## More on Deflategate

By converting football pressures to ball temperatures under the Ideal Gas Law, it is possible to conveniently show Colt and Patriot information – transients, simulations and observations – on a common scale. I’ve done this in the diagram shown below, and, in my opinion, it neatly summarizes the actual information. Commentary follows the figure.

Figure 1.  Transients as digitized from Figures 25 and 27 converted to temperature transients using Ideal Gas Law. Red- Patriot, blue- Colt; thick – dry; thin – wet; solid -Logo, dashed – Non-Logo.  Simulations shown in open circles: large –  Logo 67 deg F initialization; small – NonLogo 71 deg F initialization.  Observed average: solid circle- Non-Logo; + – Logo.

I’ve shown the opening 48 deg F value as a black triangle for reference.

The dry and wet transients of Figures 25 and 27 were digitized and fitted with negative exponentials to an asymptote.  For each permutation (pregame Colt pressure 13 psig, Patriot 12.5 psig; Logo bias of 0.38 psi; pregame 71 deg F for Non-Logo; 67 deg F Logo), the moles contained in the football based on pregame initialization were calculated.  From the prescribed moles, the corresponding temperature transients are calculable under the Ideal Gas Law.   There is no reason why the temperature gradients for dry footballs (thick lines) should differ and yet there is about a 5 deg F range.  Theoretically, it seems that the temperature gradients should start at 48 deg F, but the fitted curves all start higher.  Curiously, the range for wet footballs (thin lines) is much narrower. I don’t exclude the possibility of an error in my calculations as these calculations are not routine for me, but I’ve checked carefully and they seem right.

Next, I’ve converted the simulation results (Non-Logo 71 deg F initialization – large circles; Logo 67 deg F initialization – small circles) to temperature.  Again, there is no reason why there should be any difference in temperature transients, but there is.

Finally, I’ve plotted the observed half-time average measurements at plausible average measurement times (Patriot 3.75 minutes; Colt 8 minutes) based on Logo initialization (+) and Non-Logo initialization (solid dot).  Note that these calculations do not depend on which gauge was used for half-time measurement, only which gauge was used for pre-game initialization.

From the figure, the implied temperature of Patriot balls at half-time based on Logo initialization and of Colt balls at half-time based on Non-Logo initialization both correspond well with simulation and transient calculations and with each other, while the implied temperature of Patriot balls based on Non-Logo initialization and Colt balls based on Logo initialization are both inconsistent with the other information.

I think that this single figure encapsulates all of the relevant information in Figures 24, 25, 26, 27, 29 and 30 in a single location.

