# Gender and Balance in Aviation

In which we use math and physics to show that the pilots of my flight from Toronto to Albany this past weekend were full of crap.

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As previously noted, I was in Waterloo, Ontario this past weekend for the Open House at the University of Waterloo’s new Quantum Nano Center. My talk went very well, save for some new-building technical glitches, and video of both the talk and the panel discussion should be posted to their web page soonish– I’ll post a link when it goes up.

The trip back from Toronto was slightly marred by the fact that the gates from which the tiny little prop planes they fly on the Albany-Toronto route are probably the worst of any airport I’ve flown through: cramped, confusing, understaffed, and with less charm than the average bus station. This did, however, provide material for this blog post.

At one point, when I went to the counter to ask when they’d be boarding, the agent at the gate told me they’d be boarding “ladies first,” for reasons having to do with “weight and balance.” That was a little surprising, to say the least, and I was somewhat relieved when they didn’t actually ask us to sort by gender before boarding the plane. When we got to the plane itself, however, the pilots told us to disregard our boarding passes, and had all the women sit toward the back of the plane, and the men sit toward the front. This, again, was explained as something that “helps us balance the aircraft.” This sort of statement obviously triggered my physicist instincts, making me wonder just how much of a difference sorting passengers by gender could possibly make.

So, how could you quantify the effect on “balance” of the gender-split seating arrangement? Well, the tiny little prop plane in question was a Beechcraft 1900, with the seating diagram seen at the top of this post, and also below:

The passenger cabin contains 9 rows of seats, eight of them in pairs, and the ninth with a third seat in the middle. This gives it a capacity of 19 passengers, and the cabin length is 10 meters.

The way to quantify issues of balance is through considering the center of mass of the object. The center of mass is not a physical thing, but is sort of the “average” position of the mass of an object, where we can imagine all the mass of the object being concentrated, when we do the physicist thing of treating it as a spherical featureless point mass. If we’re thinking about trying to balance an object, what we usually think about is making sure that the center of mass is located directly above or below the position where the object is supported.

Mathematically, we find the position of the center of mass of a composite object by taking the position of each individual component, multiplying by the mass of that component, and adding that product for each of the pieces making up the object. We then divide the sum by the total mass, which leaves us with the position of the center of mass. In equation form, it looks like this:

$x_{CM} = \frac{m_1x_1 + m_2 x_2 + ...}{m_1+m_2+...}$

So, if we distribute a bunch of passengers through that seating chart, we can find where their center of mass is, and see how much it moves when we sort the passengers by gender. For the sake of mathematical simplicity, we’ll consider 18 passengers to start, so the whole thing is symmetric– leaving the middle seat in the last row empty.

With a 10-meter cabin containing 9 rows of seats, each row gets 1.1 m of space. We’ll take the origin to be at the position of the middle row (5th from the front), so there are two seats each at 0, ±1.1m, ±2.2m, ±3.3mn, and ±4.4m. If the passengers all had the same average mass, with the third rear seat empty, the center of mass of the passengers as a group would be located right at the zero point, in the fifth row of seats.

Of course, the passengers don’t all have the same mass, which was the point of the pilots sorting us by gender. So, how much of a difference does the mass make? According to Wikipedia (which is plenty good enough for this sort of back-of-the-envelope calculation), the average American man has a mass of 88.6kg, while the average American woman has a mass of 77.2 kg. To account for personal belongings and make the mass a little easier, I’m going to round these to 90kg and 79kg for men and women respectively. We’ll further assume an equal split: 9 men and 9 women on the plane.

The odd number for each gender leaves us with four rows of two men, a row with one man and one woman, and then four rows of two women. These situations are exactly symmetric, and plugging in numbers for the position along the length of the plane, we find that the men as a group have their center of mass at -2.58m, and the women as a group are at +2.58m. We can use those positions and the total masses of each gender to find the position of the center of mass of the passengers as a whole, and we do, indeed, find that it is shifted forward. By 0.168m, or about 6.6 inches.

(Leaving the last seat empty doesn’t make much difference, by the way. If you fill it with a woman (giving 10 women and 9 men), it shifts the center of mass from 17cm forward of the middle seats to about 6cm behind them. Even if you add another man, making it 10-9 M-F, you still fill the last seat with a woman, and the center of mass is still about 6cm behind the middle seats.)

