  Direct measurements of Gini coefficients by conventional arithmetic
calculations are a poor estimator, even if paradoxically, they include the
entire population, as because of super-additivity they cannot lend themselves
to comparisons between units of different size, and intertemporal analyses are
vitiated by the population changes. The Gini of aggregated units A and B will
be higher than those of A and B computed separately. This effect becomes more
acute with fatness of tails. When the sample size is smaller than entire
population, the error is extremely high. The conventional literature on Gini
coefficients cannot be trusted and comparing countries of different sizes makes
no sense; nor does it make sense to make claims of "changes in inequality"
based on conventional measures. We compare the standard methodologies to the
indirect methods via maximum likelihood estimation of tail exponent. We compare
to the tail method which is unbiased, with considerably lower error rate. We
also consider measurement errors of the tail exponent and suggest a simple but
efficient methodology to calculate Gini coefficients.