Appendix

I’ve used the following scripts to produce the above figure. In this iteration, I didn’t use Exponent’s conversion formulas to Master Gauge scale, as the failure of these formulas to conserve the differential between gauges troubles me.

```#IDEAL GAS LAW RELATIONS pV=nRT
pascal= 6894.76 #pascals per 1 psi
R= 8.31441 #gas constant J K-1 mol-1
Vf= .004237 #m3 volume of football 4237 cm3
cent =function(x) (x-32)*5/9
fahr = function(x) 9/5*x +32
coltp=13.0
bias=.38 #between Logo and Nonlogo Gauge

Nx=function(targetp,pregame=71) (6894.76*(14.7+targetp))*Vf/(R*(273+cent(pregame)))
#function to calculate moles from pregame pressure(psig) and temperature (deg F)
Nc= (6894.76*(14.7+coltp))*Vf/(R*(273+cent(71)));Nc # .3308858
Pf= function(tem, n=Nc) (n*R*(273+cent(tem))/Vf)/6894.76 - 14.7 #in psi
#function to calculate pressure in psig from temperature in deg F and moles
Pf(48) #11.84 at 13.05
Tf= function(P, n=Nc) fahr( (Vf* ((P+14.7)*6894.76))/(n*R) -273 ) #in psi
#function to calcuate temperature in deg F from pressure in psig and moles
Tf(10.9567,n=Nplogo)#47.99
Tf(11.49-bias, n=Nplogo) # 51.02595

##INPUT FIT TO DIGITIZED TRANSIENTS IN FIGURES 25 (Non-Logo) and 27 (Logo)
#setwd("d:/2015/football")
B\$pregame=rep(c(71,67),each=4)
B\$P=12.5+0.5*(B\$team=="Colt")-0.38*(B\$gauge=="Logo")
B\$n = (6894.76*(14.7+B\$P))*Vf/(R*(273+cent(B\$pregame))); # .3308858

##PLOT FIGURE
ind=seq(0,13.5,.25);
o=F

if(o) png("implied_ball_temperatures.png",w=1080,h=810)
if(o) par(list(font.lab=2,font.axis=2,cex.lab=1.4,cex.axis=1.4,cex.main=2))
if(o) cex0=3 else cex0=2
plot(0,type="n", xlim=c(-.25,14),ylim=c(42,75),axes=F,xlab="Halftime Minutes",ylab="Ball
Temperature (deg F)")
axis(side=1); axis(side=2,las=1); box()

for(k in c(1,3,5,7)) {
f=function(x) B\$A[k]+B\$B[k]*exp(-B\$C[k]*x)
lines( ind,Tf( f(ind), n=B\$n[k]), col=6-2*as.numeric(B\$team[k]),lwd=3,lty=as.numeric(B
\$gauge[k]))
} #dry transients

for(k in 1+c(1,3,5,7)) {
f=function(x) B\$A[k]+B\$B[k]*exp(-B\$C[k]*x)
lines( ind,Tf( f(ind), n=B\$n[k]), col=6-2*as.numeric(B\$team[k]),lwd=1,lty=as.numeric(B
\$gauge[k]))
} #wet transients

#plot observed averages converted to temperature
points(0,48,pch=17,cex=cex0)
#Patriot
points(3.5, Tf(11.11,n=Nx(12.5-bias,pregame=71) ),pch="+",col=2,cex=cex0) #51.02595
points(3.5, Tf(11.11,n=Nx(12.5,pregame=71) ),pch=19,col=2,cex=cex0) #51.02595
#Colt
points(8, Tf(mean(foot\$Nonlogo[12:15]),n=Nx(coltp-
bias,pregame=71)),pch="+",col=4,cex=cex0)
points(8, Tf(mean(foot\$Nonlogo[12:15]),n=Nx(coltp,pregame=71)),pch=19,col=4,cex=cex0)

#plot simulations converted to temperature
points( sim\$Pat_avg, Tf( sim\$Pat_pressure,n= Nx(12.5-bias*(sim\$Case=="Logo"),sim\$Pregame_T
)), pch=1,cex=2*as.numeric(sim\$Case),col=2)
points( sim\$Colt_avg, Tf( sim\$Colt_pressure,n= Nx(coltp-bias*(sim\$Case=="Logo"),sim
\$Pregame_T )), pch=1,cex=2*as.numeric(sim\$Case),col=4)

legend("topleft", lty=c(1,2),lwd=c(3,3),legend=c("Logo","Non-Logo") ,cex=2)
legend("bottomright", fill=c(2,4),legend=c("Patriot","Colt"),border=F,cex=2)
title("Implied Half-time Ball Temperatures")
if(o) dev.off()
```

1. medge
Posted Jun 26, 2015 at 6:12 PM | Permalink

Great work Steve..Simply amazing..Thank you!

2. Henry
Posted Jun 26, 2015 at 8:15 PM | Permalink

The ball boy who for years had the nickname the DeFlator, that was just a coinkadink.

Like the flurry of emails between the ball boys and Brady after the scandal broke, just another coinkadink.

Ball boys and NFL quaterbacks with net worths over \$100 million plus married to super models always text each on regular basis.

Honestly what was the point of this by Steve??????

The New England Patriots and more specifically Belichek, are repeat offenders and known unrepentant cheats.

Busted for years of filming the signals used for play calls in the 1st half of the game, then the video was brought into the locker room deciphered and used in the second half of the game to get a humongous unfair advantage.

Filthy crooks and cheats, they belong with the Micheals Manns of the world and I see no point in defendeding them.

And I love the argument of what difference did it make, obviously it did or why would Brady have gone to such lengths to deflate the balls to his liking.

Anybody who has played football competively knows that a ball that isn’t rock hard is easier to grip, catch, throw and hold.

Only a kicker wants a rock hard ball which is why amazingly another coinkadink only two of the almost dozen balls were within the proper PSI.

• MikeN
Posted Jun 27, 2015 at 8:30 AM | Permalink

>Like the flurry of emails between the ball boys and Brady after the scandal broke, just another coinkadink.
There were no e-mails revealed. Trying to find out what happened is a logical action.

>Ball boys and NFL quaterbacks with net worths over \$100 million plus married to super models always text each on regular basis.
They didn’t. That’s why Wells saw ill intent in the texts after the game. Alternate explanation is that Brady was trying to find out about what happened from the people responsible.

>Honestly what was the point of this by Steve??????
to show the scientific experts did not get it right. Without the scientific evidence, the texts alone are not conclusive.

>Busted for years of filming the signals used for play calls in the 1st half of the game, then the

No, they filmed them for future use. It was actually within the rules, and the NFL got it wrong in the memo the Patriots violated. Belichick actually defended this in his press conference six years later.

