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There's a lot of garbage statistics going around the internet regarding the Patriots, but I was just curious what the well-versed in statistics have to say about this.

The main question is - in what cases would distribution of season fumbles (y) by NFL team (x) be a normal distribution? In what cases would it not?

My thoughts are simply -- it would be a normal distribution if the fumble rate for each team per game/ season is equal (which is a big assumption).

It would appear less and less normal the more of an influence that non-random, independent factors --- such as player skill, play-calling, coaching, incentives, home field qualities, potential cheating, or whatever else -- had on fumble rates.

Is my thought process correct or not?

I've also heard a lot of people commenting that the distribution of fumbles per season by NFL team is actually a Poisson distribution.

I've dealt with Poisson distributions before, and this seems preposterous and completely off-base, though I could be wrong. Isn't a Poisson distribution usually uses in entirely different cases? I thought it was used for modeling when a call might come in within the next hour, or when a dice might come up 6 after N tosses. I can see this modeling fumble distribution of N plays, but comparing the NFL teams season fumbles?

Any ideas appreciated --- I'm not super into this media issue --- I'm barely into football -- I was more interested in the numbers. I don't even think the data in question here (that the Patriots fumble rate is an outlier) was even gathered or appropriated managed to answer the right question in any case.

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    $\begingroup$ What or who are the NFL and the Patriots? I don't really care, and it appears to be just detail here, but I feel obliged to flag that this is an international forum and that it's best not to assume that what is well known locally is well known globally. $\endgroup$
    – Nick Cox
    Commented Jan 27, 2015 at 20:39
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    $\begingroup$ Also, are you referring to the distribution of an individual teams fumbles per game per season, or of all teams per game in a particular season, or of all teams per game over time? $\endgroup$
    – Wayne
    Commented Jan 27, 2015 at 20:47
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    $\begingroup$ Essentially, for context, data was produced (possibly bogus) showing that a sports team, named the Patriots, were an outlier in terms of "number of fumbles per season." A fumble is basically a significant event in the game where a player carrying the football gets hit and loses the football. It's considered a relatively random, or unpredictable, event -- although that is more of what's in question. The "# fumbles" charts by # teams depicts a relatively normal distribution, with one team - accused of cheating -- as a significant outlier. Some claim this isn't meant to be a normal distribution. $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 21:16
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    $\begingroup$ I believe this is referencing these two blog posts: 1 and 2 $\endgroup$
    – Affine
    Commented Jan 27, 2015 at 21:31
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    $\begingroup$ Poisson distributions model counts of rare events. ("Rare" would typically be less than a $10\%$ success rate out of each trial, often orders of magnitude less.) Historically the Poisson distribution came to fame when Ladislaus Bortkiewicz used it to model occurrences in which Prussian soldiers were kicked to death by horses. I see a striking parallel with how NFL football is played :-). $\endgroup$
    – whuber ♦
    Commented Jan 27, 2015 at 22:37

2 Answers 2

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Theoretically, a normal distribution has a nonzero probability of negative numbers. So that's right out. A normal also has fully continuous distribution, whereas fumble rates would be discrete or rational.

It could be very close, and good enough, for example, the sum of many binomials (had a fumble or didnt with x% chance, summed across 100 games) approaches what looks like a normal bell curve.

People go to poisson because it is a discrete counting variable, with integer results defined from independent results; that is to say if each play had a consistent fumble probability, then over 100 plays the final outcome fumble count would be poisson distributed.

If there's any correlation within the ranks then it wont be any theoretical (clean) distribution. If for example having a lot of fumbles reduces your total number of plays in that game, then it's a self-correlated score and things get messy. I do believe if your first dozen plays all had a fumble (not likely but possible), then you might not get any more. It's definitely not an independent sum of even probabilities.

If the coach is allowed to remove a player who has had several fumbles, then the rate would decrease from that point on, another non-independence of the score.

The real observed distribution sure could look a lot like a normal in any event. Do you have any data we could play with?

EDIT: We see some data at this link: http://www.sharpfootballanalysis.com/blog/2015/the-new-eng;land-patriots-mysteriously-became-fumble-proof-in-2007 Thanks Affine for finding that.

And in that article the claim is made more explicitly: "Based on the assumption that plays per fumble follow a normal distribution, you’d expect to see, according to random fluctuation, the results that the Patriots have gotten since 2007 once in 5842 instances."

Which is a malformed hypothesis, you'd never care about the probability of an exact answer, the question of interest is how likely is any result this extreme OR HIGHER, combined. A point result has an extremely rare probability, but if there's a fat tail to the distribution, then perhaps more extreme results can happen, and the outlier event is really not so extreme. As this is an inverse distribution, Touches per Fumble, consider both variables as random poisson, you get so many touches per game and you see so many fumbles per game. The ratio will have a long tail, because it's possible to have many many touches with few fumbles. The outlier is to be expected, even looking at the previous decade's results, there was an outlier at 56 TpF which didn't get any comment from the blog author.

