The Tales of Tails

When we enter a space our natural inclination is to do a quick visual survey of who’s around. When we walk into a first grade classroom for instance, we expect to see a whole lot of 5 and 6 year olds and one or two adults.


Now if we were to plot the heights of the people in the room we would get a narrow ‘bell curve’ like distribution centred around 3.5’ and a point or two sticking out somewhere in the range of 5’and 6’.

Let’s say I wasn’t there and all I had in front of me was the distribution that you put down. I could pretty easily guess what kind of situation this was. Where else do you have a bunch of small people with just one large person? Now instead, what if I showed you a distribution that looked like this?




There are still thirty kids of the same size range. But now there are around 30 grown ups as well. Where would you find that kind of situation?

If you guessed that it was one of the kid’s 6th birthday parties, you would probably be right. So every distribution has a tale to tell. And the tale runs deeper. It tells us something about the nature of height itself. The bell shaped curve suggests that the height of one child in the class is virtually independent of the height of the others. Indeed for each of us, our heights play out from a highly similar genetic program with small variations arising from random environmental events.

It’s fairly easy to make sense of this kind of bell shaped distribution since we’ve all seen it in school. But what about the distribution of incomes? What tale does it tell? Why does it look so different? (To understand what income distributions look like, please read my earlier post Who cares about the average income!). These heavy tailed distributions and the power relationship that they often reflect, tells you that there’s no one size fits all, no meaningful ‘average’. More significantly they reveal a story of high interdependence, a story of a dynamically changing complex system. It is a reflection that my income does depend on yours in some very complex way. The distribution of income is a static view of a constantly changing interdependent process. Money is changing hands all the time and every so often we take a picture of how it is looks before it is passed on. This is fundamentally different from the story of the bell curve. And every small turn and kink in the distribution offers another twist to the tale.

Who cares about the average income!

Many of us think of statistics as basically taking the average of some numbers. Maybe we even think of the normal distribution or ‘the bell curve’ and the concept of standard deviation. In this context, if you say the average height of people in India is around 5’ 5” with a standard deviation of 5”, it’s pretty intuitive what that means – that when you arrive in India you will find most people around 5’5” with some variation this way and that way of mostly 5”. In large part we all look similar and can fit in the same seats, sleep on the same size beds and fit through the same doorways. Instead, imagine if the distribution was not a bell curve but rather looked like this. A decreasing function with a heavy tail.



What this would mean is that most people are less than a foot tall while a few are absolutely enormous 50’ giants (in the heavy tail) with the rest somewhere in between. In this scenario there is no one size fits all and it would be impossible for a randomly selected group to be comfortable sitting at the same sized table. The enormous giants the height of multi-storey apartment buildings would have to live in a different sort of world than their underfoot counterparts. The average height in this distribution is still around 5’5” but knowing this average, and even the standard deviation, would be completely uninformative. Rather what could prepare you for what to expect is to know something about the rate at which the distribution decreases, basically the exponent that describes the relationship.

Now, unlike height distribution, which are bell curves, virtually all over the world the distributions of incomes and wealth display heavy tails. What differs between countries is largely the exponent. So if you knew nothing about India and you wondered what single number would best prepare you for the economic landscape, average income is really quite uninformative. Rather if you knew the exponent, or better still had a good visual of the distribution, you would come prepared to find Mukesh Ambani in a 27 story house on Altamount road surrounded by millions of abysmally poor people with barely a roof over their heads (and a little bit of everything in between). So, particularly from the perspective of understanding poverty, what we should care about is the exponent. Who cares about the average income!

In a separate post I will talk about what gives rise to these different types of distributions and why income and wealth distributions looks like this. Stay tuned.

**Note that if you do decide to delve further into this, most people show heavy tailed distributions in log-log or log-linear scales to emphasize the properties of the tail so make sure you look carefully at the axes.