The real trouble with this world of ours is not that it is an unreasonable world, nor even that is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite.
G. K. Chesterton, “ Orthodoxy ”
The Italian poet Trilussa said that Statistics is the science according to which if you eat two chicken a day and I eat none, on the average you and I eat a chicken a day. This is an emblematic case of misuse of Statistics through overworked averages. In fact, the statistical science is based not on one, but on two main concepts:
i) the assessment of the central value of a set of numbers, the average;
ii) the measure of how the individual numbers distribute themselves around the average. Do they cluster closely about it, or they scatter widely? Which is their variability?
Thus, the degree of variability of a set of numbers around the average gives material significance to the average itself.
As there are a few types of averages so are there a few types of variability measures.
Since Trilussa used the arithmetic mean in his discourse, let us introduce a variability measure commonly used in similar cases: the coefficient of variation, ranging from zero – for flat homogeneous sets – to 100% – for the extreme heterogeneous sets.
Going back to the chicken’s story, should we be told that Tom and Dick eat a chicken a day on the average, with a coefficient of variation equal to 100%, anyone who had a minimal statistical knowledge would understand at once that either Tom or Dick grasp always two chicken.
If in the country of Tom and Dick the GDP should grow by 2 – 3% on yearly basis, both of them knew, by experience, that their income would not do the same, since averages are working again.
For simplicity sake let us consider a case with no GDP growth. In a given year four people have an income of £15,000, £20,000, £28,000 and £37,000 respectively. By making the arithmetic mean such a group has an average income per capita equal to £25,000, with a coefficient of variation of 33.2%. Should in the subsequent year the same people have an income of £13,000, £18.000, £30,000 and £39,000 respectively, the average income per capita would be again £25,000, but with a coefficient of variation equal to 40.7%. It means that the income diversity, i.e. the economic inequality, has grown by more than 20%.
Life evolves by breaking crystallized symmetries so as to generate a new life and new orders. Likewise countries and nations progress and flourish in building up more and more structured societies by giving merit to the more talented and industrious people. For this reason economic inequality is endemic in evolving human communities as it is asymmetry in Nature. But economic inequality can trigger deprivation, illiteracy and antisocial behaviour, which undermine the standard of life of any community.
Adam Smith, ( 1723 – 1790 ), father of economic liberalism, stated in The Wealth of Nations that man is possessed of a certain “fellow feeling ”: How selfish soever man may be supposed, there are evidently some principles in his nature which interest him in the fortune of others, and render their happiness necessary to him, though he derives nothing except the pleasure of seeing it.
Since the respect of rights of the diverse ones – as deviating their way from recognized averages – is taking ground in mature democracies, we need to know better how diversities work.
It is said that we are living in the Age of Statistics. Actually we are living in the age of overworked averages. In order to tackle new challenges democracies have to deal with diversity and variability in a knowledgeable way. Otherwise people distrust statistical data not reflecting their effective situation. However, it is not duty of statisticians to assess the limit allowed for a kind of diversity, or an economic inequality. The only task of statisticians is to provide the public with the best tools for debating about diversities, or sustainable economic inequalities. Therefore we need to be more acquainted with the neglected twin brother of the abused average, named variability.