Q & A: why are distributions non-Gaussian?

Q:
If you see non-Gaussian distribution in Physics, what does it imply? What is the cause/mechanism of such distribution pattern? For example, I heard skill distribution in human population is skewed to the right(positively skewed). Why is this? and what interventions(or factors) can make it look more like a normal bell shaped or even skewed to the left (negatively skewed) (->is this achievable and stable in real world in skill distribution in human population? why/how?)? Could you provide other good examples of non-Gaussian distribution (pattern) and why/how such distribution manifests (spontaneously?).
- Anonymous
A:

In a way, that's like asking "when you see a non-elephant animal, what is it?" OK, it's not that extreme because there is a fairly common class of problems- observables which are the sum of many independent contributions- that gives Gaussian distributions. There are also some other special cases that have Gaussian distributions. The generic case, however, is non-Gaussian. Since for some decades I mainly made a living by studying a variety of non-Gaussian statistical variables in physics, you probably don't want to get me started telling their many diverse stories.

Here's an example. Maybe you're looking at resistance in a small sample of a material with an insulating and conducting phase. Disorder in the sample breaks it up into little domains.  Right near the phase transition a small numebr of domains fluctuate randomly between the two phases. The distribution of resistances may be bi-modal, or 4-modal, etc. The explanation is obvious.

With regard to the skill distribution, things are more subtle than for simple physical variables. For starters, if you define skill as some sort of positive number, the distribution couldn't be exactly Gaussian, because Gaussian distributions have tails extending infinitely far in both directions. There's another basic problem in describing a "skill distribution". There's no natural measure. Say A can use only a hammer, B can also use a scewdriver, and C can use Allen wrenches as well. What's the ratio of the skill increment from A to B to that from B to C? The numbers you assign are an arbitrary choice. The same problem comes up in designing, for example, IQ tests. There the test designers chose to define the differences to make the distribution Gaussian, just for convenience.

So let's pick some variable where the facts tell us what the distribution is, not where we tell the facts what distribution we'll choose to describe them. Income is a nice variable with a fairly clear measure. It's highly non-Gaussian. In order to survive, some minimum income is needed, especially if the dollar value of assistance is included. So the distribution not only stays positive but doesn't go down to zero. It's highly skewed upward. Sometimes people claim that many factors independently affect income multiplicatively. That would give a Gaussian distribution not of incomes but of logarithms of income. Actual distributions have much bigger tails at the high end than even that skewed distriibution. Social scientists discuss why that happens, often with ideas along the lines of "for whosoever hath, to him shall be given".

Mike W.

(published on 11/05/2013)

Follow-Up #1: non-Gaussian distributions in physics and income

Q:
1)What are the "simple, clean" examples for positively/negatively skewed non-normal distribution in Physics? For example, a nice wave packet (a photon in vacuum etc.)has a clean Gaussian distribution, right? Under what simple condition/influence is it skewed toward right or left cleanly? Does the extreme difference in gravitational field (maybe near a black hole etc.) between the head and the tail of the Gaussian photon wave packet make the Gaussian distribution skewed? Other examples? 2)I am aware of "non-Gaussian income distribution." let's pick income distribution as it might provide a clearer variable and act as an interesting/relevant example (for people). I have seen positively skewed income distribution (US, UK and others) but are there countries (maybe Scandinavian?) where income distribution looks more like normal distribution or negatively skewed? If so why? And, If you see "positively skewed income distribution," does it indicate something is wrong with the society or country? Is the income distribution that looks more like Gaussian distribution (or negatively skewed?) healthier? Some say "education" is an important factor to increase income but I heard it did not change income distribution in UK. why? (On the other hand, what are the cases/countries education "did" change income distribution? and why in these cases?) What are other factors/cases/examples that "did" make a change in income(or resource) distribution in human (or animal/plant/bacteria etc.) history?
- Anonymous
A:

1) There are countless examples to pick among, but here's one of the simplest. Say you're counting radioactive decays from a big block of weakly radioactive material. The decays are independent of each other. The distribution of counts in a given time interval is Poissonian. After enough time to expect N counts, the chance of getting n is P(n)= Nne-N/n!. That's skewed upward. Only after N becomes very large does the meat of that distribution start to look Gaussian.

What about one skewed toward the low side? There must be something more obvious, but here's the first that crosses my mind. Look at the fluctuating EMF's as you magnetize a piece of iron with some impurities in it. (These fluctuations are called Barkhausen noise.) The magnetization changes are all of the same sign, so you can take their logs. The distribution of the logs is skewed downward. There's a pretty good theory describing this process. (Google "Barkhausen ABBM".)

2) I've never heard of any negatively skewed income distribution on a national scale. Usually one keeps track of width of the distribution of logarithms of incomes, which is related to but not the same as the skewness of the income distribution. These widths vary a lot between countries and over time, even in my lifetime. My personal opinion, which I believe is supported by a lot of data, is that most indices of health (life expectancy, etc.) tend to be higher in societies with narrow distributions. You could scrounge around among social science research for good data. (some key words: "OECD", Saez, Gini index)

These distributions are changed by many, many factors, including election results. Perhaps the most important historical event, however, was the development of agriculture, which allowed stored surpluses and large inequalities.

I've already been blabbing beyond my expertise, so I won't address the role of education. The OECD keeps good data on changing inequality over time in many countries, and you could try to see if that reflects changes in educational systems.

Mike W.

(published on 11/08/2013)