Should Denver’s Dominique Rogers-Cromartie learn some maths to avoid future injury?

Football players in America’s NFL don’t often inspire confidence in their intellect and self control. Yes, that’s Danny Trevathan dropping the ball to get into his celebration early, without actually crossing the goal-line for the touchdown. Whoops.

no TD

But a game lost from a seemingly unassailable position, and a week on the sidelines with a shoulder injury? When star NFL cornerback Dominique Rogers-Cromartie was staring out the window during maths class, or perhaps idly taking three hours to scratch his full name into his desk, I bet he didn’t think this would be the result.

“Cornerback” isn’t a typo, by the way. Even many Australians now know that in American football, the quarterback is the star – standing behind a protective wall of teammates (lineman), throwing passes to other teammates (wide receivers) streaking down the sidelines. And that’s where the cornerbacks come in. They’re the ones who ruin everyone’s fun by knocking the wide receivers over, batting the ball to the ground and occasionally snatching the ball away for themselves:

Actually, that’s more fun than a touchdown pass. In fact, an interception by Pittsburgh’s James Harrison was recently acclaimed as the fifth-greatest play in Superbowl history (treat yourself).

(If you’re not convinced that watching interceptions is fun, you probably don’t like NFL football. Please don’t stop reading the article over a trivial thing like that – remember, maths is the focus.)

So, what about all these intercepting heroes? Players who can get a bunch of interceptions must be pretty valuable to their teams, right?

Well, yes and no. Getting the ball back from the opposition with an “int” definitely is valuable – it’s just not necessarily all thanks to the guy who ends up catching the ball. For instance, there could be help from a teammate, or maybe just a bad throw.

Ints happen a few times per game, but individual players don’t get them very often – the NFL league leaders get 1 int every 2 games or so… far behind NFL’s touchdown-scoring leaders. The regularity of ints per game lags behind even the rate of goals per game for strikers in the English Premier League (as I write, the leader is Luis Suarez with 15 from 10 games).

You can easily insert your own examples of why intercepting is a team effort. You’ll get more int opportunities when your teammates don’t leave receivers wide open, or when your other teammates put heaps of pressure on the opposing quarterback; if you’re the best cornerback on the team, maybe the quarterback will just prefer to try and find a receiver somewhere as far from you as possible.

And yet… all this seems a bit lost on the NFL, its fans and its players, who can’t tear their focus away from the one guy who comes up with the ball, as if he’d have done it without his ten teammates. Dallas’ Larry Brown won the 1996 Super Bowl MVP after making two interceptions in the game, the second one at a key late stage, when the ball was thrown straight to him, with barely an opponent in sight (see video from 1:20).

Over the long term, you’d certainly expect better players to accumulate more ints. But ints for a single player, in the context of the length of an NFL season, occur some distance apart. And you’ll be pleased (or possibly confused) that a Poisson distribution can be used to illustrate it.

Poisson distributions are cool, for two reasons:

1)      You can use them to analyse easily understood real-life situations – like the number of phone calls you get during your lunch break, how many seagulls arrive within 10 minutes of you laying out a parcel of fish and chips and walking away, or the number of people who queue at a traffic light. You’ll have noticed that all of these are whole numbers as low as zero or as high – in theory – as infinity. (Anyone who’s tried to get off the Eastern Freeway at Hoddle Street knows what I mean. Sorry interstate / international readers – Melbourne joke.)

2)      They have just a single variable: lambda, which is simultaneously the average and the key to the distribution’s variation, its density function and everything else you need to know about it.

This, of course, leads us to Tim Jennings. In 2012, Tim led the NFL with 9 interceptions. Let’s assume for a moment that this is the average that he could expect from his level of play in 2012.

Based on that assumption, here’s a graph of the distribution of intercepting performances by Jennings – basically, if he played a large number of seasons, how often he would make a given number of ints. As you can see, the most likely outcomes are clustered around his average, 9, and tail off towards zero, and even more rapidly towards 25 and beyond (the probability of 22 interceptions is 1 in 10,000):


Jennings’ nine closest rivals for the interception title gathered 8, 8, 7, 6, 6, 5, 5, 5 and 5 interceptions:

int leaders

As a point of comparison, here’s a similar graph for Byrd, Fletcher, McCourty and Samuel, all of whom had 5 ints:

2012ints ave 5

Assuming that each of the top 10 have performed to their average: going into a season, what’s the probability of Jennings beating them all?

