Further to my series of posts on Deflategate, reader chrimony observed that my statistical analysis had shown that it was possible that there had been no tampering, but had not excluded the possibility of tampering. This is a sensible observation, but raises the question of whether and how one could use the available statistical information to exclude tampering. This is analysis that ought to have been done in the Wells Report. I’ve done the analysis in this post and the results are sharper than I’d anticipated.
For Logo initialization, any manual deflation exceeding de minimis of say 0.1 psi can be excluded by observations. For Non-Logo initialization, statistical information rules out “high” deflation scenarios i.e. deflation by more than the inter-gauge bias of 0.38 psi plus uncertainty, including deflation levels of ~0.76 psi reported in Exponent’s deflation simulations. Remarkably, for Non-Logo initialization, the only manual deflation that is not precluded are amounts equal (within uncertainty) to the inter-gauge bias of ~0.38 psi. Precisely why Patriots would have deflated balls by an amount almost exactly equal to the bias between referee Anderson’s gauges is a bizarre coincidence, to say the least. I think that one can safely say that it is “more probable than not” that referee Anderson used the Logo gauge than that such an implausible coincidence.
As discussed in a previous post (see here), half-time ball pressures can be converted to ball temperatures using the Ideal Gas Law and knowledge and/or assumptions of pre-game initialization conditions.
The half-time ambient temperature was 48 deg F (black solid dot). The average Colt half-time pressures (using relatively unbiased Non-Logo gauge) convert to an average ball temperature of approximately 58.1 deg F (blue + sign). Based on the information that the referees only measured 4 Colt balls because they were “running out of time”, I’ve estimated the average Colt measurement time at 12.5 minutes, just before the end of half-time at 13.5 minutes. This yields the negative exponential transient as shown below. Because Patriots had substantially more ball possession, especially towards the end of the first half, their mix of balls would be wetter than the Colt mix and thus, if anything, below the Colt transient. Note that this transient is for a mix of wet and dry balls – not dry balls or wet balls.
Figure 1. Half-time ball temperatures. Dry transient is fitted negative exponential to Colt half-time average at estimated average measurement time of 8 minutes. Wet transient differential is based on information in Figure 27. Implied ball temperatures for average Patriot half-time pressure measurements is shown for three cases: Logo initialization and no deflation; Non-Logo initialization and no deflation; Non-Logo initialization and 0.72 psi deflation – matching average deflation in Exponent simulations of rapid deflation.
Exponent’s simulations of surreptitious deflation all yielded an average deflation of ~0.76 psi (with very little variability – see note in Appendix.) If Patriot balls had been deflated after measurement by the same amount as Exponent’s deflation simulation (0.76 psi) – a plausible comparison – then the implied ball temperature for Patriot half-time average pressure of 11.11 psi (Non-Logo gauge) is 58.4 deg F – higher than the corresponding temperature for Colt balls measured later in the half-time- and well above the transient at plausible average Patriot measurement times. The implied hiatus contradicts the possibility of surreptitious deflation in the amount of the Exponent simulations. The only deflation values (Non-Logo gauge initialization) that are consistent with the transient to observed Colt values are values in an interval centered (curiously) at ~0.38 psi – the precise value of the inter-gauge bias.
For comparison, I’ve also shown the corresponding ball temperature assuming Logo gauge initialization at 71 deg F (red + sign), almost exactly on the Colt temperature transient. If, after Logo gauge initialization, there had been manual deflation of 0.38 psi, the ball temperature corresponding to 11.11 psi at half-time would be almost exactly equal to the 0.76 psi deflation case for Non-Logo initialization – contradicted by the resulting hiatus.
Discussion
These results are considerably sharper than results in earlier discussion. For the Logo gauge initialization of Patriot balls, it is not just that observed values are consistent with Logo initialization, but any noticeable (in some sense) manual deflation would yield ball temperatures at 11.11 psi (Non-Logo) that were too high relative to the Colt measurements later in the half-time. Any manual deflation greater than ~0.1 psi or so would be inconsistent with observations. Exponent’s simulations did not show that such small deflation could be achieved, nor is there any sensible reason why anyone would bother trying to deflate footballs by ~0.1 psi.
For Non-Logo initialization, any manual deflation greater than ~0.38 psi plus uncertainty allowance, the observed half-time pressures of 11.11 psi (Non-Logo) yields ball temperatures that are too high in comparison with later Colt measurements and are precluded. Similarly, manual deflation less than ~0.38 psi minus uncertainty allowance yields ball temperatures that are too low in comparison with later Colt measurements.
The remarkable result is that only manual deflation (Non-Logo initialization) that is not precluded are amounts equal, within uncertainty, to the inter-gauge bias of ~0.38. It surely passes all understanding why the Patriots would set a deflation target that so exactly matched the inter-gauge bias of referee Anderson’s two gauges. And then executed a surreptitious deflation program exactly implementing this implausible objective. Or even why they would bother with ~0.38 psi deflation rather than more substantial deflation of 1-1.5 psi or more.
I think that one can safely say that it is “more probable than not” that referee Anderson used the Logo gauge than that such an implausible coincidence.
Appendix – Exponent’s Deflation Simulations
Exponent’s deflation simulations involved three different employees attempting to deflate 11 footballs in 1 minute 40 seconds. The results were very consistent: each employee deflated the balls by an average of 0.76 psi, with very narrow deflation ranges both intra-employee and intra-employee. The average deflation ranged from 0.75-0.79 psi and standard deviations of ~0.1 psi. Note that the Wells Report had arm-wavingly attributed Patriot pressure variability to variability in deflation, but their own deflation simulations did not yield anything other than negligible variability – an inconsistency not addressed in the report.
