Reading the Wells Report


Most people have not read the Wells Report.  But most people have an opinion on DeflateGate.

Even fewer have read the scientific portion of the Wells Report prepared by Exponent.  Even fewer than that have read it critically.  But again most people have an opinion on it.

So this is based on a critical reading of the Exponent report.

What specific evidence led the Exponent Investigators to their conclusion that someone had deflated the balls?   Why do the Patriots still insist it must have been natural causes and the weather?

The Scene of the Crime

The primary evidence in this case was destroyed at halftime by the officials.  This was not intentional.   Simple physics tells us that if the amount of air (i.e. the number of air molecules)  in the footballs does not change, and the temperature and dryness return to the state of the original measurements, the pressures will also return to the original pressure. Simply by taking 3 or 4 balls with the lowest pressure and locking them away in a cabinet for later analysis would have been sufficient to prove if there was any cheating going on.  Instead, the officials pumped them up with more air, and sent them back out to the field.

For those of you who like Crime Scene Investigations, this would be like bystanders at an alleged crime scene taking a series of photos of what they think is evidence, then bundling up the critical evidence in a trash bag, and throwing it away to be incinerated.

What we have left is a set of pressure measurements taken while the balls were transitioning in temperature from the ball field to the warm locker room.  There are questions of which gauges were used, how accurate the gauges were, the exact timing of the measurements, the climate in the locker room, and whether a comparison of the pressures in the Patriots balls to the Colts balls is even meaningful.

And then this was given to the Exponent Investigators to sift through to determine if a crime had even been committed.

The Investigative Report

The Exponent Investigators came up with a condemnation of the Patriots.  Or at least the statement that it is “probable” that there was human intervention in the ball pressures, unless some other effect was discovered which might change their conclusion.  And we are left with some serious questions as to how accurately their simulation of the Game-Day Scenario matched the actual conditions of the day.

The Exponent conclusion was based on an average pressure difference of just .2psi.  In this situation, this amounts to a temperature difference of just 4oF.  With a temperature change of 72F to 48F going from the locker room to the playing field respectively (and a corresponding pressure decrease by natural causes of 1.2psi) this seems small.  The allowed starting range by the NFL rules is 12.5psi to 13.5psi.  A .2psi difference is small compared to this range of pressure and is very unlikely to benefit a quarterback.  It is even smaller than the .3psi inaccuracy of one of the gauges used in the measurements. But by doing a careful statistical analysis of the pressure,  and using a simulated environment of the ball field and then the locker room the Investigators came up with this conclusion.   The NFL leadership then decided “probable” was enough, and doled out a punishment to the Patriots, and singled out Tom Brady to suffer the brunt of that punishment.

The Denial of Guilt

The other side  claims that there were other scientific factors missed by the Exponent Investigators, and there was just too much uncertainty in all these factors to come to any conclusion at all, and in fact the Patriots balls were close to the expected values.  In addition their own investigation has failed to come up with a guilty party.

Our Review

We will look at some of those scientific factors which appear to have been missed by the Exponent Investigators, and how they might affect the conclusion.

We do this by looking at temperature primarily,  but keeping track of the pressure.  Temperature is a little more intuitive to people. According to the Ideal Gas Law, Pressure and Temperature are directly proportional to each other as long as the Volume and amount of air in the football does not change.  In this particular case, a change in Temperature of 2oF results in a change in Pressure of about .1psi.

We will avoid discussion of most of the statistical calculations.  These are very important to do, but do not really make sense unless you have all the significant factors identified and accounted for.  They are also complex, and I generally find that if you are arguing because different interpretations of the statistics allow for different conclusions, the results are inconclusive anyway.  There have been several different arguments about statistics in this case, and they do go either way, primarily because the result was so close statistically that Exponent could only issue a “probable” rating  that there was a crime.

So we start with the Temperature Differential that was used to condemn the Patriots.  This is shown in black.  Then we look at problems with the report.  Where they benefit the Patriots, they are shown in red, with an estimate of their size.

4oF (.2psi).  The Critical Difference.

That is the average temperature difference between exonerating the Patriots, and condemning them. Just four degrees.  The Exponent Investigators graph is shown below:

The coup de grace of the Wells Report. In the green circled area the Patriots average football pressure(shown as horizontal brown line surrounded by a shaded uncertainty area) does not overlap with the possible values of the Patriots wet-dry balls as determined in the Game-day simulator. The rising curves represent the ball pressures rising as the balls warm up in the Locker Room at half-time.  The time of 4.5 minutes is the average time it may have taken the referees to make the measurement.  A change of just 4F, either raising the Patriots average pressure, or lowering the rising pressure curves would allow them to overlap, and would cause the conclusion to be “inconclusive”.

Figure 29 of the Wells report was meant to be the coup de grace for the Patriots that led the Exponent investigators to their conclusion that it was “probable” that human intervention caused the deflation of the footballs. The bottom line is the average of the Patriots football pressures, and the shaded area is the uncertainty (2-sigma) in the pressure as determined by the investigators. The wet-ball transient curve for the Patriots at 4.5 minutes lies about .2psi or 4oF above that uncertainty area. In fact, by this investigation, the transient curve pressure lies about four standard deviations above the mean.  Assuming no uncertainty in the transient pressure curve as measured in the model, this is statistically significant, but still reasonably close, which is why the determination was “probable, and not a certainty.

The actual conclusion within the report after showing this graph is: “Therefore, subject to the discovery of an as yet unidentified and unexamined factor, the measurements recorded for the Patriots footballs on Game Day do not appear to be completely explainable based on natural causes alone.

-1.5oF to -7.0oF. Gauge problems.

Referee Walt Anderson had two gauges in his bag when he measured the pressures in the balls before the game. He recalled using the “Logo” gauge, but wasn’t 100% certain. Both the Patriots and the Colts measured their own balls using their own gauges to be 12.5psi, and 13.0psi respectively. After that measurement, the balls from each team were brought to the floor of the officials locker room and left there. Walt Anderson a short time later measured the balls, and agreed with their measurements. The “Logo” gauge was later found to read about .3psi high, and the “non-Logo” gauge was about .07psi low. The temperature was estimated  in the vicinity of where the balls were to be between 67F and 71F (probably a later measurement).  There are a lot of permutations to choices of temperatures, gauges, balls,  including switching gauges in the middle of the measurement.

Despite Walt Anderson’s recollections, the Exponent investigators concluded that it was likely that the non-Logo gauge was used. The Figure 29 graph in the previous section was done using the 71F temperature, and the “non-Logo” gauge.

A different conclusion could be reached by assuming that when the Patriots and Colts measured their own balls, they did so in locker rooms where the temperature was between 72F and 73F – the setting of the thermostat.  (note — we do not know the settings of the thermostats in the Colts/Patriots locker rooms, however this was the setting in the officials locker room, and it stands to reason that it would be the same in the other locker rooms.)  The referee’s measurement then confirms that they measured them at about the same temperature because the pressures were consistent by his gauge.  ( the temperature of the balls in the officials locker room was then 66F-67F or 73F-74F for the logo or non-logo gauges respectively)  Then let’s not worry about which gauge the referee used. The Colts  gauge was later concluded by Exponent to be accurate, and that should be good enough.  So instead of using a starting temperature of 71F, we should use a starting temperature of 72.5F.

This effectively brings the Patriot average pressure up by .075psi, or about 1.5F, closing the gap to 2.5F.

Another possibility is that Walt Anderson may have switched gauges in between measuring the two sets of footballs using the Logo gauge for the Patriots balls, and the non-Logo gauge for the Colts balls.  This is a reasonable claim, especially since the Patriots have a protocol called “gloving” (rubbing the balls hard) which would have warmed the balls up prior to the Patriots final measurement of the balls. This could have set the Patriots balls at a pressure roughly .37psi lower than the Colts right from the opening gate, causing a 7oF shift in effective temperature of the Patriots balls.

The Exponent Investigators did do another graph depicting a starting temperature of 67F, and using the Logo gauge (Figure 27.).  Some people claim there is a significant error in that graph setting which is described in a report in another blog by Steve McIntyre.  Fixing that mistake may result in a similar effect as to the gauge switch described in the previous paragraph.

Unfortunately, there is a counter argument to every choice of which gauge was used for what, and we will never know the truth here.  Rather than show a preference for one configuration or the other,  the Wells Investigators need to rule out each possibility.

-1.0oF – 4.0oF. Evaporative cooling, and questionable climate control.

This will be confusing, and not very intuitive.  It also requires a bit of advanced atmospheric physics and thermodynamics to understand.

An example of evaporative cooling is what happens when you sweat.  A little bit of moisture on your skin cools it down.  I think we all recognize that this cooling is dependent on humidity and wind speed.  If it is humid, sweating is just not as effective cooling you down.  Your body compensates and makes you sweat more.  And a little bit of wind — a hand fan or a small breeze can be very refreshing and cooling.

 The atmospheric terms we use here are wet-bulb temperature, and dry-bulb temperature.  The dry-bulb temperature is the one we are used to, and we just call that the temperature.  The web-bulb temperature is the temperature that something can cool down to if it gets wet.  In fact to measure it, people will take a regular thermometer, wrap it in wet gauze, and wave it around in the air.  If it is humid, the wet-bulb temperature will not be much cooler than the dry-bulb temperature, in fact at 100% humidity, it will be the same.  In a low humidity situation, the temperature can be quite a bit lower, even as much as 20 or 30 degrees.  Some of you may be familiar with a “swamp cooler”, or “evaporative cooler”, which in low humidity areas of the country can be a very effective cooling mechanism for a home using this principle.