>Anybody who has played football competively knows that a ball that isn’t rock hard is easier to grip,
Yet Aaron Rodgers tries to inflate them high, going deliberately over and hoping the refs don’t notice.

>Only a kicker wants a rock hard ball which is why amazingly another coinkadink only two of the almost dozen balls were within the proper PSI.
The kickers use a different set of game balls. Pats were accused of trying to get a kicking ball into the game. Turns out a ref was stealing, but perhaps you wish to declare Brady guilty of this too.

• Henry
Posted Jun 27, 2015 at 2:49 PM | Permalink

The person caught filming testified that he filmed the first half of the game and brought the video into the locker room during half time.

That was what he was suppose to do, it wasn’t OPTIONAL.

Yeah I agree with you it was for future use……like the second half of the game.

If your delusion supposition was true why wouldn’t they want video of the second half of the game “FOR FUTURE USE”.

Like a record of how the opposing team reacted to changes made during half time might be useful???…..If indeed this was for “FUTURE USE”.

Your position is total nonsense they were filming the hand signals of the opposition, then the defense or offense deployed.

You need to be laughably obtuse or a member of O’s Iranian nuclear negotiating team to believe such laughable nonsense.

• MikeN
Posted Jun 27, 2015 at 5:43 PM | Permalink

Of course that’s what they were filming. Belichick said he felt it was legal if used for future use but not the same game. Somehow the Steelers contradicted their own Senator and said they thought the taping had no effect. If it was for halftime adjustments they would have reached a different conclusion.

• Henry
Posted Jun 27, 2015 at 3:02 PM | Permalink

>Like the flurry of emails between the ball boys and Brady after the scandal broke, just another coinkadink.

There were no e-mails revealed. Trying to find out what happened is a logical action.

Yeah that’s why Brady refused to hand over his phone, which would of provided exonerating evidence of a POOR BEWILDERED BRADY ASKING HIS BALL BOYS “QUE PASO MY FRIENDS QUE PASO????”

>Anybody who has played football competitively knows that a ball that isn’t rock hard is easier to grip,

Yet Aaron Rodgers tries to inflate them high, going deliberately over and hoping the refs don’t notice.

Very good you got me there all professional athletes are exactly the same in their preferences and abilities, there are no differences between them whatsoever…….

Never mind Brady himself before the scandal said he likes a softer ball, whats important is what Aaron Rodgers likes.

• MikeN
Posted Jun 27, 2015 at 5:48 PM | Permalink

Any texts he sent them would have been on the phones that the NFL did look at. As would the record of phone calls.
Brady’s refusal to hand over his phone is not proof of guilt, and it is possible the Players Association wouldn’t allow such a precedent.

• Glenn999
Posted Jun 29, 2015 at 7:15 PM | Permalink

approaching with an open mind….

3. Posted Jun 27, 2015 at 4:36 AM | Permalink

The New England Patriots and more specifically Belichek, are repeat offenders and known unrepentant cheats.

You’ll have to excuse me for living in the UK as it’s not known to me. But let’s assume what you say is true. Perhaps it would have been better not to use garbage to try and prove the Patriots were cheats, as this would undermine precisely what you knew to be true in the minds of ignorant people like myself who didn’t?

• Henry
Posted Jun 27, 2015 at 6:15 AM | Permalink

Yes your right the NFL should of required 300 footballs to of been provided before every game to provide a better sample size for statistical analysis.

And yes of course these footballs should of been tested by lab coated technicians with environmental conditions rigorously controlled.

Sheeesh………Most things in life go by the “Good Enough” rule.

There was more than enough evidence when ALL things were considered that these unrepentant cheater Belicheat and the Patriots were up to no good AGAIN!!!!!

• mpainter
Posted Jun 27, 2015 at 6:27 AM | Permalink

The post and comments make it clear that no sound conclusions can be drawn from the half time pressure measurements without corresponding ball temperature data.
The “more probable than not” conclusion of the Wells Report is shown to be unjustified.

Posted Jun 27, 2015 at 3:14 PM | Permalink

“300 footballs”

• RichG
Posted Jun 28, 2015 at 10:46 AM | Permalink

Richard – to understand the irrationality of posters like Henry, imagine the intensity of the rivalry between Manchester and Liverpool – except Liverpool hasn’t won since 1968.

Now imagine if the Premiere league management was based in Liverpool, run by former officials of Liverpool, with game day league officials formerly from the Liverpool team, all with personal ties to the coaching staff of the opposing team on the day in question.

Now you can understand why this is such a mess.

