In statistics and econometrics, the mean log deviation (MLD) is a measure of income inequality. The MLD is zero when everyone has the same income, and takes larger positive values as incomes become more unequal, especially at the high end.
The MLD of household income has been defined as
where N is the number of households, is the income of household i, and is the mean of . Naturally the same formula can be used for positive variables other than income and for units of observation other than households.
Equivalent definitions are
where is the mean of ln(x). The last definition shows that MLD is nonnegative, since by Jensen's inequality.
MLD has been called "the standard deviation of ln(x)", (SDL) but this is not correct. The SDL is
and this is not equal to the MLD. For example, for the standard lognormal distribution, MLD = 1/2 but SDL = 1.
The MLD is a special case of the generalized entropy index. Specifically, the MLD is the generalized entropy index with α=0.
- Jonathan Haughton and Shahidur R. Khandker. 2009. The Handbook on Poverty and Inequality. Washington, DC: The World Bank.