Footballs can be cooled by evaporative cooling.   Although Exponent never named it as such, we can see this cooling in their graphs.  Here is their Fig 21:

Game Day scenario. When the balls are brought to the field, the temperature drops from 72F to 48F. The pressure inside the Patriots balls (in brown)drops exponentially from 12.5 psi to about 11.3psi with a time constant of 15 minutes. The wet balls drop to a lower pressure due to the evaporative cooling, with a wet-bulb temperature of  45F.  This accounts for the .15psi difference between the wet and dry balls.

Following the curves here, when the balls are brought to the field, they experience a temperature drop from 72F to 48F.  The balls are dampened in this simulation by spraying them and wiping the water off with a cloth.  Wet balls are subject to evaporative cooling.  The humidity on the field was estimated to be 75%, so we can look up on a chart (a Psychrometric chart – see Appendix D) the wet-bulb temperature which would be 45F.   This 3F drop in temperature corresponds to a .15psi drop in pressure, and this agrees with the difference in pressure between the dry balls and the wet balls as they descend in pressure seen in this figure.

When returning to the locker room, the balls come into a room where the Exponent Investigators claim the temperature matched that of the thermostat at 72F-73F, and the humidity was 20%.  This is a very low humidity situation, but not uncommon in the winter.  According to a Psychrometric chart, this corresponds to a wet-bulb temperature of 50F (see Appendix D).  The final temperature for the “wet-ball” could then be somewhere between 72F (dry – 12.5psi), and 50F (wet with good air circulation – 11.4psi)!   We overlaid the theoretical transient curves corresponding to these two extremes onto Figure 29, and that is shown here. (See Appendix C for details) The Exponent investigators don’t say anything about air circulation or climate control in general except that they tried to match the temperature and humidity of the air at the ball field, and at the locker room.

Overlaying the possible range for transient evaporative cooling as computed theoretically using the same starting pressure as the Patriots wet ball.

These theoretical transient curves are surprising, and perhaps even shocking!  Walking into a room where the temperature rises from 48F to 72F, could cause very little increase in temperature or pressure in a wet football, as evaporative cooling takes over in the low humidity environment, and keeps the ball cool.  Of course this requires good circulation — a wind.

Let me repeat that:  The wet ball curve could be almost flat!   Clearly this was not what Exponent measured!  Their curve follows almost exactly a “dried” wet-ball curve, and ascends rapidly.  Their measurement is not out of line with the theory, but the expectation would be that their curve would lie below the “dried” wet-ball curve, because the balls were wet, and evaporative cooling was observed to have an effect on footballs.

A theoretical derivation of these curves, and a thorough discussion of the Exponent measurement is found in Appendix C.  It is clear from some of the Exponent charts that they had trouble with their climate control, in particular, controlling humidity and temperature in a consistent way.  There is no discussion of this, nor an uncertainty in the their results due to these experimental factors which may have artificially increased the rate of rise of the ball pressure.

Also, in the Exponent report there was no discussion of air circulation in any of their simulations, but it may have played an significant role in the actual situation.  Air circulation is both what helps warm the footballs, but also increases the rate of cooling from evaporation.

Is it possible the Exponent simulation day footballs were almost dry in the locker room?  This goes against what was said by the officials who stated that some of the balls were “damp, but not waterlogged”.  Do we have an estimate for the wind velocity and air circulation in the officials locker room?  And did the Exponent Investigators provide circulation of that magnitude?  Note, based on HVAC regulations for public locker rooms, it must have been at least .1mph (1.5 inches per second),  and could have been 1-2mph if near a ventilation port.

So what do we do with this? Realistically, because the variation can be large here, the argument is incomplete until a more reliable measurement can be made, or the Exponent Investigators provide more explanation.  One good step would have been for Exponent to calibrate their simulation chamber against the actual Officials locker room in Foxboro.

So right now, we keep it as a range.  I do not expect it will be at the bottom of the range, nor at the top.  I do expect that at least some drop in the experimental curve would occur if the climate control in the room was better (see Appendix C).  That is the reason for at least 1oF lowering the curve.

-1oF to -5oF. Ball Bag Contact with the Cold Turf.

The bottom of the ball bag was sitting on AstroTurf which was sitting on a cold earth. Preceding game day, there were 18 days of January during which the Temperature rarely got out of the 20’s. On the day of the game, the temperature rose about 30 degrees, starting at 20F in the early morning to 50F at game time. We assume the ball bag is a quality athletic bag. Cooling is all about exposed surface areas, and the entire bottom of the bag was in contact with the turf for a couple hours leading up to halftime. Somewhere down below that bag, inside the turf is a boundary layer which was probably at 32F. The cooling power is brought to the surface through the wet nylon grass bristles as shown in the picture.

The turf consists of nylon grass rising up through a layer of sand and rubber pellets. Not only would the sand have been cold, but the ground below would have been frozen from the January cold leading up to the day of the game. Six hours of above freezing weather would not have warmed up the ground to the air temperature. Add to that the presence of water from the rain sitting on the grass for long periods of time, and it translates into a cold, wet ground that comes in contact with the ball bag, and the balls on the playing field.

Now, this is pure conjecture, but it is hard to believe that there would not have been some effect here. 1 degree feels very conservative.

Imagine instead of a 50F day in January, this was a 90F day in May, where average temperatures in New England are in the 60’s.   To cool off, you lay down on the ground in the shade where you can easily take advantage of the cool feel of the ground. The Physics is the same in January, however, the temperatures have been shifted by 40F.

This is easy to test, and next January when the field is frozen again, and on some day when the air temperature is significantly greater than the internal field temperature, perhaps someone could just put a bag of balls on the field, and do some measurements.

-1.4F. Balls in Contact with Cold Turf

There were balls that were used on the playing field.  In between each play the ball sat at the line of scrimmage.  To begin a play, the center grabbed the ball, twirled it in the wet grass,  pressed it to the turf, and then hiked it.  About 15 seconds later it (or a new ball delivered by the ball boy) was back at a new line of scrimmage.  In the previous section we saw how the turf was cold.  It was also wet.  Anyone who has been at Gillette Stadium in the rain will tell you how the water hangs on the artificial grass and glistens.  Well in January, that cold water that was glistening was being cooled from below for a long time before it touched a football.  And then the football itself, about 10% of its surface area sat in direct contact with the grass, and most of the time on the field it was sitting on the field.  All of this served to cool the ball down.

One of the biggest questions is how several of the Patriots balls were at a pressure/temperature much less than the ambient air temperature of 48F.  Well, first the wet-bulb temperature on the field was 45F in a wind of 10-15mph.  Then there is the field itself.  The cold water, and the contact with the turf.  This is conjecture, but I would not be surprised to learn that the 3 balls at 10.85psi (40F equivalent temperature) or below were that way because of play on the field.  This was particularly true for the Patriots who had a long drive just before half-time, which included 4 time-outs after the two minute warning where the balls sat there and cooled while we watched commercials.  These balls would not have had time to equillibrate back to the “wet-ball” temperature in the ball bag, before being measured in the locker room at half time. These balls were also probably the “wettest”, which makes them most susceptible to slow cooling because of evaporative cooling. (See Appendix B).

Because this only involves a few balls, it only affects the average by about 1.4oF.  However, it does explain the wide distribution in the balls of the Patriots, except for the one ball that was measured to be 11.85psi, which seems too high.

-1.0F.  Ball Bag Evaporative Cooling

The ball bags would have been wet sitting out in the rain and the wind of the field.  Just like footballs, these bags would cool down as well, so the 45F wet bulb temperature applies to them, rather than just the 48F dry-bulb temperature of the ambient air above the field.  This effect would also last as the bags were carried into the locker room as well, probably at a walking speed of about 3mph, slowing down the warming of the bag in the locker room air.  This really means that all the balls should have been cooled to at least this 45F temperature regardless of wetness.

Now in the simulation, the balls did not sit inside a wet ball bag.  The temperature was brought down to 48F.  Some balls were wet and others were dry, so some balls, would have been subject to evaporative cooling, but all should have been cooled.  We’ll say 1/3 of them were not cooled, so that brings down the average temp by 1.0F instead of 3.0F

Another interesting aspect of this is the Colts use of trash bags to keep the balls as dry as possible.  Putting wet balls inside a trash bag can reduce evaporative cooling.  Not only does it cut off the wind, but it also puts the balls in a confined space, so evaporation from the surface of the balls will raise the humidity inside the bag, lessening the cooling.

0 minutes to +2.0 minutes. Unzipping the Bag

When were the ball bags at half time unzipped? There is a considerable difference between warming up in open air and warming up in an enclosed space like a zipped ball bag. There is even more difference if your balls are further wrapped in a plastic bag in an enclosed space.

Just think of the cold beer analogy. Imagine you have 3 sixpaks of beer starting at 45F, and you place 1 sixpak in the open in the shade at 72F on a table, another inside a zipped ball bag sitting on the table, and another wrapped inside a heavy duty garbage bag inside a zipped ball bag sitting on the table. An hour later you want a cold beer. Which beer would you choose? Why is the beer wrapped in the bag inside the ball bag the coldest?