So, how much of an effect would you expect this to have on the plane? I’m not a pilot, but based on the stats on that aircraft site, the total shift is about 1/10th of the width of the wings of the Beechcraft. That’s not a huge difference.

And even that overstates the case, because the plane itself has a large mass– about 4900kg empty. If we assume that the center of mass of the plane is near the middle (the wing-mounted engines presumably being the heaviest parts), moving the passenger center of mass forward by 17cm shifts the center of mass of the loaded plane forward by only about 4cm, less than two inches. The passengers don’t really contribute very much to the calculation at all.

So, why were our pilots so insistent on separating passengers by gender? Most likely superstition, rather than physics. They might very well believe that putting the lighter women toward the back changes the balance in some significant way, in which case doing so makes them feel better about flying.

Personally, I believe in math and physics. And I just don’t see anything in the numbers suggesting that re-sorting the passengers would make any significant difference. And if the tiny little shifts that we see from the above calculation are important to the safe operation of the aircraft, well, I don’t think that’s a plane I want to be flying on.

(Not that I’m much of a fan of the Beechcraft in the first place– the seats are tiny, and the “headrest” doesn’t even reach the top of my shoulders. The only reason I took that flight (which I’ve done twice before) was that it’s the only direct flight from Albany to Toronto, and the alternative was taking two flights on tiny uncomfortable planes, with a layover in Chicago in the middle.)

1. #1 Richard Simons
October 2, 2012

I came across a similar request once on a Saab 340, a somewhat larger aircraft (33 passengers). Perimeter Airlines flies the Metroliner (configured for 12-14 seats) around northern Manitoba and I’ve never heard them asking people to sit with the men nearer the front, although once with 4 passengers we were asked to sit more forwards (there is less engine noise nearer the back). Of course, many of the passengers are Cree and Ojibway and the men and the women can be equally large.

2. #2 Alex
October 2, 2012

Based on the title, I assumed that this would have something to do with childcare and family leave issues for airline employees.

3. #3 Stephenk
October 2, 2012
4. #4 Stephenk
October 2, 2012

The type certificate for the aircraft type in question can be found here
http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgMakeModel.nsf/MainFrame

Looks like for a Beechcraft 1900 the allowable CoG varies
185 to 196.4 inches (from datum*) at 12500lbs
181 to 196.4 inches at 11279lbs or less

IE total allowable range is 11.4 inches when fully loaded and 15.4 inches at some lesser loadings.

Now do those _total_ ranges jive with the numbers you worked out?
(I don’t think it was superstition :-) )

*I don’t know what the datum is, but it isn’t that relevant, the pilots will be doing sums to keep within those bounds they will also have the data for that specific plane and be able to work out moments for fuel, bags etc.

October 2, 2012

Based on the title, I assumed that this would have something to do with childcare and family leave issues for airline employees.

This is not an accident… I picked the title because it amused me to create that sort of confusion.

http://en.wikipedia.org/wiki/Air_Midwest_Flight_5481

That says that the flight was overweight by 260kg, and had its center of mass shifted backwards by 5%, which would work out to a bit less than a meter, using the full length of the aircraft. The shift I calculate here is less than 1%.

And even the overweight/unbalanced problem was not enough to bring the plane down, in the FAA’s judgement, but needed an additional miscalibration of the controls to cause the crash. So I don’t think the cases are really comparable.

October 2, 2012

And, correcting myself a little, the not-quite-1% shift I mention was the shift in the center of mass of the passengers. The shift in the center of mass of the plane would be more like 5cm (2 inches), or about 0.3%.

7. #7 Stephenk
October 2, 2012

I think the 5% referred to was 5% of the allowable range. Ie less than an inch.

8. #8 Eric Lund
October 2, 2012

Your calculation also assumes that passengers have no baggage, either checked or carry-on. On that type of plane (one of several turboprop types I have been on), carry-ons which are too large to fit under the seat in front (or for those pax who don’t have a seat in front of them) are placed in the luggage compartment and claimed planeside when you reach your destination, as there are no overhead compartments. The plane may also have been carrying cargo.

