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Around SBN: The Amateur Mathematics Of Linsanity

How do you solve a problem like McFadden, part II

WARNING: ERRONEOUS ZONE. LAWYER PLAYING WITH LOADED MATH.

This morning we looked at the fear that is running rampant through the hills of Tennessee this week concerning Darren McFadden's upcoming visit to Neyland Stadium. The fear's well-founded and based primarily on the following facts: (1) Arkansas' rushing offense is No. 2 in the nation; (2) McFadden and partner McFelix Jones hung 487 yards on Steve Spurrier last week; and (3) Tennessee's rush defense is ranked 74th in the nation.

But here's the thing. The rushing defenses of six of Arkansas' opponents this season are ranked 92nd or worse in the nation. Even allowing for the fact that part of the reason they're ranked so low is that they had to play Arkansas, that's still not very good. When Arkansas played Alabama (38th in rush defense) and Auburn (24th in rush defense), they lost. They also lost to the next worst rushing defense, Kentucky (92). So maybe, just maybe, that No. 2 ranking for rushing offense is a bit inflated.

ARKANSAS
DATE
OPPONENT
RESULT/TIME
RUN DEFENSE RANK
RUN DEFENSE YARDS (per game)
09/01 Troy 46-26
103
205.78
09/15 at Alabama 41-38
38
126.44
09/22 #21 Kentucky 42-29
92
192.67
09/29 N Texas 66-7
107
209.38
10/06 UT-Chatt 34-15
(I-AA)
(I-AA)
10/13 #22 Auburn 9-7
24
114
10/20 at Ole Miss 44-8
105
208
10/27 Fla Int'l 58-10
99
204.11
11/03 S Carolina 48-36
104
206.50

And what about Tennessee's national rank of 74th in rushing defense?

TENNESSEE
DATE OPPONENT RESULT/TIME RUN OFFENSE RANK RUN OFFENSE YARDS (per game)
09/01 at #12 California 45-31
T-50
166
09/08 So Miss 39-19
28
190.78
09/15 at #5 Florida 59-20
29
189.44
09/22 Arkansas St 48-27
37
175.33
10/06 #12 Georgia 35-14
40
173.67
10/13 at Miss St 33-21
66
149.78
10/20 at Alabama 41-17
62
153.33
10/27 #16 S Carolina 27-24
95
121.60
11/03 La Lafayette 59-7
8
241.56

Hmm. Again, we've done our part to make those stats what they are, certainly. But is it too far out of line to conclude that our poor run defense ranking is at least partially a result of playing against good running teams while Arkansas' exceptional run offense ranking is at least partially a product of playing against teams with a poor running defense? You tell me. I'm thinking that Arkansas is still an excellent running team -- the best we'll play this year, no doubt -- and that we're still not very good at stopping the run, but the numbers aren't as bad as they seem at first blush and the situation isn't as hopeless as it seemed earlier in the week.

Wouldn't it be wonderul if we could determine how many yards we should expect the Hogs to put up on us this week? That's what I thought. I figured it would be easy enough to look at the yards gained by the Hogs against their opponents and extrapolate the numbers to determine what we should expect the Hogs to gain against our No. 74 rush defense. Eventually, I figured out that I didn't want to "extrapolate" but "interpolate." No problem, I thought to myself, how hard can that be? Well . . .

Okay, then. Look, it's no great secret that lawyers chose law due to acute allergies to mathematics, so let's do the poor man's interpolation, where poor = "dumb at numbers".

Test Case! Tennessee's currently ranked (very!) roughly midway between Alabama and Kentucky in rush defense nationally. Alabama's giving up 126.44 yards per game and Kentucky's giving up 192.67 yards per game. So we should be giving up roughly, what, 166 yard per game? Checking . . . it's actually 167.67.

TEAM
RANK
YARDS
Best (Boston College) 1 58
AR opponent (Auburn) 24 114
AR opponent (Alabama) 38 126.44
Tennessee 74
in sealed envelope . . . 167.67, which is very close to 166 according to my calculator!
AR opponent (Kentucky) 92 192.67
AR opponent (Florida Int'l) 99 204.11
AR opponent (Troy) 103 205.78
AR opponent (South Carolina) 104 206.5
AR opponent (Ole Miss) 105 208
AR opponent (North Texas) 107 209.38
Worst (UAB) 119 248.7

Okay, so so far so good, and my, that's a lot of sos. But the system works sort of. So now for the real thing. Alabama gave up 301 yards to Arkansas, and Kentucky gave up 338. And boy am I glad that Kentucky didn't actually give up fewer yards than did Alabama or my head would have exploded and this post would have been entitled: Using Math to Paint Yourself Into a Corner.

Anyway, the magic number for the Vols this weekend appears to be . . . drumroll . . . about 318. As in yards the Razorbacks should get against our defense on the ground. Which is not good, but not entirely not good. The three teams that have rushed for the most yards against us are the Bears (230) (beatdown), the Gators (255) (blowout), and the Ragin' Cajuns (234) (blowout in the other direction). So although it's unlikely, Tennessee could win the game even if Arkansas gets 318 yards on the ground. Remember, Georgia is currently averaging 173.67 yards per game, and we held them to 69. Let's do that again, shall we?

