## Uncertainty in trading passengers for fuel

I had an interesting experience recently while preparing for a flight from Los Angeles to Baltimore. It was a completely full flight– initially, at least– with myself and 174 other passengers who had already boarded the Southwest 737-800, seemingly ready to push back and get on our way.

However, after a delay of several minutes, a flight attendant came on the PA and asked for two– specifically two– volunteers to give up their seat, in exchange for a flight later that afternoon. Two people immediately jumped up, left the airplane, and then we were ready to go… now with two empty seats.

The problem was weight: due to a changing forecast of bad weather, both in Baltimore and en route, we had taken on additional fuel at the last minute (e.g., to allow for diverting to a possibly now-more-distant alternate airport), resulting in the airplane exceeding its maximum takeoff weight. Something had to go, and apparently two passengers and their carry-on bags were a sufficient reduction in weight to allow us to take off.

What I found interesting about this episode was the relative precision of the change– 175 (or even 174) passengers bad, 173 passengers good– compared with the uncertainty in the total weight of the passengers, personal items, and carry-on bags remaining on board. That is, how does the airline know how much we weigh? Since Southwest does not ask individual passengers for their weight, let alone ask them to step on an actual scale prior to boarding, some method of estimation is required.

The FAA provides guidance on how to do this (see reference below): for large-cabin aircraft, the assumed average weight of an adult passenger, his or her clothing, personal items, and a carry-on bag is 190 pounds, with a standard deviation of 47 pounds. The figure below shows the resulting probability distribution of the total weight of all 175 passengers on the initially completely full flight:

Distribution of total weight of 175 passengers on a Southwest Boeing 737-800.

It’s worth noting that the referenced Advisory Circular does provide a more detailed breakdown of assumed average passenger weight, to account for season of travel (5 more pounds of clothing in the winter), gender, children vs. adults, and “nonstandard weight groups” such as sports teams, etc. But for this summer flight, with a relatively even split of male and female passengers, the only simplifying assumption in the above figure is no kids.

The point is that this seems like a significant amount of uncertainty in the actual total weight of the airplane, for less than 400 pounds to be the difference between “Nope, we’re overweight” and “Okay, we’re safe to take off.”

Reference:

• Federal Aviation Administration Advisory Circular AC-120-27E, “Aircraft Weight and Balance Control,” 10 June 2005 [PDF]
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### 4 Responses to Uncertainty in trading passengers for fuel

1. This reminds me of a great stand-up bit about airplane weight distribution:
Brent Pella – Why You Shouldn’t Fly on Spirit Airlines

• I just heard Spirit take a beating by another comedian recently as well, but I don’t forget who. Something about the flight attendant asking over the PA, “Is there a doctor on board?” And thinking, “Not on *Spirit* there’s not.”

2. Sic says:

The small change in distribution can actually have a big effect on the probabilities in the tails.

For example, consider the P(Passenger total weight > 35000) in both distributions. Assuming I did not make a mistake, it is roughly .25% for the 175 passenger plane and only .03% for the 173 passenger plane.

Likewise for P(Weight>36000), we have roughly 1 in 200,000 for the 175 passenger plane and ~1 in 5,000,000 for the 173 passenger plane.

• Right; both of your sets of example calculations are correct. This gets at the heart of the question: what is the decision criteria used by the airline? That is, given n passengers, presumably the decision is of the form, “If axn>b, we’re overweight,” but what are a and b? Is b=35,000 or 36,000 as in your examples? It isn’t even clear whether a=190; see Section 4-217 of the referenced AC, describing “segmented weights,” which “involves adding a portion of the standard deviation to an average weight to increase the confidence that the actual [total] weight will not exceed the average weight.” The appropriate table entry in this case would be a=198, which would suggest a threshold of b=34,400 or so…?

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