## The fat tail

*June 12, 2009 at 5:17 am* *gabrielrossman* *
2 comments *

| Gabriel |

At Slate XX, Virginia Postrel has an article explaining why women’s clothing sizing hasn’t kept up with the increasingly large American woman herself. This is often explained as an indulgence of taste (designers don’t like making clothes for people they find unattractive) or a Podolny-esque status thing (serving stigmatized customers kills your brand). Postrel doesn’t buy any of that, taking the Gary Becker economics-of-discrimination line that some entrepreneur should be filling this demand unless there is some good business reason not to. (I think the “taste” and “status” things are very plausible at the high-end, but I agree with Postrel about “the customer is always right” for the mass market). Her argument is that it’s about the cost of fabric (boring) and much more interestingly, the right-skewed distribution for size. I think this is worth unpacking and ruminating on because it’s a good example of how it’s more useful to think about distributions than just central tendencies (as people often find more natural).

While (within gender) height follows a normal distribution pretty closely, weight has a right skew. Here’s what that means. The medical people tell us that the ideal BMI is about 22, give or take a few points, which works out to about 130 lbs (+/- 20) for a 5’4″ woman. Now a woman at the first percentile for BMI is going to weigh about 90 lbs. On the other hand, a woman as fat (in percentile terms) as Victoria Beckham is thin is going to weigh over 220 lbs, probably more.

So relative to healthy, extremely thin is about 40 lbs off whereas extremely fat is at least 90 lbs off. If “healthy” is a reasonable approximation for the median, then for any given point of weight to the left of the median there’s going to be about twice as much density (in the statistical sense) than a comparable point to the right of the median. In other words (to paraphrase Tolstoy) thin women are all alike; every fat woman is fat in her own way. Postrel’s argument is that to the extent that clothing is meant to be tailored to be a pretty close fit for your body then any given plus size will fit few people even if many people fit some plus size.

If you think of clothing sizes as analogous to binning a distribution to plot it as a histogram, any given bin on the skewed side of a distribution will have less density than a bin at the opposite percentile. Postrel notes that there are certain design costs and inventory costs associated with keeping a size in stock and so if it takes sizes 16 and 17 combined to equal the sales just of size 5, then it’s rational for companies to consider dropping their plus sizes, even though in the aggregate they serve a lot of paying customers.

This makes a fair amount of sense, but I wonder about the extent to which it relies on the assumption that the breadth of a size is always a constant range, say +/- 3lbs from some target customer. For all I know this is how clothes *are* sized and *ought* to be sized, but I wonder if it is the practice for sizing to have a wider tolerance at higher weights. In quant work when we have a right-skewed distribution we often log the variable. What this effectively does is make the raw scale bin width a function of x, so as you get higher on x the bins get wider on the raw scale even though the bins are all the same width on the log scale. I can think of two substantive reasons why it might be appropriate to imagine any given plus clothing size encompassing a wider range of weight than any given petite clothing size.

First, there might be a taste difference where thin people tended to prefer tighter-fitting clothes and fat people looser fitting clothes, especially for things like jeans. Since loose clothes are more forgiving of fit then it would make sense to have broader plus sizes. You see a similar thing in that people with short hair get it cut much more often than people with long hair. I have very short hair and when I think “I need a hair cut,” I’m thinking something closer to “my hair is 30% longer than it should be” rather than “my hair is one inch longer than it should be.”

Second is that maybe we shouldn’t be thinking about clothing sizes in pounds at all, but something like inches (which is how men’s clothes are sized). In this case, geometry diminishes the skew. If you imagine the radius of a human being in cross-section, that person’s weight is approximately pi*r^{2}*height whereas that person’s waist size is approximately 2*pi*r. The squared term for weight means that weight will be more right-skewed than circumference. These are the same people, but depending on how you measure “size” the distribution may be skewed or it might be symmetrical. This is actually a big deal generally in statistics since assuming the wrong distribution for a variable can lead to weird distributions for the error term. Hence good statistical practice either transforms skewed variables as part of the data cleaning or uses “count” analyses like Poisson and negative-binomial that are designed to work with skewed distributions. There’s also a more basic theoretical question of whether the skewed variable is even the right operationalization. If we’re interested in the size of a person is weight better than waist size? If we’re interested in the size of an organization is number of employees better than length of the chain of command? In both cases the answer is that it depends on what you’re trying to explain.

Anyway, if either for reasons of taste or reasons of geometry, a size 0 has less tolerance (as measured in pounds) than a size 8, which in turn has less tolerance than a size 16, then this could partially compensate for the dynamic Postrel is describing. However note that, unlike me, she actually talked to clothiers so take her data over my armchair speculation.

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1.flanuese | June 16, 2009 at 1:58 amMy other concern about women’s clothing is that one assumes the sizes themselves stay the same, that an 8 is always the same 8. Whereas men’s sizes are often actual measurements – a particular arm length, a particular neck circumference – women’s clothes go by numbers that are supposed to correspond to waist, hip, and bust measurements but they often do not.

Retailers take it upon themselves to redefine sizes. The Gap is notorious for this. They seem to have decided that a good way to sell clothing is to assume that most women will be flattered if they are able to fit into smaller sizes than they are accustomed to wearing. The strategy is based on the premise that if a woman thinks she’s a size 8 and suddenly fits into a size 6, she’ll be so deliriously happy that she will buy more clothing. Thus, all the sizes at the Gap are not only unlike the same numerical size at other retailers, but they keep getting bigger and bigger. The “oh-my-goodness-I’m-skinny-and-happy” trick is a one-time thing…unless the sizes just continue to expand. This practice is frustrating to people at the lower end of the size distribution because they end up either not being able to find small enough sizes or they have to fit into numerically demeaning sizes like 0 or 00. It’s existentially a little unsettling to think of oneself as a ‘zero’ or, worse, a ‘double zero’.

Your blog suggests the appropriate solution which is to use a non-linear scale to plot the difference between one size and the next. At the lower end, a half inch difference might be meaningful and appropriate whereas at the higher end the same measurement may need to change by more like six inches.

2.gabrielrossman | June 16, 2009 at 2:05 amIn a previous article, Postrel talked about vanity sizing and related issues.

A clarification, I was saying that even a linear inch-based scale will translate to non-linear in terms of weight because an extra inch of waistline translates to fewer pounds when you’re going from 30 to 31 than when you’re going from 40 to 41.