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Deflated footballs and UOM

posted Feb 13, 2015, 2:54 AM by Frank O'Brien   [ updated Feb 16, 2015, 2:06 PM ]


Deflated footballs caused quite a stir during a semifinal championship game of the American National Football League (NFL). Based on a check by officials during halftime, it’s alleged 11 of 12 footballs were under inflated by 2 psi (pounds per square inch) each. NFL regulations state that balls shall be inflated to a pressure of 12.5 to 13.5 psi. The team at the center of the controversy, the New England Patriots, have said they inflate to 12.5 psi. This was verified as within specification by officials at game start. Whether any air was let out to reduce pressure after game start is an ongoing investigation.

If the NFL is worried about a 2 psi drift in their football pressure, they would be well served to learn from the uncertainty of measurement (UOM) guidance used by test laboratories. Despite what Judge Judy might say, sometimes UOM (pronounced “um”) is an answer.

What follows is a fun analysis to show that the measured 2 psi under pressure can be attributed a lack of control over the sources of uncertainty of measurement (UOM); specifically measurement temperature, and meter accuracy.

Control over UOM

Product safety test laboratories must comply with the following quality management and operational standards, which include UOM requirements:
  • IEC 17025:2005, General requirements for the competence of testing and calibration laboratories; 
  • IEC System of Conformity Assessment Schemes for Electrotechnical Equipment and Components (IECEE) CB test laboratory rules and operational documents, including OD-2005:2014; and 
  • For the USA, A2LA document P102 – A2LA, Policy on Measurement Traceability. 

To support the application and reporting of UOM, product safety test laboratories have:
  • ISO/IEC Guide 98-3:2008, Uncertainty of measurement – Part 3: Guide to the expression of uncertainty in measurement (GUM); and 
  • IEC Guide 115:2007, Application of uncertainty of measurement to conformity assessment activities in the electrotechnical sector (based on GUM:1995, which was reissued as Guide 98-3:2008, with minor corrections); and 
  • IECEE CB Test Laboratory decision sheet, CTL DSH-251B:2009, meter accuracies 

The key concept is the measured value is specified together with a range on either side, within which there’s a confidence factor that the actual value falls. Recall that for a normal (gaussian) distribution of measurements, 95% of the values fall within 2 standard deviations of the average. With UOM, this 95% confidence is specified as a coverage factor, k=2. For example, a voltage measurement might be given as 123.4 Vrms, +/-3%, k=2; or a length measurement might be given as 1.22 m, +/-0.03 m, k=2. The expanded UOM can be specified as a percentage or absolute range.

When calculating a mission to Mars a small deviation in desired boost direction at the start of the journey, could result in a large error in location when arriving at Mars. For this case, the UOM can be specified as needing to be very small. In other cases, such as buying cold cuts at the deli, perhaps a error of 10 or 20 g, when buying 500 g, is acceptable. If the NFL is concerned about a 2 psi drift, they need to specify a maximum UOM (minimum error) consistent with this concern.

Within Guide 115, 2 methods are offered for determining UOM. There’s Type A, statistically calculating this based on a series of measurement observations. Alternatively, there’s Type B, using judgement to estimate the major sources of uncertainty. With product safety, the Type B method is more common, and controls are placed over major contributors to UOM. For example, there’s a tight range for test environment, tight range for power source, meters have a minimum accuracy, test methods are defined, and technicians are experienced.

Considering football pressure measurements, and using a Type B analysis, major sources of uncertainty include: the measuring temperature, meter accuracy, experience of the person performing the measurement, and with analogue meters, the bias of the person reading the meter. The measuring temperature and meter accuracy can play a large role in UOM. If experienced officials are measuring the football pressure, we might assume that the dispersion of measurements will be minimal.

Temperature effects

Thermodynamics tell us that with a drop in temperature we’ll have a drop in pressure. The relationship as specified by the ideal gas law is PV=nRT. For the pre-game temperature T1, where we’ll assume 70 F, 294 K. For the half time temperature T2 when the deflation was detected, based on Accuweather, we’ll use 50 F, 283 K. For the 2 temperature points, the nR/V parts cancel. The relationship becomes:

P2/P1 = T2/T1, where P is pressure in Pa, and T is temperature in K.

