Inequality Measures

Atkinson  inequality     ZAtkinson = 1-ZMacRae[1] ≥ 1-exp(Σi=1..N(Ei*ln(Ai/Ei))/Etotal)*Etotal/Atotal  nosniktA  inequality   ZnosniktA ≥ 1-exp(Σi=1..N(Ai*ln(Ei/Ai))/Atotal)*Atotal/Etotal  Theil-T  redundancy   RTheil  = -ln(1-ZAtkinson) = -ln(ZMacRae)     ≥ ln(Atotal/Etotal) - Σi=1..N(Ei*ln(Ai/Ei))/Etotal  Theil-L  redundancy   RliehT  = -ln(1-ZnosniktA)     ≥ ln(Etotal/Atotal) - Σi=1..N(Ai*ln(Ei/Ai))/Atotal   Theil-S      redundancy         Symmetric      redundancy     Rsym  = -ln(1-Zsym) = 2*ZPlato*artanh(ZPlato)     = (RTheil(E|A)+RTheil(A|E))/2 = (RTheil+RliehT)/2     ≥ Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal))/2   Symmetric   inequality   Zsym  = 1-exp(-Rsym) = 1-√((1-ZAtkinson)*(1-ZnosniktA))     ≥ 1-exp(Σi=1..N(ln(Ai/Ei)*(Ei/Etotal-Ai/Atotal))/2)   Hoover   inequality   ZHoover ≥ Σi=1..N|Ei/Etotal-Ai/Atotal|/2   Coulter   inequality   ZCoulter ≥ √(Σi=1..N(Ei/Etotal-Ai/Atotal)2/2)  Gini  inequality   sort data:  Ei/Ai>Ei-1/Ai-1  ZGini ≥ 1-Σi=1..N((2*Σk=1..i(Ek)-Ei)*Ai)/(Etotal*Atotal) EU  inequality   1:a = (1-ZGini)/(1+ZGini) is the SOEP "equality parameter"  therefore: ZEurope = 2*ZGini/(1+ZGini) Plato  inequality   inverse functions:    Zsym = 1-((1-ZPlato)/(1+ZPlato))ZPlato    Rsym = 2*ZPlato*artanh(ZPlato)  approximation:    ZPlato ≈ 1 - arcsin((1-Zsym)(0.06*Zsym+0.61))*2/π,  error < 0.002 for Zsym < 0.75  fast recursion (there is a better version in onOEI-1.0.5.py):    initialize: ZPlato ≈ 1 - arcsin(exp(Rsym(0.06/exp(Rsym)-0.67)))*2/π    repeat:       Zlast = ZPlato       ZPlato = tanh(Rsym/(ZPlato+Zlast))       until  2*ZPlato*artanh(ZPlato) - Rsym  is small enough.  format for comparison to the "Pareto Principle":    a : b = (ZPlato+1)/2 : (ZPlato-1)/2    ZPlato = |2a-1| = |2b-1|  Inequality  Issuization  RA  = (RTheil+RliehT)/2 - ZHoover = Rsym - ZHoover     = Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal) - |Ei/Etotal-Ai/Atotal|)/2

Any city however small, is divided at least into two,
one the city of the poor, the other of the rich;
these are hostile to each other.
 (Plato, Politeia, 370 BC)