The reason has to do with the insulating quality of air. Exponent demonstrated convincingly that a dry football in the open air will equilibrate with the surrounding air with a time constant of 15 minutes –i.e. after 15 minutes it will be more than 63% equilibrated. This is because in the open air, there is circulation, and the air cooled by the football constantly gets replaced by more warm air. Inside a ball bag, the air is trapped. Now the only way to warm up is to have the air warmed by the inside surface of the bag, travel across the bag, or heat the air surrounding the football. This is a slow process. The thermal conductivity of air is much lower than leather.  And then if you wrap the balls in plastic (which apparently was done by the Colts), you need to start questioning how long a time it make have taken for their dry footballs to reach the outdoor equilibrium temperature. Of course once you expose them to the open circulation they will cool down or warm up quickly according to the 15 minute time constant.

The Exponent Investigators did use ball bags as part of their game day scenario.  So this may merely be a question.  They obviously concluded that the bags were not a factor — although that does feel surprising.  But they may know something about the timing of opening the bags, and just chose not to report it.  So we give a range here including 0 minutes, recognizing that this may not have had any effect.


 All these effects together add up to 4.9F – 16F ( .25psi  – .8psi) of adjustment to the graph in the favor of the Patriots — well beyond the 4.0F needed.  You even get an additional minute or so if the bags were unzipped late in the process that went on in the Officials Locker room, and was handled differently in the simulation. And there are more items that others have noticed and questioned.  Each one of these things may be individually dismissed as being insignificant, but all together they may cause a very different conclusion. This takes the question of are the Patriots “probably guilty” to why are we wasting time on this when it is clear nothing really happened, except Mother Nature acting the way she always has.

Mother Nature

One obvious result of this exercise is that people have learned that football pressures in cold air will fall well out of the range suggested by the football rule book.   Mother Nature does not play by the football rule book.

The next question is do the Patriots gain an unfair advantage due to the cold weather in New England “deflating” the footballs? Perhaps it is so. Perhaps this is also the reason for the success of Green Bay.  But what about Buffalo?  Other teams get some advantage due to their home conditions. Consider the thin air at Mile High stadium in Denver. Or the benign weather guaranteed by a closed stadium.

Appendix A.  Ideal Gas Law

This section is intentionally left blank.  Everybody knows this by now.

Appendix B. The Pressures in the Footballs at Half time.

This section is intentionally left mostly blank.

Instead, look at the pressures in the Wells Report.  Note the cluster of 3 Patriots balls at lowest pressure, the next cluster of 5 balls (which are near to the Ideal Gas Law Pressure), and finally a cluster of 3 higher pressure balls (presumably dry).  Our guess is that these clusters represent respectively: the balls most recently used in the game during the drive leading up to half time;  the balls used in the game at other times; and the balls which were dry and not used in the game.

Appendix C.  Evaporative Cooling

Theoretical Background


First a back of the envelope calculation.  Should we worry about evaporative cooling as the balls come into the locker room at half time at all?   We can calculate the amount of heat energy that is needed to raise a 450 gm leather ball back to room temperature using this simple equation:

(1)         Qleather = M * (T1 – T0) * Cp

where Q is the energy, M is the mass of the football, (T1-T0) is the temperature difference (24oF), and Cp is the heat capacity of leather(1.5kJ/kg/oC).  Doing the temperature conversion, and inserting in the values, we find the result:

               Qleather = 9 kJ

If we assume that the amount of water that covers the football is about 2 gm (about 1/2  teaspoon), then using the heat of vaporization of water 2,257kJ/kg and assuming it dries up, then the amount of cooling due to evaporation will be about 4.5kJ.

  (2)          Qevap = 4.5 kJ

These are roughly the same order of magnitude.  Evaporative cooling will have a noticeable effect, unless we are wrong in our estimate of the amount of water that is evaporating around the ball — which for this experiment could be a highly uncertain number.  (To be fair, in a non-scientific way, I did put some water on a leather ball — not a regulation NFL football — just to see roughly how much water might be there.  I think two to five grams is fair, and most of it was gone in 15 minutes in an atmosphere of 45% humidity and 76F with no noticeable wind — and it cooled the ball down quickly to 67F -68F.  By Psychrometrics, the wet-bulb Temp of this little experiment was about 59F )

Transient Curves

So what effect should this cooling effect look like as a function of time:

First, let’s make sure that the theoretical curves actually describe what we should expect.

Physics tells us that the rate at which heat is added to a dry ball is proportional to the temperature difference between the ball and the surrounding air.  This leads to an exponential curve. If we describe it in terms of Pressure.

(3)            P(t) = P1 – (P1 – P0)*exp(-t/tau)

The constant tau is 15 minutes, as measured by the Exponent investigators.  Po and P1 are the initial and final pressures respectively.

When the ball is wet, the heat transfer rate is the sum of the dry ball heat transfer rate minus the cooling transfer rate due to evaporation.  The cooling rate for evaporation is a complex function of temperature, humidity and wind velocity, which we will leave alone, and also depends on the amount of water on the ball which will run out over time due to evaporation.   However whatever the solution here is, the pressure will be lower than the Pressure observed in the dry ball case.  Also in the case of high wind velocity, we expect the temperature of the ball to approach that of the “wet-bulb” temperature, so we can substitute the wet-bulb temperature / pressure into equation (3) above for P1, and get an approximate lower bound to the pressure.

These limits are what appear in the Fig 29 graph shown again below:

Figure 29 again, showing the calculated curves which should bound the region of evaporative cooling for the wet balls as they enter the locker room. This uses the starting point at the same starting point (about 11.2psi) as measured by the Exponent Investigators. Do not confuse the “dry” here with the “Patriots – dry” curve. If we use 11.4psi as the starting point, it would more closely follow the “Patriots – dry” curve.

Systemic Problems with Climate Control

So how could Exponent have gotten the wet-ball curve that they did in the above figure, where the theoretical transient curve for a “dried wet-ball” sitting right on top of their wet-ball curve, when their ball should not have been dry.

One explanation  is that in their simulation, the Exponent wet-balls were almost dry, perhaps with only water left in the seams.  This might be suggested by how closely the Exponent curve matches the theoretical “dry-ball” curve. However, this would be contrary to the observations of the Ref’s at halftime who described the balls to be “damp, but not waterlogged”.  Surely they would have done better than that.

Another explanation is that the humidity in the simulated locker room was too high, or the temperature was pushed high by the HVAC system controlling that room to compensate for the arrival of the balls and people.  We don’t know the size of the simulation room or how well it acts as a infinite reservoir of dry heat.  It is a physical reality that it is difficult to manage climate control in a small room where you hope to measure transient phenomena like the kind Exponent was trying to measure.

Taking another look at Fig 21, we see the following:

Fig. 21 again. Highlighted region A is the the region of this chart which is depicted as a close up in Fig. 29. Region C shows clear exponential behavior of the downward curves, as they asymptotically reach their lowest temperatures. Region B shows questionable non-exponential behavior, as the dry-balls reach their final temperatures in 15 minutes and flatten. Was the heating in the room set too high, and then suddenly turned off? This would suggest that the balls were artificially accelerated in their rising rate due to problems controlling the climate in the simulation chamber.

Not surprisingly there is evidence in the Exponent report that the Exponent curves could also have been caused by climate control problems within their chamber.  Figure 21 shown above shows the non-exponential behavior as the pressure curves for the dry balls rises too quickly and then abruptly flattens in region B.  It also rises and flattens too quickly for the Patriots dry-ball curve (not highlighted).  It is hard to speculate what did this without knowing the apparatus, but it certainly is not indicative of a smooth temperature transition.

The Exponent investigators never mention evaporative cooling as being the source of the difference between wet-ball and dry-ball pressures. It is also evident by looking at their charts that they had a hard time with consistency between different runs of the apparatus. In the different charts containing wet balls and dry balls, these range between a separation of .15psi (Fig. 21) to .23psi (Fig. 29) to .46psi (Fig. 27), at the lowest pressure ranges, and we expect that separation to become larger as the balls move into the locker room, with warm dry air (see Chart 21). These should all be the same. The only explanation here is that they had a hard time controlling the humidity in their simulated rooms.   At 48F, with 75% humidity, the dry bulb temperature was 45F, corresponding to a separation of .15psi.  .46psi is simply too large a separation, and must correspond to a lower humidity.

The few data points they put on the Fig 29 seem to track closely to their experimental wet-ball pressure curve. Of course, you expect they would. The curve was calibrated in the same conditions the balls brought in would be. The bigger question is what that curve would look like in the actual Patriots locker room, with some balls covered in moisture on their entire surface, and the unknown climate control within that setting.