Apologies to hooper, our resident nuclear engineering grad student, who must be banging his head against his monitor right now. Somebody get him some BC.

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It's a convincing argument
but then I saw the algebraic formula and I went into a math-induced coma for a second.
Proud member of the EBMCC (Eric Berry Man Crush Club) and Josh McNeil's Videographer

by Volorado on Nov 7, 2007 12:22 PM EST reply actions  

One thing to remember
Anytime I talk to people about Arkansas' running game I always bring up this point (people never seem to take it into account):  EVERYONE knows they are going to run the ball but they are still able to do it successfully.  To me, that's one of the main arguments regarding the whole McFadden for Heisman thing is that everyone on defense knows he is getting the ball.  He doesn't have the luxury of playing on a team where defenses have to respect the pass.  Every single team stacks the box, but he still gets his yards and still has a high YPC average.  But in regards to your above assessment, does a high rushing output always equal victory for Arkansas?  The answer is no.  To me, it's more about Arkansas' defense and how they perform.  In all 3 SEC losses, the team had a 4th quarter lead and the defense folded, losing two of those on the last play of the game.

by flahog on Nov 7, 2007 2:23 PM EST reply actions  

The way I see it....
... the bottom line to me seems to be that as well as our defnse must play, our offense must play even better.

I really think the key to winning this game is simply to keep Arkansas' offense on the sideline. That means long, drawn out drives with lots of really good intermediate routes and excellent running. Perhaps some screens or draws. Wait a minute, sorry, I forgot who I was talking about.

Either way, the defense must play exceptionally well to keep McFadJones in check, but if the offense plays like they should, we shouldn't see too much of them anyway.

Oh yeah, and please, for Pete's sake, DOUBLE UP ON MONK. I hope he doesn't become DJ Hall pt. 2.

All hail Jim Bob Cooter!

by XRayVol on Nov 7, 2007 3:13 PM EST reply actions  

Here we come
Hey guys- I'm the yankee from NYC that Joel posted on a few weeks ago that's heading down to Tennessee with a few friends.  Can't wait to get down there and experience an SEC tailgate and a good game.  Just wanted to wish you guys luck.  I'll be rooting for you (although I'm a Michigan fan)
Thanks-

by RockyTopbound on Nov 7, 2007 4:22 PM EST reply actions  

Waaaaaaaahhh!
I just peed a little thinking of McFadden.

by ghostofneyland on Nov 7, 2007 4:33 PM EST reply actions  

Good analysis :)
Joel, it's actually a good job with the numbers.  My head is intact, and my monitor is still in good shape (though I enjoyed a good laugh at that comment).

The geek in me just can't help but point out that Arkansas has an unusual breaking point at 200 rushing yards a game.  If their opponents average more than that, they've won.  Less than that, they've lost.  (The Kentucky game really makes a fine line of it.)  If you believe that, then Tennessee is on the golden road.

Another P.O.V. - we must be really concerned about this game if a lawyer is reduced to doing math.

by David Hooper on Nov 7, 2007 10:25 PM EST reply actions  

Good Result, Erroneous Math
(Boring Math Explanation Forthcoming)

The assumption made when interpolating is that the data behaves in a linear manner.  In the first case you use, run defense ranking vs. rushing yards allowed, this assumption is correct.  Plotting this data will show that it behaves linearly until we reach the data at the edges of the set, i.e. the top ten defenses and the bottom ten defenses.  However, when plotting the rankings of the defenses Arkansas has played vs. Arkansas rushing yards, it is readily apparent that this is not the case.  This can more easily be seen by just looking at the results from Mississippi and Florida International who allowed fewer yards against Arkansas than either Alabama or Kentucky did.  

(End Boring Math Explanation)

Now for the good news, or bad news for you Joel because you did all this work, Tennessee allowing a little over 300 yards rushing is a very reasonable estimate.  With the limited sample size, it is impossible to use numerical methods.   Your best bet would have probably been to exclude the Auburn and South Carolina games, due to them being outliers, and then just calculate the average.   Doing this gives you a result of 319.14 yards per game rushing for Arkansas.  This simple method gives a result that is just as mathematically significant as interpolation in this case without all of the unnecessary work.  

by Vol Navy on Nov 7, 2007 10:26 PM EST reply actions  

Math fight! Yay!
Ensuring Joel never posts numbers again:

Even including the ends, the linear model is decent enough (R-square of 0.56 and F-test of 0.03).  At any rate, there's simply no getting around the fact that playing with stats on a sample size of 8 is sketchy at best.  The ranking is likely to be a poor choice for predictor because it takes a perfectly good continuous predictor (yardage), removes the relative differences, and turns it into a boring ordinal.

So we agree, though I'm more tolerant of the linear model.

by David Hooper on Nov 7, 2007 10:37 PM EST up reply actions  

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