You seem to be confused. There's no claim that a mean of 2 is acceptable or anything like that. The fact is that in the last 31 years, we've had 62 mass shootings, and that comes out to an average of 2 per year. You don't get to compare to a Poisson distribution with a mean of zero because you have to use the actual mean, which is 2.
This analysis has nothing to do with whether or not there are too many shootings. It's about whether the shootings occur at a fixed rate, and are independent of each other.
You have it correct in your last sentence although your previous sentence had it quite wrong.
You are correct that the analysis is about whether the shootings occur at a fixed rate. The author had set the expected value of 2 at a given time period, which is to say the author assumes that the 'natural rate' of a mass shooting murder is 2 a year.
The mean across the years has got nothing to do with this IMO. It may have been a good rule of thumb to compare against, but it's not a good one to do an analysis upon. It leads to all sorts of mistaken conclusions, like the one the author made.
You're all partialy wrong. The author wants to test if there is an incrase of mass murders, so he wants to test if the large number of mass murders that occured in 2012 is likely to be due by chance or not. he wants to know if 2012 is an exceptional year or not regarding the average. We dont care what the average is, we want to know if this observed number of mass murder is not likely to be observed regarding the average. His pvalue is small but not small enough to conclude that the distribution of observed mass murders does not fit the model.
No, there's no assumption of "natural rate" or anything like that. The ordinary way of estimating the mean parameter for a Poisson distribution is by taking average value over the sample period (http://en.wikipedia.org/wiki/Poisson_distribution#Maximum_li...).
>The mean across the years has got nothing to do with this IMO.
Well, that's all well and good, but your opinion is irrelevant. This is simply how this statistical tool is used. And crucially, the data fits this model, so objecting to its completely standard parameters is very silly.
This analysis has nothing to do with whether or not there are too many shootings. It's about whether the shootings occur at a fixed rate, and are independent of each other.