To summarize this discussion of evaporative cooling, there are a number of big areas of concern:

  1. The calibration and control of the humidity and temperature within the simulation seems to have led to an artificially rapid rise in the pressure as measured during the locker room phase of the simulation.
  2. Air circulation and humidity during cooling may have caused inconsistent pressure differences between wet and dry balls. From the Exponent data when the ball is simulating being on the field and cooling, it is apparent that there is enough air circulation and dampness on the balls to achieve the wet-bulb temperature/pressure because of the difference in pressures between a dry-ball and a wet-ball at the low point in pressure.  However there is no mention of the air circulation or windspeed in the report.  By their report, they did try to maintain the humidity at 75%, but the variability in the separation between dry and wet balls suggest they may have had problems controlling this.  There should be some uncertainty calculation in the startoff point of the transient curve, or a calibration of the separation between wet and dry balls, that should show up in their report.
  3. Dampness of the balls.  The balls will dry pretty quickly in a warm dry environment (e.g. 72F at 20% humidity).  Based on the simple theory of evaporative cooling, the amount of water on the balls could make a large difference in their measurements here, and could be substantially different from what happened during the actual time in the locker room.  Some sort of measurement of dampness, and its effect on the curve should be in the report.

The air circulation and evaporative cooling problem greatly confuses the expected temperature rise (and pressure) of a wet ball. We also don’t know how wet/dry the balls were. What percentage of the surface area of the ball was really wet? Did the Exponent drying with a towel really correspond to what the ball boys did on the field? Was there a significant problem with managing the climate control — the humidity and temperature of the simulated locker room?  This result from Exponent is at best incomplete in this area, and make it difficult to make conclusions.

Appendix D.  Wet Bulb and Dry Bulb Temperatures

Psychromatic Chart — The wet-bulb temperature can be determined using this well known chart.  Look for it on the Internet and watch videos on its usage.  The lines shown are used to determine the Web Bulb temperatures relevant to the DeflateGate situation.

Psychrometric chart. The heavy red lines intersect at 72F and 20% humidity, showing a wet-bulb temperature in the locker room of 50F. The heavy green lines intersect at 48F and 75% humidity, giving a wet-bulb temperature outside of 44F – 45F.  The dew point can be read to be 41F.

Appendix E. Control Groups and Calibration (added on later)

Some people have asked about the Colt’s footballs.  What happened to them?  The Wells Report makes a big point of using them as the “control”.

In an experiment a control group is the group that you don’t perform the experiment on, but in all other ways is like the group you are performing the experiment on.  The difference between the control group and the experimental group then shows the effect of the experiment.

In this case the Colt’s balls were considered the control group, and the Patriot’s balls were the experimental group — the experiment being that someone along the way may have deflated the Patriot’s balls.  Presumably that was the only difference in their treatment.

But wait a moment.  There is a laundry list of differences in their treatment:

  1. The two groups of balls may have been initially measured with two different pressure gauges, resulting in a .37psi difference right at the beginning.
  2. Only 4 balls of the Colt’s 12 balls were measured. All the Patriot’s balls were measured.
    • Four of the Colts balls had never been used in the game, and were dry.  Was it these four that were measured?
    • Or were the four those that had  been sitting at the top of the bag in the locker room, and most exposed to the warmer air, and least exposed to sitting at the bottom of the bag on the cold turf?
  3. There special procedures for handling the balls on the ball field taken by the Colts, including putting them in plastic garbage bags to help keep them dry, and potentially less exposed to evaporative cooling.
  4. The Colts offense sat on the bench for 30 minutes before halftime, and their footballs were less exposed to the weather and the cold field.
  5. The Colts balls were measured much later at half-time than the Patriots, and there seem to be some issues with measuring what was happening with evaporative cooling during that time.
  6. How dry did the balls boys for each team really get the footballs?

With all this, and perhaps there is more, it is hard to think of the Colt’s balls as a reliable control group, particularly since, as we have discovered, footballs are quite sensitive to temperature and evaporative cooling.

Because of the uncertainties in the ball handling, I find it more sensible to just evaluate based on the basics laws of physics. And just the simple measurement assuming the Logo gauge was the measuring device and some slowed warming due to evaporative cooling, or some additional cooling due to the cold field seems to put us there.

If you want to use the Colt’s balls as controls,  all those differences above need to be accounted for.

It is a dilemma for the Wells investigators however.  Without a control group, or without returning to Gillette stadium and doing some calibration measurements in cold weather, all they will have done is proved that their experimental apparatus in Menlo Park, CA differs in a statistically significant way from whatever happened on January 18, 2015 at Gillette Stadium in New England, rather than the other way around.


The NFL’s NEW Football Pressure Rules — Hooray!

Congratulations to the NFL for instituting rules which should remove the mystery from DeflateGate, even if it is at the risk of embarrassing the NFL and exonerating the Patriots.  This is a bold and correct move.   It is fair.  It is tedious, but should ultimately clear the air and determine if the weather is the culprit in DeflateGate.  However, by the time we have the correct answer, it will be well past deadlines placed on Brady’s suspension.  That part seems unfair.

The good procedures to be implemented:

  • Measure the footballs before the game, presumably at room temperature and with the balls dry.  This is no different from what has been done in the past, except they have doubled the number of balls, and will record the pressures.
  • Measure the footballs after the game. If there has been no “cheating”, the balls should return to the original temperature and pressure. (This return will take about 1 hour after the game, and should be done in the same room as the initial measurement, and in full knowledge that wet balls will tend towards the “wet-bulb” temperature and pressure, and dry balls will tend towards the original temperature and pressure).
  • Measure the footballs at halftime.  (This is what may exonerate the Patriots)

What they will find with the halftime measurements:

  1. On hot days the ball pressure will be high.  For example, on an 80F day, the ball pressure on a dry, unused ball will go up .5psi.  If it is sunny on that same day, balls on a turf field where the field temperatures (not the air temperature) may reach 120F could find pressures up by 1.0 to 2.0 psi.  This will lead to a wide variability in ball pressures depending upon their usage on the field.
  2. On wet days, there will be variability in the ball pressures which are wet, particularly if the humidity is significantly less than 100%, and it is windy, allowing for significant evaporative cooling.
  3. On cool wet days, particularly in January when the ground is frozen and the air temperature is warmer than the ground temperature, the ball pressures will be low, and vary with wetness and recentness of use on the chilled wet field. (similar to the Patriots-Colts AFC Championship game).  If what the Patriots have been saying is true, then the pressures in these balls will be similar to the Patriots balls in the halftime measurement of the AFC Championship game.
  4. On days when it is 70F and dry, the ball pressures changes will be remarkably unremarkable.

So can this exonerate the Patriots?  It won’t happen until we find a game which fits item #3 above, and the NFL decides to do the measurement at that game.  Then it will either condemn the Patriots, or exonerate them.  My strong bet is on exoneration.  We will either find halftime measurements which fit the profile of Patriots balls at the AFC Championship game, or we will not.

Because of statistics we will never be able to absolutely rule out a nefarious occurrence in the bathroom with the game balls, but we should be able to demonstrate convincingly that the weather could have been the complete culprit all along.  And, if one or more balls were deflated, it was not by a significant amount.

The following information on the new NFL procedures can be found at:  NFL Finalizes New Rules for Football Inspection

Among the new procedures:

  The “kicking ball coordinator” at each game, who previously only handled the six “k balls,” will now take custody of all footballs once they have been approved by the officials, and maintain control of the footballs until 10 minutes before kickoff. The coordinator, a member of the officiating crew, and a security representative will bring the 24 approved game footballs (12 for each team) to the on-field replay station, at which point the footballs will be distributed to each team, in the presence of the league security personnel. The 24 backup footballs (12 for each team) will remain secured in the officials’ locker room.

 The NFL will designate random games in which to test the football PSI at halftime and after the game. The kicking ball coordinator will collect the footballs from both teams at halftime, will be escorted to the locker room by the league’s security personnel, will measure and record the PSI of all 24 footballs, and then remove the footballs from play. The 24 backup footballs will then be used in the second half.

 At the end of each randomly selected game, the kicking ball coordinator will again inspect all game balls from each team and record the results. All recorded information will be reported back to the league office.

 Before the game, the referee will designate two members of the officiating crew to inspect the balls pregame. The officials will number the balls 1-12, and record all PSI data. Previously, the balls were not numbered, the data was not recorded, and only one member of the officiating crew inspected the footballs.

 The footballs still need to measure between 12.5 and 13.5 PSI. If a ball comes in above or below those numbers, it will be adjusted to 13.0 PSI.


Source: NFL Finalizes New Rules for Football Inspection

DeflateGate: The Wells Report Crumbles

As time goes on, the more the Wells Report crumbles.  Despite some significant scientific work***, it is clear now that the statistics are suspect, there are unexplained variations in ball pressures which just don’t make sense with a systematic attempt by an individual to lower the pressure for the benefit of a quarterback,  there are large variations in wet/dry ball pressures that are improperly accounted for and might be attributed to evaporative cooling,  and the Wells Report investigators failed to account for pressure drops in some of the Patriots ball pressures due to the long final drive up the field by the Patriots played on a cold turf field.

Their original conclusion is negated.  It can no longer be said that it is “probable” that the Patriots cheated.  The combination of these problems in the scientific analysis make it highly unlikely that cheating of any sort went on.

This result is easy to check.  Next year, or some year in the future when the weather conditions and field conditions match those of the Patriots-Colts game, just measure the halftime pressures.  This would be a “real game time simulation”, not the simulation cooked in an LA lab, which missed important game time factors. This would be the ultimate vindication, and unless the NFL changes its tune immediately, they will have to explain how they could have been so wrong.

