Pardox of the average salary

You have in reality the case that sounds like a joke or a figure of speech:

Nearly everybody is below average.

Huff, D. (2010). How to lie with statistics. WW Norton & Company. P. 31.

Majority of individuals earn less than average salary. Why? There is a simple explanation for this fact. Firstly, income distribution could be described with Lorenz curve.

Secondly, the shape of this curve is not the same for different countries and timepoints. Surprisingly, it can be easily characterized by Gini coefficient. Here (link) you can find dataset containing Gini coefficients for different countries. For my calculations, I will use Gini coefficient measured for Belarus (G=26.5%).

Proper Lorenz curve could be calculated using only the Gini coefficient:

Lorenz curve formula: f(x)=1-(1-x)^(1-1/a) (1)

as soon as G=1/(2a-1) (2)

we can calculate a=0.5/G+0.5 and

express formula with G only: f(x)=1-(1-x)^((1-G)/(1+G))

Now let's arrange all our employees using their wages. If we want to find the salary of a person at 50% (median value, 50% will earn less and 50% will earn more) we need to find "y(50%)" for "x±0.5bin" (bin will be a tiny fraction of only 0.0001%).

y1=1-(1-(0.5-0.5*0.000001))^(1-1/(1/(2*0.265)+0.5))≈0.3315121843

y2=1-(1-(0.5+0.5*0.000001))^(1-1/(1/(2*0.265)+0.5))≈0.3315129611

In order to continue, now we will need to know the average net salary (S) for the selected land. For Belarus, it will be 310 EUR (S=310 EUR). Let's continue now. The typical (average) salary for the selected tiny bin would be equal:

y=S*(y2-y1)/bin

Therefore y(50%)=310*(0.3315129611-0.3315121843)/0.000001≈241 EUR

Similarly,

y(10%)=188

y(20%)=198

y(40%)=223

y(50%)=241

y(72.7%)=310

y(80%)=354

y(90%)=473

y(95%)=632

y(99%)=1240

Thus, only 22% of people (100%-72.7%) earn average salary or more. While 19% earn not less than 500 EUR and 5.4% earn not less than two average salaries (2*310=620 EUR). Top 1 % (y(99%)) earns 1240 EUR/month or more. As you could mention, at given y, x  depends only on G (y∼x^(1/G) ⇒ x∼y^G). It brings me to 2 findings:

1. As soon as Gini coefficient in real life is always higher than 0, more than 50% will earn less than the average salary in their country.
2. Rise of inequality (expressed as Gini coefficient) confers increase in number of people earning less than average salary.