4. Henry
Posted Jun 27, 2015 at 6:59 AM | Permalink

The opposing team’s balls I believe was the de facto control sample and I don’t think its a leap of faith to assume all the balls temperatures would be close enough for the PSI reading to be generally considered accurate.

• mpainter
Posted Jun 27, 2015 at 10:46 AM | Permalink

You apparently have not read the comments nor the post very carefully. There were, at half-time, only four Colt’s balls measured for pressure. I don’t see much control there.
The Patriots balls were gauged first, the four Colts balls last. Then time ran out.
Indeed, a “leap of faith” is required if we are to assume that all of the balls were of the same temperature when measured at half-time.

5. Posted Jun 27, 2015 at 7:05 AM | Permalink

Go metric

6. Henry
Posted Jun 27, 2015 at 6:02 PM | Permalink

“MikeN says:
Of course that’s what they were filming. Belichick said he felt it was legal if used for future use but not the same game. Somehow the Steelers contradicted their own Senator and said they thought the taping had no effect. If it was for halftime adjustments they would have reached a different conclusion.”

Why was the guy filming instructed to bring the video TO THE LOCKER ROOM DURING HALF TIME OF THE SAME GAME!!!!!!!!!!!!!!!!!!!!!!!!!

What part of that don’t you get?????

Why didn’t he film the entire game and deliver the video some time after the game was over??????

IT OBVIOUS WHY!!!!!!!!!!!!!!!!

What does the Steelers opinion have to do with it????

Belicheat did it for many years against all teams for many years, Steelers opinion who gives a Sh_ _??????

• MikeN
Posted Jun 27, 2015 at 10:12 PM | Permalink

>What part of that don’t you get?
The part where that’s what happened. First I’ve heard of it, and I can’t find it mentioned anywhere.

• RichG
Posted Jun 28, 2015 at 11:06 AM | Permalink

Yes – more urban legend. As if it is even possible for some one to take film covering 90min of football, edit it to match up with the actual plays being run, deduce the system, and train up the coaching staff during the 15 min halftime. Recording signals for later use was not only legal but done by every team in the league. Not too much common sense here.

Thread is certainly being hijacked away from Steve’s very cogent analysis here. Showing the data normalized to temperature is brilliant.

• MikeN
Posted Jun 28, 2015 at 12:56 PM | Permalink

Before Bill Belichick, under Pete Carroll the Patriots played Miami 3 times in 5 weeks, and by the playoff game the defensive line knew the audibles, and were laughing at the Dolphins.

7. Henry
Posted Jun 27, 2015 at 6:20 PM | Permalink

You have obviously never played competitive football to realize how busy coaches are during half time.

Why would the coaches want some guy in the stands surreptitiously video taping the coaches signals to the opposing team and the subsequent deployed defense or offense, then have this video brought into the beehive of activity of the half time locker room unless they planned to use it right then.

If Belicheat planned to write a book or movie of the week with this info from the video tape sometime in the next ice age he wouldn’t of had the guy doing the filming anywhere near the locker room during half time of that same game.

This is plain stupid obvious to anybody who has played the game…..

• Navy Bob
Posted Jun 27, 2015 at 8:11 PM | Permalink

Henry – that was 8 years ago. Let it go.

8. lloydr56
Posted Jun 28, 2015 at 7:46 AM | Permalink

Apart from the usual wonder at Steve’s skills and attention to detail, why are this post and comments of interest here in climate land?

Because an apparently first-rate crew of consultants (engineers?) performed the same kind of nonsense that famous climate scientists have been performing for years, and continue to perform. Credentials don’t necessarily mean what we’d like to think they mean, i.e. following best practices, double-checking one’s work. In the case of climate, it is pretty clear why consistent deviations occur: there is a narrative to support, and governments with their money to satisfy. Something similar has happened with diet and nutrition: fat, cholesterol, salt.

But this exercise for the NFL, with all its money, and presumably its ability to hire the best of the best? Is sloppy work, with some combination of ignorance and dishonesty (for example, dishonesty in failing to ask the hard questions identified in intro stats, followed by honest but unimpressive ignorance–we didn’t find out the answers to those questions) now accepted fairly widely? Or was there some agenda working with the NFL that helps to explain the result? Did the NFL want to make an example of the Pats, and bring mighty “science” to bear on one side of the scales?

• mpainter
Posted Jun 28, 2015 at 8:43 AM | Permalink

My first impression was Well’s conclusion:
“More probable than not” that the balls had been intentionally deflated. This weak conclusion becomes unsustainable in light of the post by Steve and the comments.
It will be interesting to see how this plays out.

9. David Jay
Posted Jun 28, 2015 at 11:30 PM | Permalink