[EM] Simulations with social welfare functions
David Cary
dcarysysb at yahoo.com
Thu May 25 23:45:21 PDT 2006
--- bql at bolson.org wrote:
> To answer my own question, I think the attached perl script nicely
> shows
> the difference between std-dev and gini by this output:
The Gini Coefficient is invariant under scaling, but not under
translation. Standard deviation is invariant under translation, but
not under scaling. If you want a better comparison between the two,
you might try comparing Gini to stdev/mean. Also, Gini, like
standard deviation, can be calculated for a population or a sample.
The Perl code is inconsistently using the formulas for the Gini of a
population, but the standard deviation of a sample.
The Wikipedia article is confusing to the point of being erroneous.
To calculate the Gini coefficient for a population, use the Brown
formula with:
X_k = k / n
Y_k = S_k / S_n
where:
S_k = sum of w_i for i = 1 to k
(w_i) i = 1 to n, is the sequence of values (e.g. income) for each
member of the population, sorted in increasing order.
With this setup, the Brown formula can be restated as:
T = sum of w_i * (n+1-i) for i = 1 to n
G = 1 - 2*(T/S_n - 0.5) / n
This is algebraically equivalent to the formula given earlier on this
list.
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