The Statistics

Using p-values like the Wells Report does, only makes sense if you can control for all the variations in the history of two populations, in this case the Colts’ footballs, and the Patriots’ footballs.  There are three large questions that suggest that the control did not exist:

1. We don’t know which pressure gauge was used to measure the two populations. (This they worked to account for.)

2. Whether a ball is wet or dry is shown to significantly impact the rate of warming, and the starting temperature.  We simply don’t know if the small sample of measured Colts’ balls were dry or wet, and we don’t know how long each population was allowed to warm inside the locker room before measurement.  Even the Wells Report experiments show wide unexplained variations in the pressures of wet and dry balls depending on which of their experimental charts you look at.  And we have also found out independent of the Wells Report, that the Colts’ balls were kept in trash bags to keep them dry, the Patriots were not.  The Colts’ balls simply cannot be used as a control in this situation.

3.  Some of the Patriots balls spent more time on the playing field just prior to the half time measurement because of a lengthy Patriots drive up the field.  There is strong reason to believe that the field was significantly colder than the air temperature, and would have caused cooling on those balls.

And even with this uncertainty, another uncertainty involves the way the Wells Report scientists computed their statistics.  While they knew the ball pressures were rising in the locker room, they chose to use the stats from the starting pressures.

As many people have now pointed out, just points 1 and 2, coupled with the way the scientists computed their statistics is enough to render any conclusion other than “inconclusive” impossible.  With all three, the conclusion is completely negated.

In their “Game Day Simulation”, the Wells Report forgot to include the Game.

The 2015 AFC Championship Game was played on a cold artificial turf field, which caused wide variations in the Patriots football pressures measured at half time.

The Artificial Turf was colder than the air temperature, and during the game balls sitting on the cold turf at the line of scrimmage during the would have cooled off more than other balls.  In particular a number of Patriots balls were rolled around and sat on that wet cold turf during that long final drive leading up to halftime that included 4 timeouts after the two minute warning.  That drive, and the colder temperature explains the scatter in pressures seen in the Patriots footballs, as well as the lower average pressure.  This effect was not considered by the Wells Report Investigators.  (see  for details)

That the Artificial Turf was cold must not have been obvious to Wells Report Investigators in LA.  However, to us who live in New England, we can remember that during most of January, the temperatures hovered at or below 20 degrees, but on that one day of the AFC Championship, the temperature jumped from well below freezing in the morning to a balmy but wet 50 degrees at game time.  The sub-freezing ground and sand under the field would not have had time to warm up, and would have still been at a 20 degree to 25 degree temperature.  Turf fields are designed to bring their cooling power to the surface, so the ground would have been significantly colder than the air.  Moisture collecting on those artificial grass blades would have been cooled, and that cold moisture would have been transferred to the balls by contact.

This is not an effect that can be ignored.  It is not an effect that can be simulated by spraying air temperature water onto the ball every 15 minutes and then wiping it off. (This was the Wells Report attempt at simulating the game). This is an effect that could easily have lowered the Patriots measured average ball temperature by .2psi to .6psi, which by everyone’s determination is more than enough to vindicate the Patriots.

Evaporative Cooling — The Difference Maker between Wet and Dry Balls

Another effect is evaporative cooling, which should have been more obvious to the LA investigators.  Indeed, you can see it in the results of several of their charts in the report itself, one where it appears as a .1psi shift in pressure, another where it is closer to a .2psi shift,  and another where it appears as a .5psi decrease in pressure of a wet ball.   Which is it?  Or, on the actual day in question, was it even more, and did it depend upon how the balls were handled? And why isn’t this mentioned as a uncertainty in the actual results?  The scientists obviously noticed it, but did not explain it or try to control for it. Some of the Wells Report scientists would do well to consult a psychrometric chart, and see how dramatic those effects really are, and actually use them to predict the warming and cooling patterns for the wet and dry footballs.  In actual fact, the wet balls should have barely changed in pressure when brought back into the locker room.  At a temperature of 72F, and humidity measuring 20%, the wet-bulb temperature is 50F.  If the balls were at 48F (which they were not, because of other factors), then only a 2 degree shift (about .1psi) would have occurred.

Patriots Exonerated

The Physics done by the Wells Report investigators, combined with the missed fact of a cold field on which the Patriots balls sat during the game, uncertainty about wet and dry balls, and questionable statistics removes most any doubt of the innocence of the Patriots.  We no longer have to accept the bizarre and unlikely scenario of a locker room attendant casually entering the bathroom, deflating random amounts of air from some but not all of the balls, and emerging 90 seconds later as if he had just been to the bathroom for the usual reasons.  If in fact he did remove some air, it was already small without the consideration of the cold field, and with that consideration it must have been negligible, or perhaps he even added some air.

As Physicists we have done our jobs and explained the facts of the situation.  The weather can be held responsible for the ball pressures.  We can no longer accept the label that it is “probable” the Patriots cheated “unless some other unforeseen conditions are found” as given by the Wells Report.  A least a couple of unforeseen conditions for the Wells Report has been found, and there is nothing more to be said.

Beyond the Physics

From here it should be simple.  However, it is not.  People have moved beyond the consideration of physics. It has long become a struggle between public opinion, an atmosphere of suspicion promoted by the NFL leadership,  people and press who have staked their reputations on the guilt or innocence of Tom Brady, and just a bone weary public who doesn’t want to hear any more.  Despite the truth of the science, what matters now is the perception of truth held by the football commissioner who has placed himself in the uncomfortable position of being judge, jury, and executioner despite his own lack of knowledge of science.  And he has his own personal stake in the outcome.  He has accepted the Wells report as gospel truth and has already acted on it, without protecting himself or the NFL with an independent process for verifying the report, or even appealing its validity.  Even though an apology is called for, he himself would look foolish coming forward with it now.

The Future

There is one other fact about Physics which may ultimately carry the day.  It doesn’t change.  It can and will repeat itself.

It is a problem for the NFL that this scenario (the set of weather conditions — frigid temperatures leading up to a relatively warm 50 degree rainy afternoon) may repeat itself next year at some other field, and for a different set of football teams.  If someone on one of the teams complains about football pressures, and they are measured more carefully at halftime,  will the same punishments be imposed as were done this time — maybe to both teams?

The legacy of this NFL leadership is at stake.  Will they be remembered as the leaders who seriously handicapped the top team in the NFL for the next season?   Will this put an * beside any team that wins the AFC, or perhaps even the Superbowl, to say they won, but they were mistakenly given a leg up by the NFL commissioner to get there?

Or will the NFL admit their mistake, publicly apologize, and quickly right the situation?  And then they have some work to do at the NFL to make sure processes are in place to insure fairness to individuals both on and off the field.  Yes– guilty parties should be dealt with swiftly and severely, but there should be a way to appeal for a TV replay to make sure the correct call was made.


*** The Wells Report Scientists did try to do a complete and thorough job.  That is evident from their report.  However, as is often the case in scientific endeavors, other scientists review the work, and find problems that should be explained or taken into account.  That did not happen here, or at least is only now happening haphazardly through the press.  It was the process that was flawed, not the scientists, and the best that can be said for the Wells Report is that it is based on an inconclusive and incomplete scientific analysis.

If a similar problem occurs in the future, it will be important for the the NFL to develop different processes to avoid jumping to incorrect conclusions.  In this case, this led to the NFL to try to shoehorn this data onto a bizarre theory of deflated footballs, a nefarious locker room attendant, cryptic text messages, and a 243 page confusing explanatory report, rather than letting the facts themselves lead to a more natural theory that fit the facts — i.e. the weather did it.


Some more references:

Three Key Areas Where the DeflateGate Report is Blinding us with Science


#DeflateGate – Physics in Review – The Wells Report


The Wells Report, the NFL’s official report on the DeflateGate scandal, was written to try to end the scientific discussion on the football pressures as measured at halftime of the AFC Championship game between the Colts and the Patriots.

It still causes a lot of confusion.

If one were to give a peer review** of the scientific side of the report (the report was done by the consulting company Exponent), and to suggest areas that might clear up some concerns with further investigations or result in an entirely different conclusion, these are the problem areas:

  1. Artificial Turf Field Temperature — Surprisingly, this was not considered in the Wells Report at all. Under the artificial grass would have been cold sand and cold ground, frozen by the cold January weather leading up to the extraordinarily warm (for January) 50oF temperature at game time.  The turf fields are designed to bring cooling from underneath to the surface.  This would have cooled balls sitting on the grass — in particular the Patriots balls as they had a long sustained drive occurring within 20 minutes of the halftime measurements.  This may explain a wide distribution in pressures of the Patriots balls, and the overall lower average pressure.  Consideration of this may totally exonerate the Patriots.  The investigators must look into this.
  2. Uncertainty — A great deal of the report deals with the uncertainty in the pressure measurements themselves, which gauges were used, the temperature of the locker room, and the timing of those measurements.  In the midst of all this uncertainty, the conclusion reached was based on the Patriots average pressure having an uncertainty that was just outside the range of their model, and timing of measurement that was just outside their model as well.  The closeness with which these measurements were to the uncertainty range, not only requires accurate accounting of the averages, but accurate accounting of the uncertainty as well.  This is true of the measurements by the gauges, and the  uncertainty of how well the model and apparatus the investigators put together actually matched the game time conditions.   Regarding the model, there is not enough discussion about this, nor any experimental error bars that show on their charts.  The control on this was the measurement of the Colt’s balls.  Unfortunately, only 4 of the Colt’s balls were measured, and we don’t know if those balls were used in the game and were wet, or if they were not used in the game and were dry.  We don’t know if the ball boys held them in their hands and warmed them, or what happened to them.  These are difficult to consider as a proper control.  The investigators should present their results as an investigation of how their game time simulation with its proper experimental uncertainty, and the Ideal gas law match the results of both sets of balls, rather than try to adjust by making the Colt’s balls a control group.
  3. Evaporative Cooling and Uncertainty — A systemic difference between the pressure on a wet ball, and the pressure on a dry ball can be noted in the Investigators results.  Although it is not explained in the report, this might be expected to reflect the “dry-bulb”, and the “wet-bulb” temperatures found in meteorology.  The root of this is really evaporative cooling.  This is a “systemic” result, and not an experimental uncertainty error as is suggested in the charts shown by the investigators.  The charts should have reflected an uncertainty in starting temperature, and any other uncertainty that might occur in their model which might be observed when repeating the experiment, or trying out various other conditions as they describe.  Since the uncertainty is critical in coming to the final conclusion of this report, this should be dealt with correctly.  Even this change in uncertainty modeling — because the Patriots average pressure was just outside the range shown — might lead to a different conclusion.