If the only player Jennings was competing against for the int title was Byrd, Jennings would win just 82.7% of the time (the sum of all the purple cells in the grid below) and he would win or tie 88.8% of the time. 80% more interceptions, and he wins only 5 times out of 6!*

*(For comparison, if the Jennings and Byrd averages were doubled to 18 and 10 respectively – Jennings still superior by 80% – Jennings would win 92.5% of the time, reducing his loss rate by more than half. Byrd would only win 5.1% of the time. Or, if we go the other way and make the average 4.5 and 2.5 – Jennings once again superior by 80% – his win rate drops to 71.4%. The rarer interceptions are, the less likely you’ll get the right winner.)

ints comparison

But Jennings is competing against more than just Byrd. He has to beat 3 other players with 5 ints, 2 with 6 etc:

jennings prob

To beat everybody:  0.8884 x 0.74122 x 0.645 x 0.5472 = 0.049, or 4.9%.

To at least tie: 12.6% – and you can see how much difference just 1 int makes in the calculations.

So yes, you do need to be lucky as well as good. And this is before Jennings tries to beat every other player who’s outside the top ten, and before you factor in everything we already know, about the accumulation of interceptions being more than a solo activity.

An NFL regular season lasts just 16 games. The longer the term, of course, the more likely that a truly talented interceptor will rise to the top. Shorten the term, and the opposite will occur. If 4.9% is the probability of prevailing at the end of a season, you can imagine how the odds get squished even further over the course of an individual game, where the average number of interceptions for even the best player is going to be not much more than 0.5. A player who average 0.6 ints per game has just a 38% chance of having more ints in a game than someone who averages half of that, just 0.3 (of course 44% of the time, neither player will get an int).

In short, in a single game anyone (that is, anyone good enough to make it onto an NFL field) could find themselves suddenly in possession of a winning lottery ticket (cue: Larry Brown looking up to find a ball flying unhindered into his chest).

This should be obvious to most people who follow the NFL closely – and even more obvious to coaches, and the players they coach. And yet, Dominique Rogers-Cromartie is getting a blog entry here all of his own: why?

On November 24, “DRC” travelled with his Denver Bronco teammates to Foxboro, Massachusetts for a game against the New England Patriots.

It was a much-anticipated game. Denver has had a great season so far, but New England, led by their talismanic, laser-pass throwing, Gisele-dating quarterback Tom Brady are more than most teams can handle.

However, in a bizarre first half, Denver tore New England to shreds, leading 24-0 – a margin rarely overcome in American football. DRC was having a great game, part of a high-performing defense which was keeping the much-vaunted New England offense from scoring.

On the last play of the first half, New England found themselves stranded near midfield. Brady attempted a nothing-to-lose “Hail Mary” pass to the end zone as time expired, in the hope of finding a teammate via a fluke or a defensive mistake. However, he stuffed it all up and instead threw a weak pass short of the end zone. It was about to fall harmlessly to the turf, ending the half, when (and I’m exaggerating just a tad here) the stadium briefly glowed as DRC’s eyes lit up at the prospect of a cheap, stat-padding interception. He ran forward, dived for it, and couldn’t quite make the catch – but he did succeed in smashing his shoulder on the turf, and missed the rest of the game. In his absence the Broncos managed to lose, getting swamped by the dynamic, scrambling, still Gisele-dating (just looked it up, they’re actually married now) Brady. It was a magnificent spectacle – well worth checking out the highlights.

DRC’s untimately painful decision to go after the int was possibly the turning point of the game, but video footage – despite my scouring efforts – cannot be found. (Anyone have it? I could retire the lame description above.) It certainly wasn’t lost on the fans:




He ended up missing a further game with his injured shoulder. If only he’d known how lame the int statistic is!

I can only hope he spent the week studying the Poisson distribution, so that on his return to on-field action he could concentrate on what he does brilliantly– stopping the opposition from completing passes, and intercepting when the chance presents itself – and not waste any more time on embellishing a meaningless statistical resume.


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