ESPN's Chris Mortensen has reported that the Patriots are found to have used 11 deflated footballs out of their allotted 12 balls. The rule states that the footballs must inflate to between 12.5 and 13.5 pounds per square inch (PSI). The belief is that these footballs were 2 pounds per square inch under the minimum limit.
The Patriots say that they are cooperating with any and all league requests.
Here's the brute math behind it:
Pressure at start * Temperature at end = Temperature at start * Pressure at end
There are adjustments for units of measurement that I will take into account (gauge pressure, K).
If the temperature when the referees measured took place indoors, it would likely be ~70 degrees.
The temperature at kickoff and at halftime was ~50 degrees.
The 2 PSI below the minimum would be ~10.5 PSI.
Pressure Gauges have an accuracy of +/- ~0.5 PSI.
Plug all of this into the equation and the Patriots were using footballs that should've measured at roughly 11.5 PSI at weigh in. Statistically speaking, this falls within the realm of expectation, but is worth reviewing.
So this means one of two things:
1) The Patriots deflated their balls after weigh-in and there would have to be proof that the Patriots did so.
2) The referees didn't perform their due diligence by actually measuring the footballs correctly.
Aaron Rodgers has stated that the refs don't do the best of jobs measuring the PSI as he slips in overinflated footballs. So signs should point to the latter, even though the conversation will obviously steer towards the former.
It's also standard procedure for the refs to inflate balls that fall below measurement.
As for the penalty?
NFL game ops manual calls for at least $25K fine for person who alters footballs, and "if appropriate" head coach or other club personnel.— Kevin Seifert (@SeifertESPN) January 21, 2015
Roughly equivalent to wearing the wrong colored cleats. Don't overreact.
Any physicists who would like to counter, please do so in the comments.