There appear to be three competing theories to describe the football pressures as measured at halftime of the AFC Championship game between the Colts and the Patriots:

  1. Someone let air out of some of the Patriots footballs.
  2. There is so much uncertainty surrounding the measurements that no definite conclusion can be reached.
  3. There are physical processes not accounted for in the Wells Report which caused the temperatures of some of the balls to be lowered to the point to which measured pressures were achievable.

The Wells Report concluded that theory #1 is the most “probable” explanation of what occurred.  They came to that conclusion by accounting for all known sources of uncertainty, and by doing a comprehensive review of any and all physical processes which might have accounted for changes in the ball pressures.  They created an apparatus to model the “game day scenario” to demonstrate what might have happened at Foxboro that day especially taking into account transient effects due to changes in temperatures when the ball pressures were measured.  Based on the experimental results of running that model they reached their conclusion.  The data of the Patriots football measurements do not fit their model and the Colts football measurements do, so therefore the Patriots explanation is “not credible”.  Someone deflated the balls.

The Patriots contend that theory #2 is the best explanation.  The uncertain conditions under which the pressure measurements were taken, the inaccuracy of the equipment, the uncertainty in the timing and the continued claim by everyone interviewed in the case who might be culpable of or colluding with others to let air out of the balls that no such act happened, lead them to believe that this is the best explanation.  Their rebuttal of the Wells Report, including many of the non-scientific aspects of the report is published on-line here.  (Wells Report Context — From the Patriots)

Theory #3 is something that is mentioned a number of times in the Wells Report in the form of the statement:

“Therefore, subject to the discovery of an as yet unidentified and unexamined factor, the measurements recorded for the Patriots footballs on Game Day do not appear to be completely explainable based on natural causes alone. ( pg.61, Exponent Report)

This is the suggestion that some new unknown factor might cause their theory to be wrong. That unexamined factor may be the cold sand and ground under the turf field.  We start with a discussion of that factor.

Artificial Turf Temperature

The Wells report claims that the air temperature alone is insufficient to have caused the low pressures measured in the balls.  But what about the playing surface?  This concern about the temperature in and under the artificial turf falls in line with theory #3 — other physical processes not considered.

On January 18, 2015 the ground in New England was frozen from a cold January, averaging well below freezing. Despite warm temperatures that day, the turf field is expected to have been cold. A ball sitting on the wet artificial grass will cool down due to the contact. (

The cooling due to the ball sitting at the line of scrimmage in between each play, and the fact that the Patriots had a sustained drive just before half-time can explain the wide variety of pressures in the Patriots balls, and the difference between the Colts balls and the Patriots as measured at halftime.

At that time in January there would have been a block of frozen ground underneath Gillette Stadium’s field, and turf fields are designed to allow cooling from below.  In fact the turf is designed to have a layer of “high heat capacity” sand which would keep cooler on high temperature days and the grass fibers bring that coolness through a layer of insulating rubber pellets.  While in play, the balls would sit on the ball field for extended periods of time, especially the Patriots balls during a sustained drive leading up to halftime.  These balls would have experienced some of this cooling, and would have still been feeling those effects during the halftime measurements.  Was this just overlooked by the investigators, or did they know about it, and had a good reason for ignoring it?

The report should consider the playing field, and investigate the effects if any.

A separate report found here goes into more detail how this might account for all the low pressures of the balls at halftime, and even the wide variance in distribution of pressure in the Patriots balls that was found.


The Gauge:  Which one was Used?

Walt Anderson, the Referee who measured the pressure in the footballs before the game had two pressure gauges.  Subsequent tests by Exponent showed that the gauges measure about .3psi to .45psi apart.  The controversy is over which gauge was used in that first pressure measurement.   The choice is crucial.  As is pointed out by the Patriots, if we just use measurements using the gauge Walt Anderson recalled he used during the initial measurement, then the average pressure measured in the 11 footballs measured at halftime is consistent with the pressure expected by the Ideal Gas Law.

The Wells report shows this in Fig 27 of their report.  This illustrates a model which the Wells Report claims shows the time dependence of where the Ideal Gas Law for a football sitting out in the open air of the locker room warming up while being measured.  At the left side of the chart, within the experimental uncertainty of the football measurements, the Patriots balls convincingly overlap with the value expected by the model!


The Wells report contends that time dependence is the problem.  The balls took 3-8 minutes to be measured, and during that time they should have been warming up.   As shown on the chart (look at the brown curves), the flat brown line with uncertainty bars around it  shown in light brown, after 4 minutes starts to lie outside the wet-ball curve (the dashed brown line), and by the time we reach 8 minutes is almost 1.5 times the size of the uncertainty bars outside the curve.

If we use the other gauge, then the average pressure was significantly lower, and must be explained some other way.  Since the investigators fail to convincingly remove the uncertainty surrounding the gauges their conclusions should only be based on ruling out measurements made by both gauges.


Experimental Uncertainty

Key to the conclusion of the report is understanding the uncertainty surrounding the pressure measurements of the balls, and in particular determining if that uncertainty can be reduced to a level that any conclusion at all can be reached.  The investigators reached the conclusion that it was “probable” that someone had removed air from the balls.

Unfortunately a game day scenario for football manufactured in a lab would seem to have a lot of uncertainty about it no matter how carefully scientists try to reproduce, or account for the weather and other game conditions.  To show the only uncertainty in the experimental game day model as being due to dry/wet ball measurements, and then base conclusions as far reaching as they do is questionable.

Why aren’t there shaded uncertainty areas around the pressure lines depicted in graphs in Fig’s 26 and 27, on which so much of the conclusion of this report rests?   At a minimum we know there is an uncertainty in the initial measurement temperature that should be represented directly on these charts.  If that uncertainty is .1psi – .2 psi, it  can change how people view this chart considerably.  And certainly that uncertainty in the temperature measurement should not be used or manipulated  as suggested in the statement “the various temperatures were adjusted such that the measurements obtained via these simulations correspond to the Colts measurements” (pg. 57 of Exponent report).  Please, just put the correct uncertainty estimates in, and let the chips fall where they may with the data.  It is important to see that the Colts data does not align perfectly with the models prediction if that is the case.

It is also important to note that the difference between the wet-ball curve, and the dry-ball curve could be a systemic difference (see Evaporative Cooling below), and thus do not and should not represent experimental error limits for these curves!  The investigators should be able to control for these systemic effects, or suggest that some uncertainty exists because we do not know the humidity and/or air circulation that existed within the officials locker room, and representing some bound on that type of uncertainty. On some of the diagrams in the report, the wet-dry difference is about .1 – .2psi.  On other charts, like the one above, it appears to be about .5psi.

Directly related to those pressure lines is the Fig 21 graph (not shown), where the rising pressure for the dry balls does not seem to follow an exponential curve.  This suggests some other systemic problem, or at least should be explained by the investigators as to why it may be that way.

Although the investigators do a great job reducing and explaining much of the uncertainty surrounding pressure gauges, squashing footballs, stretching football leather, etc, they need to provide more accounting for uncertainty in their own model on which they base their conclusions.

Finally, it is not unusual for results to lie outside their uncertainty estimate.  With a “normal” distribution approximately one third of the time this will occur.  It is just more “probable” that by repeating the experiment it will lie within the estimated uncertainty.  Given how close this range is as shown in Fig. 27 above, the only conclusion that could be reached is that it is only “probable”, but not impossible that that the pressures in the Patriots balls could not be explained within the model.  Further experimentation including the cold turf, and showing some uncertainty due to our not knowing the humidity/evaporative cooling effects

Using the Colts Balls as Controls

A “control group” in an experiment is a set of items (in this case the Colts footballs) which are statistically similar to the the other group (the Patriot footballs) but not subjected to the changes which are being tested for.   Unfortunately, this was only a subset sample of the Colts balls.  These 4 balls may have all been dry.  We needed to add in all the balls to be sure we included balls that had been on the field getting cold and wet. Without that, we may be comparing apples with oranges.   We know there may be at least some known systemic differences in balls which were wet and dry (see Evaporative Cooling below, and not to mention issues due to their use or non-use in a game as asked in the Artificial Turf question above)?  Why not just compare everything to the measurements predicted by the Ideal Gas Law, and the Transient model.

