above a certain length is really giving more valuable information.


September 3rd, 2013 at 3:02 PM ^

Exactly.  Any player who is going to average 4-5 yards a carry or more is going to need a higher standard deviation because you just don't average >4 yards/carry (with a large sample size) without breaking long runs.  Not to be a broken record, but if a back could get 4 or 5 yards a carry every single play of the game, then you score on every drive and go undefeated


September 3rd, 2013 at 1:27 PM ^

Averages lie all the time; I like the analysis of medians.

But there is a selection-bias problem with using this kind of analysis as an indicator of a RB's ability:  RBs are definitely not randomly assigned to runs (ergo the "third down back", etc.).  

Some backs might be sent out because of their ability to make the best of a short-yardage situation, and are lucky to get the 3 yards.

(There are econometric models designed to deal with this problem, but they would bore most of mgoreaders).



September 3rd, 2013 at 1:57 PM ^

But you touched on something that has always intrigued me when looking at RB's production, specifically that the end zone gets in the way of gauging the success of some running plays.

If you assume the goal of running the ball is either a TD (e.g., when the ball is X and goal to go) or a first down when outside of the 10 yard line (of course no coach is instructing his RB to take a knee once he hits the first down marker), I have thought about success rates of RB to be a function of how close to the "goal" of the play does the RB achieve.

For example a three yard gain on 1st and 10 from mid field doesn't seem as successful as a three yard gain on 3rd and 3 from the 50.

Not all three yard gains are created equal, would you agree?

Great work. Thanks for the leg work.


September 3rd, 2013 at 2:01 PM ^

Great stuff - I also vote for this to be updated regularly as the season unfolds. A previous comment makes a great point: it would be very interesting to compare just 1st and 2nd down runs to control at least a bit for the 3rd down effect. Of course, over the season I think the coaches will figure out what they want to do and that will probably make getting truly comparable data on each back harder to get - some guys will start getting very few carries and someone will settle in as the 3rd down back, etc. But I say: MOAR STATS.



September 3rd, 2013 at 7:16 PM ^

There are other dimensions to the problem too.  What is really important is not how many yards the back gains but how many he gains in excess of what would be expected on the basis of the situation

eg.  the down, how many yards to go, the starting point (one yard line of one's own territory vs 50 yard line).  Also consider the team's tendency to run or pass (eg the degree of surprise in a single run after nine straight passes) ..  You can think of many other factors, such as the degree to which a game is close vs. a blowout, whether you are ahead or behind and at what stage of game that occurs.   Likewise, consider whether you put a runner in consistently on third or fourth down and another on downs one or two, or one runner at the beginning of a game vs strong opponents, the other in at the end in a blowout of weak opponents.

Some of these factors could be considered in a much bigger model, which separates the yerdage gain due to the situation vs. the excess gain due to the particular running back.





September 4th, 2013 at 8:30 AM ^

Nice job.  As I was reading it, I was thinking about whether it might make sense to chart each back' s yardage before contact with an opposing defender, and then after contact. The yardage before contact would seem to reflect how well the plays were blocked, and the post-contact yardage would seem to reflect what the back did on his own. But then, some backs avoid tackles by physically breaking them, whereas other backs make defenders miss--so I thought maybe it's better to chart the yardage before & after the first "tackle attempt," rather than contact.

It also seems that a back' s yardage before the first tackle attempt could eventually reflect how well the back sees and hits the right holes.  Just looking at the data for this game, Green had that 30-yard run on which he was untouched until the end, so that run would probably put him first or second (after Gardner) in that category no matter what he did on his other carries. But since the offensive line is the same, with a high-enough sample size it seems to me that a guy who consistently gets another yard, or even another half-yard, before the first tackle  attempt could well be demonstrating better vision.

Ron Utah

September 4th, 2013 at 10:35 AM ^

I hope you keep this up all season; it would be interesting to see how these numbers play out.  Thanks for all the work.

The argument about removing one or two carries and its impact on RB stats is an old, pathetic one.  Breaking a big run is vital to an offense's success--much like the deep passing game--and often has a huge impact on the outcome of a game.  Removing those carries from a player's performance to artificially shrink his YPC is silly.

That said, if a player has just one lucky carry and isn't productive the rest of the time he touches the ball, that's obviously a cause for concern.  That's why the "eyeball" test is always important: blocking and talent are easy to see but hard to quantify.

Even with the small sample size, your analysis is still valuable.  I thought Toussaint looked better than Green, even though Green's stats might have appeared stronger.  I also though Smith looked good, despite his low YPC.  Based on what I saw, the depth chart should be:

  1. Fitz
  2. Green
  3. Smith

With Rawls and Hayes supporting role players.  But that was one game against a weak opponent--seeing how these guys run against ND will help refine that list.

Overall, the stats are valuable, and I would love it if you kept up with this herculean task all season.  Thanks again.