Should the fumble rate of NFL teams be a normal distribution? 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.
 A: 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.
A: 
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.