The Distribution in the Pressure of the Patriots Footballs

The distribution of the pressure measurement in the Patriots balls is an inconvenient truth for both sides.  They are spread too broadly.  This is a glaringly obvious problem if we assume it is just the Ideal Gas Law that took the balls from their narrow distribution before the game (12.5psi +/- .1psi) to the measurements which at half time which are spread over 1.3psi.  The Exponent investigators have shown that in the lab the temperature/pressure changes are very repeatable, and follow the Ideal Gas Law.  We should not see such a wide distribution.

Even theory #1 — A person letting air out would have been greatly inconsistent in how they let air out — ignoring some balls, and letting much more out of others.  Would a random set of pressures be good for a quarterback?  Does that make sense?

And theory #2 — While averaging over the halftime pressures in these balls, and obtaining an estimated uncertainty seems reasonable, the nagging question is why did the pressures show such a large variance.  It does not seem physically reasonable.

However, theory #3 — this theory makes sense.  Very simply, the cold artificial turf theory would explain  it as due to some balls being more recently on the field than others.

Systemic Effects: Evaporative Cooling

At least one blog post showed concern about evaporative cooling causing some significant effects on the ball pressure.  In one of the figures (Fig. 27 shown above), some experimental results show two curves, one for a dry-ball curve, and one for a wet-ball curve.  They differ along this curve by about .5psi.  Although the reason for this difference is unexplained in the report, some atmospheric scientists might guess that this is the kind of difference that is accounted for by “dry-bulb” and “wet-bulb” types of measurements.  These differences arise because of evaporative cooling lowering the temperature on the wet-bulb (or wet-ball), and thereby lowering the pressure.  These effects can be larger depending upon the humidity in the air, and air circulation.

An explanation of these clearly measurable effects, whether they are due to the evaporative cooling or not, is warranted. Particularly since this difference is represented in the graphs Fig 27 and Fig 28, on which the conclusions of the report were founded.


The comments in this paper are meant as a “peer review” of the scientific evidence and experimental procedure presented in the Wells Report, in particular where that may influence the conclusion of the report.

These are some important questions that were raised in a careful reading of the scientific evidence of the report.

The expectation and hope is that the Investigators would look seriously at the effects mentioned here, and answer the questions raised.

Suggestion for the Future

Unfortunately, this experience with the Patriots-Colts game is the first in which the pressure inside the football has been called into question.  Undoubtedly, in the future there are some things that could be done better.  There could be better recording of pressure.  There could be more careful monitoring of the game balls.

In the future, to verify that the balls have not been tampered with, the pressures could be measured ahead of the game as they are now, and then measured again about an hour after the game to see if they come back to their original pressure.


**Peer Review

The scientific norm of peer review before publishing a scientific article is a time honored process which leads to fewer mistakes, a focus on critical aspects of the investigation, and wider acceptance of the results.  In this process, prior to publication, the article is sent to a handful of independent reviewers who may or may not be known to the lead investigator.  These reviewers who have general knowledge of the subject, study the report, and their comments are sent back to the investigator giving that person a chance to improve their publication, or even go back to the drawing board.

In the Wells Report, it appears there was not an outside review process.  Dr. Daniel Marlow was the lead investigator, and is clearly an esteemed physicist, but as any scientist knows, the scientific method can be fraught with making bad assumptions, failing to discover or notice key facts, and a lot of trial and error.  Discovery quite often comes through recognizing these mistakes, pressing forward, and finding the correct answers.

This, in and of itself, does not mean that mistakes were made in the investigation.  And in fact, for the topics that were considered, most scientists agree that the report by Exponent was capably done.  However by failing to have this peer review, the group did not avail itself of either a chance that others might identify problems with the report ahead of time, nor avail themselves of the wider acceptance of the results.

***Improving the Game Day Scenario Experiment

A way to further the realism of the “Game Day Scenario” is to include the football game itself.  Create a simulated field by laying some damp towels on a layer of ice and let it sit outside at 48F for an hour.  Then repeat the above experiment, but instead of dampening the balls with 48F water, periodically bring them out and push and roll them on the damp towels, and let them sit on the towels for 2-3 minutes.  Then put them back in the bag.   In the last 25 minutes — up to 3 minutes before bringing them inside for measurement, repeat the push and roll on the towels, always having at least one ball out sitting on the towels.



Official Wells Report

Wells Report Context ( From the Patriots)

MacKinnon’s Scientific Conclusion

Evaporative Cooling a Glaring Omission from Wells Report

A New Form of Peer Review

DeflateGate and the Temperature of the Playing Field

Question:    Can a cold artificial turf field explain the lower pressure in the Patriots footballs?
Answer:    It was January in New England.  The cooling from the turf due to the ball sitting at the line of scrimmage in between each play, and the fact that the Patriots had a sustained drive just before half-time totally explains the wide variety of pressures in the Patriots balls, and the difference between the Colts balls and the Patriots as measured at halftime.


Artificial turf fields can get very hot on sunny summer days. Field temperatures of 150oF to 200oF have been reported. To compensate, manufacturers put in high heat capacity sand to retain cool temperatures, and utilize the grass fibers to conduct cooling from that sand to the surface.

But what happened on the cold cloudy day of January 18th, 2015, when the Colts played the Patriots in the AFC Championship game that is now known as the DeflateGate game. It was not a hot day, and the sun was not shining.  But it was a hot day for January! The air temperature rose dramatically from a night time temperature of 20oF to 50oF, with most of that rise occurring in the last 7 hours before game time. In addition, the temperatures of the prior week had averaged less than 25oF, with the day before the game falling close to single digits. Indeed, all of January up to that point had only two or three days that rose above the freezing point.

That same high heat capacity sand which attempts to keep the field cooler on hot summer days, would have been cold as well, and would have kept the field cooler than the ambient air temperature on this warmer than normal winter day. And the ground underneath would have been frozen.

So during the Patriots – Colts game on January 18, 2015, the temperature of the turf field is expected to have been colder than the ambient air temperature.

The colder temperature would have an impact on the balls that came in contact with the field, lowering the temperature and pressure of the balls.

Using reasonable assumptions, the distribution in the pressures and the lower absolute pressure of the Patriots footballs measured at halftime is consistent with a lower field temperature.

Inclusion of this effect removes the requirement of a person removing air from the Patriots footballs.  The lower pressures and the scatter in the pressures can be  explained simply by the weather, and the play on the field.

This effect was not discussed in the NFL Official Wells Report .


Before the game 13 Patriots footballs were measured by a referee to have a pressure of 12.5psi at a temperature of 70oF.

They were put into a ball bag, and carried out to the ball field with an air temperature of 50oF which dropped to 48oF by halftime.

Using the Ideal Gas Law, and knowing that a dry football in open air exponentially will go from a warmer temperature to a colder temperature with a time constant of 15 minutes  (see Appendix A, and the Wells report), we expect the footballs exponentially approached 11.4psi, dropping  63% in the first 15 minutes, and were 98% of the way there in 1 hour.  [Note: this assumes there is no ball bag effect – i.e. balls in a ball bag might take longer to begin changing in temperature.   The bag and the air inside the bag may insulate the balls from the outside air].

The ball game started approximately 2 hours after the balls were brought to the field.

After the start of the game, with the teams on the field, a ball for the team on offense was placed on the field at the line of scrimmage in between plays.  These balls were regularly exchanged as needed by ball boys in between some plays.

The air was damp, misting, and rain was expected in a couple of hours.  The temperature had risen from 30oF to 50oF in the last 6 hours.  It was 6pm.  The temperature for the last week had been in the teens and 20’s.

Artificial Turf

So what temperature was the field?  The field is artificial turf.  An ad for it shows a side on view of small rubber pellets mixed with sand, with most of the sand sitting under the pellets, with grass like plastic feathers coming up from under.

FieldTurf is the brand of Artificial Turf used at Gillette Stadium


Undoubtedly, under the turf field was frozen earth, probably matching the average temperature of the last week, maybe 25oF or below (see Appendix B below).  The grass blades would be damp from the mist.  They are not frozen – the composite substance provides some level of thermal insulation.  In the literature the artificial turf is described as being of high heat capacity, with a heavy total system weight providing that heat capacity.  Physically, that heat capacity is probably provided by the sand.  We might expect the sand is still in a sub-freezing state, since it has been cloudy.  The rubber particles provide the springy earth like feeling and are probably insulating.  The blades of artificial grass pass up through the sand and rubber pellets.  They are designed to bring some of the cooling power of the sand to the surface, keeping it cool on high temperature days.

So what is the temperature of the field?  It is probably a gradation of temperatures, starting near the temperature of the surrounding air near the top of the grass, but quickly falling to match the temperatures of the substrate underneath.

Water collected on the grass, and would have been cooled by the presence of the cooled substrate below.

Painting the Football with Cold Dampness

Would the damp grass have painted cold water on the ball as the ball hit the ground or the center spun it into position?  Did contact with the ground bring the temperature down faster than the exponential law for the open air?  It seems easy to answer yes to these questions.  We just don’t know what the temperature was, because we do not have a temperature analysis of the field, or that of a football sitting on a cold, damp artificial turf field.

