OK. I analyzed the data for 2015. To make my analysis reasonable, I only looked at Passers with over 1,000 yards, which gave me a universe of 39 passers.
I then looked at two data series.
The first (and obvious) was "Passing Yards Lost to Penalties." This is what the data show:
Mean: 140.03 Yards
Standard Deviation: 90.23
Brady: 504 yards
Interpretation: Brady was just above four full Standard Deviations away from the mean (four sigma). On the following chart, you can see that nearly all passers cluster around the mean (140.03 yards). Brady is the single data point way, way out to the right in the category "More." (FYI: Carson Palmer is the one passer who sorts into the 250--300 yard bin. Four passers sort into the 200--250 yard bin: Winston, Ben, Dalton and Eli.)
I then added the "Yards Lost" to "Passing Yards" and looked at "Yards Lost" as a percent of the sum. This is what the data show:
Mean: 4.05% (i.e., the average "Yards Lost" for a Passer were 4.05% of "Yards Lost plus Passing Yards")
Standard Deviation: 1.95%
Brady: 9.6%
Interpretation: Brady was nearly three Standard Deviations away from the mean (three sigma).
What does this mean? Well, you always have to be careful with statistics and sample sizes, etc. And, I only looked at the data for 2015, since one can argue that the "points of emphasis" on a yearly basis make each year its own "universe" for statistical purposes. And, if I were doing this for publication, I'd be more careful in delineating my bins.
But with those caveats, there is no reasonable argument that can be made that this is Random. These data reveal a pattern of discrimination that would, I believe, stand up in court.