No, the difference may not have been that much. But it may have been enough to affect whether the plane was in a legal takeoff configuration. Yes, there is a safety margin on the legal standard, as the FAA report on US5481 implies. Tell that to the lawyers. And, this being an international flight, to Homeland Security.

October 2, 2012

Luggage is another issue, but probably produces shifts on the same order or smaller as the shifts due to the passengers. The luggage compartment is farther back than most of the seats (behind the wings, below the seats), but the total mass of luggage is smaller than the total mass of the passengers (using the masses above, the maximum allowed would be around 1200 kg, though I suspect in reality it was much less).

There’s also fuel to be considered, but I’m assuming that whoever designed the plane thought about the balance issues associated with the fuel. Otherwise, they’d be in danger of crashing when they’re empty.

Also, note that the way this is set up pretty much maximizes the shift you could get. In reality, some of the male passengers (the skinny high school kid across from me, for example) were almost certainly outweighed by some of the female passengers, and mixing up the masses would tend to reduce any shift (assuming the mixing was more or less random, and the actual seats within the plane were determined by the order of boarding, not anything else). If they really cared about keeping the center-of-mass from moving, they would’ve seated me and the one other big guy on the plane in the center, and put the scrawny people toward the outside. Or if they needed a shift, they would’ve arranged everybody by size, putting the biggest people on one end and the smallest on the other. Strictly gender-base sorting is more likely to be lazy superstition than any calculated shift.

10. #10 Rich Y
October 2, 2012

The weight and balance section (1.6.4) of the NTSB report on that accident is not so hard to skim, and considerably more precise than Wikipedia. They’ve done us the favor of gathering all the info in one place.

It looks like spec is 17-40% -ish of the wing chord, which is [googling] 7ft at the root of the wing. A two inch shift is about 2% of that chord.

I wouldn’t call two percent on 23% allowable range insignificant.

11. #11 Rich Y
October 2, 2012

Oh, and sorting people by a (mostly) easy to determine and non-controversial category like gender which correlates to weight is a lot less fraught from a customer service point of view.
I wouldn’t be the one to have to say “Ma’am, you look chunky. I’m sorry, but could you please move to the front of the plane?”

12. #12 Stephenk
October 2, 2012

Once again from the type certification document
The fuel tanks datum
Mains 185 inches
Wing tip 193
Aux 204
So the fuel lies close to the allowed range (185-196.4) so changes of fuel won’t have much effect on CoG as you suppose (but not none)

However, baggage is 550lb at 325 inches so it is waaaay back past the rear CoG limit. Loading of baggage would have a very strong effect on balance of this type of aircraft. And that would have been my first guess too. the pilots knew they were cutting it fine with baggage and wanted to rearrange the passengers to give them a safer margin on the rearward CoG (it’s usually rearward CoG that causes problems)

(actually these numbers I’ve quoted from the type cert docs are for the Model 200 which was at the top of the document. I don’t know specifically what your model in question was but these numbers should be near enough to illustrate the narrow bands worked in)

13. #13 David
rainy eastern USA
October 2, 2012

Are you considering only static balance? If the airplane has to make a sharp turn/climb/descent in some emergency, the weight distribution might affect its rotational moment of inertia.

14. #14 Ron
October 2, 2012

From what I’ve seen, all of the luggage is behind the passengers in the tail of the plane. There aren’t any luggage compartments below the passenger compartment.

my living room
October 3, 2012

This is a rare scienceblogs entry that is somewhat in my wheelhouse. Back when I flew for the Air Force I actually ran into this issue once on a fully loaded jet when we were coming home from a deployment. We were probably close to max baggage weight and carrying close to maximum pax capacity. Storage was similar to commercial airliners (the airframe was based on a 707) with the usual below deck storage in the aft end and also a storage area above deck about 3m behind the COG. Shortly after takeoff about 1/4 of the people went aft to the galley and restroom. The plane became tail heavy and the flight crew had to call people up front to balance it out.

This was a pretty unusual event, it took a lot of people moving to one end of the jet to make a difference, and even then it wasn’t even really that dangerous, but you don’t really want to take chances at 30k’ MSL.