Is it possible the field had completely warmed up in the surprisingly warm( 50oF) afternoon of an otherwise frigid January in Boston?  I could not find a detailed analysis of modern artificial turf thermal characteristics in the literature that might give that answer.   There are descriptions of early turf fields being unbearably hot in the sun.  The modern fields talk about their high heat capacity which keeps them cool.  That is about as much as I could find.  But given the timing of the atmospheric temperature changes that day, it may be enough to assume (given the literature) that the turf is colder than the air temperature, and its high heat capacity is what keeps it cool, and the artificial grass which transmits that coolness to the surface.

The Final Patriots Drive of the First Half

The first half was played over a time period of about 2 hours.  The first half wound up with a sustained drive by the home team, with multiple balls used as they got replaced by the ball boys.  Every ball that was used became wet with the cold dampness of the field.  They sat on the field at the line of scrimmage in between each play.  The center leaned on them and spun them in the grass.

Each time a ball gets replaced, it gets brought back to the bag, where it starts to warm back up to the air temperature of 48oF.  The time constant for warming of a dry ball is 15 minutes like before, but experimental measurements done later (see an analysis of Fig 21 of the Wells report below) show the time constant of a wet ball is more like 20 minutes — probably due to the layer of water on its surface needing to heat up as well.  And those first 20-25 minutes only bring it to 63% of the way to the ambient temperature in the ball bag.

At the end of the half, all the balls were put in the bag, and are brought back inside to a 70oF locker room.  The ball bags were then unzipped, and the pressure was measured in all the balls within 3-8 minutes.

The Pressures in the Balls

So how do we model the pressure in the balls?

Shouldn’t we expect to see a distribution of ball pressures?  Shouldn’t we find that the ball most recently used in the game will have the lowest pressure, a set of balls which have had more time to warm up, but are still slightly damp having a range of pressures, and winding up with a set of balls that are still dry because they were not used at the pressure suggested by the Ideal Gas Law as applied to the ambient air temperature?

Here is the distribution of Patriots football pressures reported in the Wells report represented graphically:


Figure 1. Pressure in the Patriot footballs, Gauge 1.   This figure was created using the pressure gauge the Referee recalled as the one he used for the initial measurement prior to the game.


Figure 2. Pressure in the Partriot footballs, Gauge 2.

This top figure was created using the ball measurements using the Pressure gauge Referee Walt Anderson recalled was the one he used.  Since he had two gauges and there was some uncertainty in which gauge was used, the measurements are shown for both gauges.  The Wells report noted a .3-.45psi difference in the measurements of the balls by the different gauges.   The uncertainty in the actual measurement (using the same gauge) seems to be between .1psi and .2 psi.

Based on this distribution, we might expect that the 3 balls at the high end of the range were dry, and not used in the game.  The balls used in the game were slightly damp, and would be at the lower end of the range.  Additionally, the Wells report displayed an empirical difference of .1psi to .2psi difference between balls that were damp, and those that remained dry as they settled to the lower pressure (see Appendix A.)

What was the Field Temperature?

Predicting the absolute pressures require that we know the effective temperature that the grass will reduce the football down to (we don’t have this data, so here is a guess).  Remember that the ball sits on the grass at the line of scrimmage in between plays which sometimes can be several minutes.  The direct contact will allow for quickly reaching equilibrium faster than might be expected if it is just air.  If we assume a temperature of between 25oF and 30oF for the substrate, is it reasonable to assign a value of 38oF to the football that rests on the grass for some time?  This is 10oF lower than the ambient temperature.

An Explanation that Works!

A ball temperature 10oF lower than the ambient air temperature would bring the expected pressure to 10.75psi (see Appendix A).  This is well below all the pressures measured with the first gauge, and all but two of the balls with the other gauge.  From there, the measured pressure is entirely dependent upon the amount of warm-up time before the half-time measurement.  The amount of time, which would vary from 5-20 minutes based on the play of the game and the amount of time exposed to the locker room air would have created a wide variance in pressures, at values less than what would be expected from just the ambient air. This explains the distribution of pressures we see in the Patriots balls, and their lower pressures as well.

Four of the Colts balls were measured as well.  However, we don’t have measurements for all their balls, and we don’t know anything about whether those 4 balls were used in game conditions or not.  The 4 balls all measured at the high end of the range, to the right, and lie within measurement uncertainty at the same value.  If the above theory is correct, we might find that those balls had not been used in the game, or used very early in the game.  In any case, without having all the balls measured, we find that given the timing of the measurements, they are consistent with having been dry balls or relatively unused balls, and directly follow the conclusions of the Wells report.


There is strong reason to believe that the turf field was at a lower temperature than the surrounding air.  If so, we would expect to see a distribution of ball pressures in the Patriots ball bag at halftime due to the Patriots sustained drive leading up to halftime.  In fact, a distribution of pressures was measured at half-time. With reasonable assumptions, the actual ball pressures are consistent with the Patriot ball pressures measured in the locker room at the beginning of the game.  However, an investigation modelling this scenario must be done before making a final determination.

In addition, with limited information available, and the fact that the Colts balls were not on the field in a way similar to the Patriots drive leading up to halftime, the Colts ball pressures measured in the locker room are consistent with the pressures measured at halftime by the four Colts balls, as has already been determined in the Wells Report.

Appendix A

Analysis of Figure 21 of the Wells Report.

This graph is the result of experimentally modelling the changes in temperature experienced by balls, done in a controlled environment by the Wells Report scientists.  This is their game day scenario.  (This curve does not take into account the effect of a cold artificial turf field.)

The Wells Report game day scenario depicts the game ball pressures expected from exposure to the ambient air temperatures as the game progressed. This does not take into account sitting on the artificial turf for extended periods of time leading up to half-time at 240 minutes.


The brown curves show the Patriots balls starting at 12.5psig. The blue curves show the Colts balls starting at 13.0psig.  The pressure drop at 120 minutes (balls brought to the field) appears to be an exponential curve (see Appendix C), and doing a rough fit, has a time constant of 15 minutes (i.e. it drops 63% in about 15 minutes).  The pressure rise at 240 minutes (halftime) does not as closely match an exponential curve, but with another rough fit, using the outer parts of the curve, the wet ball curves have a time constant of 20 minutes.  This extra time is significant because of the timing of the last Patriots drive, and subsequent measuring of the ball pressures at halftime.

Extrapolating to the Presence of a Cold Artificial Turf Field

So where would a ball recently used on the field be on this chart?  Suppose the pressure was reduced by .65psi by the field (effectively a 10oF temperature drop for .5psi, and then an additional .15psi because of the wet ball effect as shown in this chart).   It should be starting out 5 to 10 minutes before halftime at the 10.75psig point (well off the bottom of the chart).

If we had a sequence of such balls, starting at the 220th minute, and rising towards  the 11.25-11.4 psi pressure, we would find a range of pressures when the ball bag is finally opened, and the pressures start to be measured in the Officials Locker room.

Appendix B

January Boston area temperatures (from The Weather Underground —

Weather chart for the month of January, 2015


Figure 3. Weather chart showing the temperatures in the Boston area during the month of January, 2015.  From Weather Underground (

Expanded view of the weather chart including the day of January 18. Note the sharp rise in temperature during the day of Jan 18.


Figure 4. Weather chart, expanding the section from Figure 3, to highlight game day on January 18th.  Note the dramatic rise in temperature (the red line) shortly before noon on that day.  The temperature leading up to that time averaged near 25oF.  The ground underneath the Artificial Turf, and the sand within the turf are expected to have been near that average temperature.

Appendix C

Explaining the exponential behavior of temperature changes.

Most objects warm up or cool down by exponentially approaching a final temperature from a starting temperature.   This is because the rate of temperature change is proportional to the difference in temperature.  I can go through the physics of this elsewhere, but it does involve calculus, so we will leave it out of this paper.   This is the resulting equation below for the simple ideal case, but it is written in terms of Pressure because in this case of no leaks and constant volume Pressure is directly proportional to Temperature via the Ideal Gas Law

P(t) = P1 – (P1-P0) * exp( -t/t0)

In words, this says that the Pressure as a function of time equals the final pressure P1 minus the difference between the two pressures P1 and initial pressure P0 times an exponential function which is decreasing in time, with a special timing constant t0.

The constant t0 is important.  If t0 is 15 minutes, that will mean that the pressures will be roughly 63% towards equilibrium in 15 minutes, after 30 minutes will be 87% towards equilibrium, and after 1 hour will be 98% towards equilibrium.That means balls on the field can be expected to maintain a significant Pressure differences resulting from the field even after 15 minutes.

Appendix D

An alternative explanation to the data was reached by the Wells report.  

The Wells Report reached the conclusion that a person must have released some air from the Patriots footballs sometime between the initial measurement and the halftime measurement.

The motive for this was to allow the Quarterback better handling of the ball.

By examining the distribution of pressures we find that this person must have let the air out of some balls and not others, and then relatively random amounts were removed, leaving the Quarterback with a wide distribution of pressures in the balls.

In their report, the uncertainty in the pressure measurements put the average pressure in the Patriots balls just outside the predicted values, close enough, so the verdict from the Wells report scientists was only “probably” this happened.

Inclusion of another source of cold temperatures (i.e. the Artificial Turf) should cause this model to be re-examined, and may change that “probably” rating.