When it comes Patriots/Colts, it’s easy to want to focus on Tom Brady vs. Andrew Luck. Or to marvel at the sheer number of star players these teams have lost in the last 12 months. If you played college in the state of Florida, you’re probably not going to be playing in this game: T.Y. Hilton is the last star standing with Vince Wilfork, Aaron Hernandez, Brandon Spikes, and Reggie Wayne gone. The Patriots also have placed Rob Gronkowski, Sebastian Vollmer, Jerod Mayo, Tommy Kelly and Adrian Wilson on injured reserve, while Devin McCourty and Alfonzo Dennard are both questionable. Also, of course, Brady is probable with a shoulder.

The Colts just put defensive starters Gregory Toler and Fili Moala on injured reserve, adding to a list that already included Wayne, Ahmad Bradshaw, Vick Ballard, Dwayne Allen, Donald Thomas, Montori Hughes, and Pat Angerer. LaRon Landry and Darrius Heyward-Bey are both questionable, and the latter’s injury caused the team to sign ex-Patriot Deion Branch.

All the injuries and changing parts make this a pretty tough game to analyze. So I’m not going to, at least not from the usual perspective. Instead, I want to take a 30,000 foot view of the game. According to Football Outsiders, the Patriots were the most consistent team in the league this season, while the Colts were the fourth *least *consistent team. Rivers McCown was kind enough to send me the single-game DVOA grades for both teams this season, and I’ve placed those numbers in the graph below with the Colts in light blue and the Patriots in red. The graph displays each team’s single-game DVOA score for each game this season, depicted from worst (left) to best (right). For Indianapolis, the graph spans the full chart, from the worst game (against St. Louis) to the best (against Denver). As you can see, the portion of the graph occupied by New England is much narrower, stretching from Cincinnati to Pittsburgh.

The Patriots were the better team in 2013, but it would be fare to say that the Colts appear to have more upside.^{1} Football Outsiders has New England as the 5th best team in the NFL (with a DVOA of 19.0%) and the Colts as the 13th best team (DVOA of 3.4%). This means that, on average, we would expect the Patriots to beat the Colts.^{2} However, the Patriots’ DVOA scores have a standard deviation of 21.4%, while the Colts’ grades have a standard deviation of 43.5%. If we want to assume that the DVOA grades are normally distributed^{3}, then we can depict the team’s projected performance levels using a pair of bell curves. That’s what I’ve done below, with the Patriots in red and the Colts in blue: New England’s peak is both higher and to the right of Indianapolis’ peak, but the Colts appear to be the better team when they bring their A game (the far right of the curve).

One way the Colts would be expected to win, of course, is to simply perform at a level that places them in the blue “banana” shape on the right. That section of the graph begins at 0.71 standard deviations above the mean. At that level, the Colts have a DVOA of 34.3% (3.4% + 0.71*43.5%), while the Patriots have a DVOA of 34.2% (19.0% + 0.71*21.4%).

Now if you type in Excel =1-NORM.DIST(0.71,0,1,TRUE), you will find out that the area under the curve represents 23.9% of the Colts’ curve. But this doesn’t mean Indianapolis has a 23.9% chance of winning. The Colts are unlikely to win when they don’t bring their A game, but it’s not impossible. If Indianapolis plays an “average” game — producing a DVOA of 4.0% — there is still a 24% chance the the Colts win, because there’s a 24% chance that the Patriots play a game that produces a DVOA of less than 4.0% (=NORM.DIST(4,19,21.4,TRUE).

So how do we determine the Colts’ odds of winning? It gets pretty complicated, so I decided to ask Neil and Doug what to do. Fortunately, they gave me the same answer!^{4}

First, we define X as the probability distribution with mean of μ_IND and standard deviation of σ_IND, and Y as the probability distribution with mean of μ_NE and standard deviation of σ_NE.

This means that X *minus* Y is normal with a mean of (μ_IND - μ_NE) and a standard deviation that is the square root of (σ_IND^2+σ_NE^2). Put in our numbers, and we see that X – Y is normal with a mean of -15.6 and a pooled standard deviation of 48.5%. So the probability of the Colts beating the Patriots is the probability of a normal distribution being greater than 0 with mean -15.6 and a standard deviation of 48.5%. If you type =1-NORM.DIST(0,-15.6,48.5,TRUE) into Excel, you will see that the answer is 37.4%.