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  • $\begingroup$ I would have to dig for clean data myself, as most of the popular stuff being passed around is not only poorly defined by also by people with an agenda. Actually, to clarify, the fumble rate is being analyzed, not the total number of fumbles. My mistake. It's the total fumbles over total plays run by a particular offense. I'm still not clear why 'fumble rate' or fumbles/ plays (y) by Team (x) would resemble a Poisson distribution. It still seems like it would be something of a binomial distribution if each team had (roughly) the same general fumble rate over a season. $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 21:24
  • $\begingroup$ It seems logical that some players are more fumble prone than others, but why would this factor not be evenly distributed among the teams over a set of years? Perhaps it is, but perhaps it isn't. A similar argument can be made for benching a fumble-prone player. Surely there are differences in decision here, but not major. Say most teams would bench a player after 2-3 fumbles. Is that minor difference ... enough to offset the, by comparison, much larger influence of random factors in shaping the normality of the distribution? $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 21:26
  • $\begingroup$ A rate can't be Poisson by definition. Poisson is a counting result, like how many goals did a soccer team get? It will be (0,1,2,3, ...) and never 0.14 as a fumble rate would be. The same argument applies for a Normal/Gaussian, because it defines a positive probability to results outside the [0,1] range where your rate should sum 100% probability. That's semantics though, the question is probably about how symmetrical are the results about the mean? $\endgroup$
    – Karl
    Commented Jan 27, 2015 at 21:39
  • $\begingroup$ @JohnBabson: Plays are not called at random. A team that emphasizes passing will tend to throw the ball away when plays fall apart, which means a fumble is not possible. We would also expect fewer fumbles from teams that run in such a way that unexpected hits to their ball carriers are rarer. It doesn't "equal out" over teams and seasons; it's correlated to playing style, the coach's strategy (and what talent they have available to pursue that strategy), plus how much the coach hates turnovers. No coach likes them, but some coaches will bench you for the game, and some for the season. $\endgroup$
    – Wayne
    Commented Jan 27, 2015 at 21:44
  • $\begingroup$ I find how popular culture in the US tend to transform statistics a bit strange ... miles per gallon ... touches per fumble, it's like we in the US prefer the 1/x transform of the more meaningful variable x. $\endgroup$
    – WetlabStudent
    Commented Jan 28, 2015 at 3:22
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would distribution of season fumbles (y) by NFL team (x) be a normal distribution?

Under no circumstances is a non-negative and discrete random variable actually normal.

In some circumstances (discreteness aside) it might not be terrible as an approximation, but it wouldn't be the first approximation I'd look at.

"it would be a normal distribution if the fumble rate for each team per game/ season is equal"

-- no, that doesn't do it ... though homogeneity might lead to a less skew distribution than otherwise.

"the distribution of fumbles/ season by NFL Team is actually a Poisson distribution"

-- well, at least it's not immediately ruled out by the domain of the variable, but (except perhaps as a rough approximation) I would think it would be readily rejected as a possibility; I expect that heterogeneity (across team make-up, opposition, conditions etc) would make it more heavily skew; there may also be a possibility of some serial dependence (outside that caused by intermittent changes resulting from heterogeneity).

"* for modeling when a call might come in within the next hour, or when a dice might come up 6 after N tosses*"

  • when a call might come is continuous, so no.

  • "when a die might come up 6..." -- again, no. Your description of what the random variable will be isn't completely clear there, but that sounds like one of "number of tosses to the first 6" (a geometric distribution), "number of tosses to the Nth 6" (a negative binomial) or "number of 6's in N tosses" (a binomial) -- but even if you meant something else, it still won't be Poisson. (Note that 'dice' is plural, 'die' is singular, so only ever 'a die'. You need at least two of them to have 'dice')

By comparison, the "fumbles per season" one being Poisson is at least plausible as a suggestion, but I think for a variety of reasons it won't be Poisson either.

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  • $\begingroup$ I think I may be confusing Poisson distribution and exponential distribution like illustrated by this chart here: statlect.com/uddpoi1.htm --- Thanks for the explanation, though, I may need to read a bit more on Poisson. $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 22:30
  • $\begingroup$ Ah. That would certainly be a plausible model for the "time to next phone call" variable. (Probably not for the die one though, since the tosses are a counting process.) $\endgroup$
    – Glen_b
    Commented Jan 27, 2015 at 22:33
  • $\begingroup$ I think the exponential distribution is more like (probability of the first six coming up in next N tosses) as opposed to a call arriving in the next N minutes. Aka the probability rises geometrically with N, is the same at each point at time, and is memory-less. $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 22:36
  • $\begingroup$ "first six coming in the next N tosses" is discrete, while the exponential distribution is continuous, so no. You can read more about all the distributions we've named between us (exponential, gamma, normal, Poisson, negative binomial, geometric) on Wikipedia (and elsewhere on the internet) $\endgroup$
    – Glen_b
    Commented Jan 27, 2015 at 22:38
  • $\begingroup$ Yeah, perhaps I made it up -- I think it just helped me conceptualize why the exponential distribution of next call arrival looked the way it did, even if that example may technically not be an exp. distribution. $\endgroup$
    – John Babson
    Commented Jan 27, 2015 at 22:40

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