Tail-heavy aircraft, tail-heavy normal distributions

Air Midwest Flight 5481 from Charlotte/Douglas International Airport to Greenville-Spartanburg International Airport is a short (30 minutes) commuter flight, operated by a very small turbo-prop Beechcraft 1900D aircraft. On the day of the accident in January 2003, the plane was filled to capacity (19 passengers and 2 crew members). Few seconds after takeoff, just after landing gear was up, the plane's nose pitched extremely steeply upwards and the pilots were not able to counter it as they found they didn't have enough elevator movement to force the nose down. The plane stalled, lost lift, and dropped to the ground like a rock, killing everyone on board.

NTSB investigation concluded that the crash was caused by a conjunction of two unrelated problems. By itself, neither would have led to the accident, but unfortunately on that day they paired up. First, a flawed maintenance repair of the plane's elevator left it with compromised down movement, which meant the pilots were not able to stop the upwards pitching of the aircraft. Second, the plane was too heavy, which is what caused it to pitch up in the first place. The maximum load of the plane was determined by the airline based on averages: 175 pounds per passenger and 20 pounds per piece of luggage. Based on those directives, the crew's calculations showed the plane to be within limits to take off. The problem was that those averages were determined based on surveys conducted in 1936. When NTSB investigators painstakingly calculated the actual weight of passengers and bags aboard Midwest 5481, it turned out that unbeknownst to the crew the plane was overloaded by 580 pounds.

Of course, using weight averages from 1936 when everyone and their mother knows people have consistently been getting heavier is outrageous negligence on part of the airlines. Due to the finding of the investigation into Midwest 5481 crash, NTSB started lobbying FAA to force airlines to use actual weights for determining whether or not planes can take off. FAA ordered the airlines to do a new survey and update their averages. The results were quite shocking: the average weight of an airline passenger A.D. 2004 turned out to be 195 pounds. (As a side note: this is interesting in itself. The average weight of a male adult in the U.S. is 191 pounds; of a female adult--164 pounds. Why is the flying public so much heavier than the general public?) At any rate, Air Midwest updated their average passenger weight to 200 pounds, which meant that the capacity of a Beechcraft 1900D had to be adjusted downwards to 17 passengers. NTSB investigators felt this was not enough, and are still advocating for regulation requiring to use actual weights.

Which finally brings me to the main point of this post: does it make sense to use acutal weights instead of averages? In other words, is the increase in safety worth the increase in costs? The answer of course depends on two things:

1) How much would it cost to use actual weights; and,

2) What is the probability of overloading a plane if only averages are used?

Knowing precisely nothing about how the airline industry is run, i can't even begin to answer (1). Would using actual weights require installing scales and actually weighing each passenger or not? If so, how much would that cost? Would it mean ticket prices would have to go up and if so, by how much? Etc-these are all questions I can't answer. So I'll just answer (2), as that one is just a bit of trivial probability. But in order to do this, we need to know one more thing: standard deviation of weight. I take it to be 40 pounds. (I couldn't find this statistic, so I'm just making it up as I go along at this point.)

So the official passenger weight capacity of a Beech 1900D is 17 x 200 lbs = 3400 lbs. By central limit theorem, total weight of a group of 17 people is a random variable that follows a normal distribution with mean 3400 lbs and standard deviation of 40 x sqrt(17) = 165 lbs. Now we need to make an additional assumption: how much over 3400 pounds do we have to go to consider the plane to be potentially dangerously overloaded? Surely 10, or even 50 pounds over the limit will not make a whole lot of difference. Let's assume then that exceeding the prescribed limit by 5% or more is unacceptable. This means that we can't allow total passenger weight to go over 3570 pounds. So the question now is: given that we're not weighing anyone and just using averages, what's the probability that, if 17 pax board a Beech 1900D, their total weight will exceed 3570 pounds?

Calculating the z-score and plugging it into the normal distribution we've just found, we can see that this probability is over 15%. That is way too high; completely and utterly unacceptably high. NTSB investigators are absolutely right: with aircraft so small that they board under 20 pax, averages simply won't do; variance is too high. Something else needs to be worked out.

However, as planes get bigger, central limit theorem does start taking care of us. Suppose we're trying to find out the probability of overloading a 60-seater. Sticking with the assumption that the weight limit needs to be exceeded by 5% to consider the plane too heavy, we get the overload probability of just under 3%. Still too high for my taste. In a 100-seater, however, it's 0.6%, and in a 200-seater it's 0.02%. That's probably acceptable. And the bigger they get, the more acceptable it'll be; plus, the bigger the plane, the higher the costs of the alternative (i.e. of actually weighing passengers).