measuring a long tail near a limit

A friend happens to be a very tall woman - 200 centimeters (just under six foot seven inches - she rounds down to six foot six inches). The natural question is "how many woman are that tall or taller""

It turns out to be a difficult question. Adult human height, for one sex within a particular ethnic group, tends to be normally distributed. For a variety of reasons the tail for extreme height tends to be a bit "fatter" than extreme shortness. Both are sufficiently rare that a reasonable first approximation to the question for my friend is "not many"...

She happens to be a professional beach volleyball player - a sport that, like basketball, favors height. I found a list of professional female player heights and plotted a histogram a fit.

These data claim to be measured rather than self reported heights. Women tend to under report height (particularly if they are near some social limit ... there are a surprising number of models who claim to be five foot eleven and almost none at six foot. Men tend to over report height. There are fewer at five feet eleven that one would expect and a surplus at six foot even. Very tall people in sports tend to over report. It is unusual that my friend rounds down.)

At about five foot eleven and a half inches, the average female pro beach volleyball player is nearly three inches taller than the average US adult male and more than seven inches taller than the average US female. (aside - Burkhard Bilger penned a great piece on height differences between Europeans and Americans in the April 5, 2004 issue of The New Yorker.)

My friend turns out to be the tallest professional female beach volleyball player. So she is about one in three hundred in that heighty group, but how about the normal population?

Here I have plotted normal distributions for adult female, male and professional female beach volleyball players. If height distributions were really normal, it would be straightforward to answer the question. She is 14.5 inches taller than the average woman and the standard deviation of adult female height is 2.58.... so she is about 5.6 standard deviations away from the mean. (there are a few data sets. The newest and largest is quoted. The range goes down to about 5.2 standard deviations ... still approximately infinite in this game)

5.6 standard deviations has her taller than 99.9999989% of the population. In the US that would give about 2 women her height or taller ... obviously wrong. Moving to the 5.2 standard deviation figure one expects about 15 women.

Clearly the real tail is somewhat "fatter" than a normal distribution would suggest.

For fun I contacted two experts on the subject of human height distributions. Both said the tail at the tall end was fatter than expected. One suggested a rule of thumb of subtracting 1 standard deviation when your are in this range. His guesstimate is about 300 women. He also said doubling that number gives a good world estimate. The other expert had some other statistics and thought the real number would be between 150 and 400 and suggested 200 as a best guess. There are some back of the envelope methods that give slightly larger numbers, but I defer to the experts.

Many products are designed to work well with about 95% of the population. Sometimes excursions can be dangerous - air bags don't work well at the lower end of the scale and the potential safety issue is large enough that many designs allow the user to disable them.

But it some products can be made for the tails if their manufacture is sufficiently custom. Automating critical steps in the manufacturing process can make this economically viable. The other trick is to figure out a distribution channel. This is beginning to happen in clothing. People have gone the high tech route with measuring booths, laser cutting and other very flexible tools, but the winners so far seem to be companies that address the distribution problem.

Five percent of a large market is attractive in many areas and we can expect new companies. A side benefit may be damage to the conventional distribution chain for the rest of us.

Comments

[Jheri]

I thought I was tall:-)

Colleen is a friend and she is an amazing person.

[sara]

6'7, my god!

i'm 5'1 and have problems finding things that fit. i'm not as far from the average as your friend, but it is really hard to find cute clothing that looks good. i think most of it is made for women who are about 5'6. at least i'm tall enough that i don't have to buy children's clothing. my mother used to have to do that. she is 4'10