Putting in the numbers we see a drop in temperature of 283/294 = 0.962, a decrease of 3.8%. The starting ball pressure P1 of 12.5 psi is a gauge pressure. For absolute pressure we add the atmospheric pressure, which is dependent on altitude and weather, but we’ll use the average at sea level of 14.7 psi. 12.5 + 14.7 psi = 27.2 psi = 187.5 kPa. The pressure at halftime, P2 = 187.5 kPa * 96.2% = 180.4 kPa = 26.2 psi. Taking away the atmospheric pressure we have 26.2 - 14.7 psi = 11.5 psi. This is a change of 1 psig.

Without controlling that the football pressure be measured at a temperature representative of the playing temperature, the UOM can easily be as large as 1 psi, irrespective of how well you control meter accuracy, test method, and experience of measuring staff.

Meter accuracy

I researched pressure gauges for footballs and couldn’t find any that had an accuracy specification. Some gauges are analogue (popup, or circular gauge with needle), while others are digital (numerical display). Illustrated below are those I found on amazon.com.
popupanalogueanother analogue
digitalanother digital
None had an accuracy specification. You can see with the analogue ones that the demarcations are every 0.5 psi. Best case this implies an accuracy of +/-0.5 psi, and for 13 psi, +/-4%. More realistically I suspect their accuracies are closer to +/-1 psi, about +/-10%. The Mikasa digital one seems to display to nearest 0.01 psi. The Tachikara digital one seems to say it measures to nearest 0.05 psi. Best case this implies an accuracy of +/-0.05 psi, and for 13 psi, 0.4%. More realistically I suspect the digital accuracies are closer to +/-0.1 psi, about +/-1%. Display resolution does not convey meter accuracy. Accuracy is a function of the pressure transducer, it’s range, resolution, and implementation within meter. The battery state could affect accuracy. There are many variables for accuracy. It must be specified by the manufacturer. The accuracy needs to include the confidence that the measured values falls within the specified UOM range, which for product safety must be at least 95%, coverage factor, k=2. 

Within the IECEE CB test laboratory operational documents, minimum meter accuracies are specified by DSH-251B:2009. For gas pressures, a minimum accuracy of +/-5%, k=2, is specified.

With the Patriots footballs there were 2 measurements. 1st the Patriots measured the ball pressure for the min required 12.5 psi. There’s a dispersion range given by the meter accuracy. 2nd the Official checking the ball pressure with another meter. Again there’s a dispersion range given by the 2nd meter accuracy. These ranges overlap but they don’t necessarily coincide. There’s a larger dispersion range given by these 2 ranges. If we assume they’re normal distributions, the GUM tells us we can combine them into an expanded UOM = 2 * sqrt( (UOM1/2)^2 + (UOM2/2)^2 ), for coverage factor k=2. The table below shows the expanded UOM and error for various meter accuracies.

Meter1, UOM1, k=2

Meter2, UOM2, k=2

expanded UOM, k=2

measured value, psi

95% confidence max dispersion value, psi

95% confidence min dispersion value, psi

95% confidence, +/-error, psi




































If we assume the meters used by the Patriots and the NFL officials each have an accuracy of +/-5%, (the middle, yellow row), which would be acceptable for IECEE measurements, we have a possible error of almost 1 psi.

There’s also the need for ongoing meter calibration. Product safety labs calibrate electrical meters every 1 year, in order to verify that meters have maintained their accuracy. Calibration meters have UOM accuracies of at least 4x better than the meter being calibrated. One can see with the 5%, 1% meters row, that the calibration meter's UOM has only a minor contribution to the meter being calibrated's overall UOM.


If the NFL measured 10.5 psi, which is 2 psi less than regulation, 1 psi can be attributed to the fall in temperature, and perhaps another 1 psi to meter accuracy. Both of these UOM factors could be minimized to desired levels, with specifications.
The Patriots went on the win the semi-final game with the disputed footballs. With the win they earned a trip to the Super Bowl, the NFL championship finals game. Again they won. Congratulations to the Patriots, the hometown team of OBCM. The Super Bowl is more than a showcase for (American) football. It’s become well known as a showcase for the best in advertising commercials. This year the playoffs provided an opportunity to learn a bit about UOM.

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