On the size of shoe racks

On a web discussion board, I conjectured the following:

The diversity of the size of women’s shoe-racks can be expressed in mathematical fashion as a distribution of a particular form, called a “power law“, meaning that the probability of a woman’s shoe-rack attaining a certain size $x$ is proportional to $(1/x)^y$, where $y \ge 1$.

When a distribution of some property has a power law form, the system looks the same at all length scales. Therefore, if one were to look at the distribution of rack-sizes for one arbitrary range, say, just racks with 100 to 1000 shoes, it would look the same as for a different range, say, 1 to 10 shoes. In other words, “zooming” in or out in the distribution, one keeps obtaining the same result. It also means that if one can determine the distribution of shoes per rack for a range of shoes, one can then predict the distribution for many other ranges.

Equally interesting, power law distributions have very long tails, meaning there is a non-zero probability of finding racks extremely large compared to the average. This finite probability of finding large racks is quite striking and can be illustrated by the example of the heights of individuals following the familiar normal distribution. It would be very surprising to find someone measuring two or three times the average human height of 5’10”. On the other hand, a power law distribution makes it possible to find a rack many times larger than average. Power laws also imply that the system’s average behavior is not typical. A typical size is one that is encountered most frequently; the average is the sum of all the sizes divided by the number of women. If one were to select a group of shoe-racks at random and count the number of shoes in each of them, the majority would be smaller than average.

A similar analysis can be carried out for women’s cloth cupboards, with $y \in [3, \infty)$.

Men don’t have shoe-racks! So an analogy between the show-rack size and the size of the cupboard would not be possible. But here is one fact that gives you the basic idea: I’ve two pairs of jeans, one is torn at several places. I have a few Ts, a few bought and a few won in different competitions held at my place. Apart from these prized possessions, I also have a couple of shorts and a track suite, a couple of towels, a few pairs of rotten socks and a few undies.