All that said, gender seems like the worst possible metric to use to balance out an aircraft. Also, I’m going to send a link to a friend who IIRC is a loadmaster on cargo aircraft to see if he can shed some light on this.

16. #16 Birger Johansson
October 3, 2012

Cabin width (m): 1.37 !
Are you kidding me? This is like being squeezed into the trunk of a car! Why not just stack the passengers on top of each other? If this was a military transport aircraft I could understand, but getting paying passengers to squeeze into the limited space pretty much rules out people with arthritis or other problems.

17. #17 Frank Wappler
testing.testing--\varepsilon_1.\varepsilon_2
October 3, 2012

Chad Orzel wrote (October 2, 2012):
> […] there are two seats each at 0, ±1.1m, ±2.2m, ±3.3m[…], and ±4.4m.
> 9 men and 9 women on the plane. […] four rows of two men, a row with one man and one woman, and then four rows of two women.
> [Assuming equal masses of all men, and equal masses of all women, with values of] 90kg and 79kg for men and women respectively
> […] we find that the men as a group have their center of mass at -2.58m, and the women as a group are at +2.58m.

Others may find that
$\frac{ 0~m * M + (1.1~m + 2.2~m + 3.3~m + 4.4~m) * 2 * M}{9 * M} ==$
$$\frac{ (0~m + 2 * 11~m){9} ==$$
$(22 / 9)~m$
which is between 2.44 and 2.45 meters.

18. #18 Eric Lund
October 3, 2012

Birger @0745: That’s plenty of room for two 50 cm seats (typical economy class width) plus a 30 cm aisle. (I also recall that the cross section is elliptical, not circular.) The back row is crowded, yes, and you can’t stand completely up if you are taller than the average American male. Not that you would want to during flight: there is no lavatory and no flight attendant.

The B1900 is used on short-haul routes (800 km or less) with small passenger volumes. I’ve seen other aircraft types that size, but most of them have been retired. If there were enough passengers to justify a larger aircraft, they would use a larger aircraft (e.g., Saab 340 or Dash-8) with a bigger interior, a lavatory, and a flight attendant.

19. #19 Dan Riley
October 3, 2012

I doubt this was superstition–regulatory compliance seems a lot more likely. The pilots have a pre-flight checklist that includes a COG calculation (with some assumptions about the passengers), and hard limits on the bounds for the COG. If they are a little out of the bounds, they will try to find a way to shift the COG into bounds, even if it means gaming the system a bit.

20. #20 Skip Huffman
United States
October 3, 2012

Very small differences in weight and balance can significantly affect both the handling of and the safe operation of an aircraft. Moving the center of gravity a few inches forward or back can make a difference.

Per FAA documentation a male passenger is assumed to weight 88kg and a female one 70kg. Assuming nine of each that gives the pilot 162 kg that s/he can move back and forth to find the right balance. This is along with any cargo shifting that can be done and also any differential loading of fuel. This really can matter for small aircraft. Especially if they are working out of a small field.

21. #21 anon
October 4, 2012

The rules are different in eastern Europe; there, regulations for aircraft stability require that all Poles sit in the left half of the plane.

22. #22 Mark Holm
Pittsburgh, PA
October 12, 2012

Perhaps you haven’t heard of the similar sized commuter plane that crashed, killing all on board about 15 or 20 years ago in the Carolinas. Accident investigation determined the plane was out of balance with CG too far aft. On takeoff, elevator did not have enough aerodynamic power to overcome the out of balance condition, nose went up, wing passed critical angle of attack, lift to drag went to hell and it all smashed into a hanger off to one side of the runway. As a direct result, the FAA revised it’s standard estimates of passenger weight upward.

23. #23 Mark Holm
Pittsburgh, PA
October 12, 2012

On the flight I mentioned, there was also an issue with incorrectly rigged elevator controls, however the accident investigation determined that both the out of balance condition (outside of the certified limits) and the out of rig elevator had to be simultaneously present to cause the accident.
http://en.wikipedia.org/wiki/Air_Midwest_Flight_5481
There was also a recommendation to actually weigh individual passengers, but you can guess how well that would go over in the US.