So, let's do a thought experiment here:
Let's assume (wrongly, I know, this is a put-up/shut-up argument here) that Sharp's numbers are accurate from here:
The problem with merely saying the Patriots are outliers, is that we don't know what the underlying distribution of data should be. We can't just take the standard deviation here because the distribution is clearly non-normal (for several reasons, one of which most stats people ignore: normal distributions cannot extend towards positive or negative infinity, which is not the case for plays per fumble, which can extend towards positive infinity) and not bell-curve shaped. We also can't really use percentiles because we don't have a large sample size. Below is a visual distribution of the data to hammer this point home:
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Although most people prefer to assume a distribution, an alternative is to construct a theoretical distribution using bootstrapping. Bootstrapping is a simulation technique that uses the empirical data to simulate a series of theoretical data sets from the same range. The procedure is fairly simple: per bootstrap one samples with replacement from the dataset (e.g. the same data can be selected twice) and calculates the metric of interest. In this case, I will be taking the average of fumbles lost per offensive play (which, again, makes no sense when you include ST plays, but save it for now, you'll see why). I ran 10,000 bootstraps here.
We can inspect the constructed distribution of values to see whether the original distribution matches our bootstrapped distribution. Below is the distribution of bootstrapped data.
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I hope one can already see the large problem here, but I'll continue.
The bias of the bootstrap distribution is quite small. This is calculated by measuring the difference between the center of the distributions (e.g. by using the mean as the centers), which gives us a difference of 0.0017 (if using the mean of the nfl distribution), or 5.0625 (if using the median instead). Since the bootstrap is somewhat-normally distributed, we can construct a confidence intervals somewhat easily, by multiplying the standard error of the bootstrap by a t score reflecting the confidence interval of choice. For 95 percent confidence intervals, the range is between 98.8 and 111.3. 23 of the NFL teams over the past five years exceed these intervals, and therefore should be defined as outliers.
One of the problems here is that using plays per fumble relies on a distribution that does not reflect the real range of values; no team will ever have 0 offensive plays but 0 fumbles is possible. No value exists if there are 0 fumbles, making plays per fumble a phenomenally stupid metric. We can simply invert the metric to get something more sensible. Once we do so we get a distribution that looks like this:
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As before we can construct a bootstrap distribution and examine the bootstrap with respect to the observed values.
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Again, we see that the range of the bootstrap values is far narrower than the range of observed fumble rates. In fact, we find the same number of outliers..
A lot of this has to do with the skewness in the data itself. One can tilt the bootstrap to better account for the skew observed, but that might create outliers in the other direction and I'm tired.
The more serious problem is that there are probably multiple distributions underlying the fumbles lost metric (as pointed out by the deadspin article and SOSH); combining them all together creates a metric that is really difficult to interpret.
In the end, I'm not sure sharp can claim the pats are an outlier, based on his data, without also claiming the same about nearly every team in the league.
Benjamin Allbright 
Brian Burke 
