Advertisement

METRON

, Volume 77, Issue 1, pp 43–52 | Cite as

The Gini mean difference and variance

  • Roberta La HayeEmail author
  • Petr Zizler
Article
  • 14 Downloads

Abstract

A quick, alternate proof is given for a previously known inequality relating the standard deviation and the Gini mean difference. The inequality is sharpened and generalized to higher, even moments. Further inequalities are derived that involve the standard deviation, higher Ginis and order statistics.

Keywords

Gini mean difference Standard deviation Variance Gini index 

Notes

References

  1. 1.
    Farris, F.: The Gini index and measures of inequality. Am. Math. Mon. 12, 851–864 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gastwirth, J.: A general definition of the Lorenz curve. Econometrica 39, 1037–1039 (1971)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gini, C.: Variabilita* e mutabilita*: contributo allo studio delle distribuzioni e delle relazioni statistiche, in Studi Economico-giuridici della Regia Facolta* Giusirsprudenza, anno III, parte II. Cuppini, Bologna (1912)Google Scholar
  4. 4.
    Glasser, G.: Variance formulas for the mean difference and coefficient of concentration. J. Am. Stat. Assoc. 57, 648–654 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kakwani, N.: On a class of poverty measures. Econometrica 48, 437–446 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kendall, M., Stuart, A., Ord, J.: Kendall’s Advanced Theory of Statistics, vol. 1, 5th edn, pp. 39–71. Oxford University Press, New York (1987)zbMATHGoogle Scholar
  7. 7.
    Piesch, W.: A look at the structure of some extended Ginis. Metron LXIII, 263–296 (2005)MathSciNetGoogle Scholar
  8. 8.
    Yitzhaki, S.: Gini’s mean difference: a superior measure of variability for non-normal distributions. Metron LXI, 285–316 (2003).  https://doi.org/10.2139/ssrn.301740 MathSciNetGoogle Scholar
  9. 9.
    Yitzhaki, S.: Gini’s mean difference offers a response to Leamer’s critique. Metron LXXIII, 31–43 (2015).  https://doi.org/10.1007/s40300-014-0057-9 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yitzhaki, S., Schechtman, E.: The Gini Methodology: a Primer on a Statistical Methodology. Series in Statistics, pp. 11–31. Springer, New York (2017)zbMATHGoogle Scholar
  11. 11.
    Zizler, P.: Gini indices and the moments of the share density function. Appl. Math. 59, 167–175 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sapienza Università di Roma 2019

Authors and Affiliations

  1. 1.Mount Royal UniversityCalgaryCanada

Personalised recommendations