Alternatively, once you see that we’re dealing with μ of -15.6 and σ of 48.5, you can convert that into the standard normal distribution where μ =0 and σ = 1, using the formula:

z = (x – μ) / σ

Here, Z would be equal to 0.32. If you type =NORMSDIST(-0.32) into Excel, you get the same 37.4%.

There’s an even “simpler” way to do it, if you find typing numbers into Excel simpler than mathematical analysis. Here’s what to do:

First, type the following into cell A1 in an Excel spreadsheet

=NORMINV(RAND(),19,21.4)

Then, type this into cell B1:

=NORMINV(RAND(),4,43.5)

Next, enter this into cell C1:

=IF(A1>B1,”NE”,”IND”) [Note: you need to use straight quotes, not curly quotes]

Then, you want to highlight the three cells from A1 to C1, and drag them down…. well, as far as you like. I did it for 50,000 rows. Finally, in cell E1, type

=COUNTIF(C:C,”IND”)/COUNT(B:B)

You’ve now created a random number generator. Every time you hit F9, the numbers will recalculate, but the answer in cell E1 should always be very close to 37.4%. This represents the likelihood of the Colts winning.

Now, what if the Colts had the same standard deviation as the Patriots? In that case, we could simply use the formula =1-NORM.DIST(0,-15.6,SQRT(21.4^2+21.4^2),TRUE) to see that the team’s odds of winning would drop down to 30.3%. So the takeaway here is that the Colts inconsistency — while allowing them to lose to a team like the Rams and nearly lose to the Texans — bumps their odds of taking down the Patriots from 30.3% to 37.4%.

- For purposes of this article, I will be assuming that variance is indeed predictive, and that a high-variance team does actually have more upside than a low-variance team. [↩]
- I am using the word “beat” as a synonym for “post a better DVOA score.” I don’t believe anything bad will happen as a result of this shorthand, but I’m open to hearing otherwise. [↩]
- Note: I am assuming that DVOA scores are normally distributed. I don’t know that this is the case. [↩]
- And in their answers, they assumed the data followed the normal distribution and that the performances by each team was independent. I have no problem with either of those assumptions. [↩]

{ 7 comments… read them below or add one }

An e-mail from a commenter………

As soon as I read:

“According to Football Outsiders, the Patriots were the most consistent team in the league this season, while the Colts were the fourth least consistent team,”

I thought, “I wonder if this makes the IND moneyline a value play but the spread not a play.”

Well, I checked, and at the open, the IND moneyline represented ~34% chance of IND winning, and some smart gamblers got some EV and bet IND. The moneyline has since been bet down and right now it represents IND having a 40% chance to win (which is basically your calculated % of 37.4 plus a vig). So now the moneyline is efficient (both sides of the represent negative EV for the gambler). Meanwhile, the spread has not changed.

Interesting that all 3 games these 2 teams played against Miami are clustered right in the middle of the graph.

Oh, I suppose I should make a prediction.

Indianapolis 24, New England 23Viva la variance!

Nice work! I would like to know what type of advanced math subjects are used to create models like this. I want to better my math skills, so I can be more in knowledgeable with this type of work.

Cool stuff as ever – as you know I enjoy reading about how to make predictions in different ways – and I like the way how it demonstrates the variance in the predicted percentage for the Colts. This one of the reasons why I have taken to cross referencing things where I can (e.g. unweighted and weighted because I didn’t like the projections some of the unweighted numbers were giving me earlier this year). Incidentally can you weight the college style SRS method (as I couldn’t think of an obvious way to do it)?

Re: this one, using the Stern/Winston method (via Excel’s Solver) unweighted we have New England by 4.5 (approx., for 64.5, 32.5, 3% tie – split), weighted (which would penalise the Patriots for not playing last week effectively I think given how its worked out) we have New England by 2.6 (approx., for a 60, 36.5, 3.5% tie – split). The 32.5% – 36.5% number on the Colts ties in fairly with a lot of the numbers above (as it is within the 30.3 – 37.4 bracket). Incidentally I wouldn’t like to make a pick on the total on this one as unweighted has it right on the the line, unweighted has it over (at about 59 points) but I think that is a reflection of the Colts game last week and I got the impression that the weather isn’t going to be that good which is always a flag.

Despite the numbers above, and what I said last week where I think I predicted the Patriots to win this tie (and despite everything, e.g. players available etc. – as there are so many things that point towards the Patriots winning this as it looks like all they really have to do is double Hilton all game) I’m going to go with a Colts win. I don’t know why, I don’t know how and it doesn’t make sense but neither did a few of the Patriots games earlier this year and I’m going to go with that logic – Colts win.

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