Mean log deviation explained

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.

Definition

The MLD of household income has been defined as^[1]

MLD = (1)/(N) \sum ^{N} _{i =1} ln (\overline{x)/() }

where N is the number of households,

x _{i} is the income of household i, and \overline{x} is the mean of x _{i} . Naturally the same formula can be used for positive variables other than income and for units of observation other than households.

Equivalent definitions are

MLD = (1)/(N) \sum ^{N} _{i =1} ( ln \overline{x} - ln x _{i) =ln} \overline{x} - \overline{ ln x}

where

\overline{ ln x} is the mean of ln(x). The last definition shows that MLD is nonnegative, since ln {\overline{x}} \geq \overline{ ln x} by Jensen's inequality.

MLD has been called "the standard deviation of ln(x)", (SDL) but this is not correct. The SDL is

SDL = \sqrt{ (1)/(N) \sum ^{N} _{i =1} ( ln x _{i} - \overline{ ln x}) ^{2}}

and this is not equal to the MLD.

In particular, if a random variable

X follows a log-normal distribution with mean and standard deviation of log (X) being \mu and \sigma , respectively, then EX = \exp\{\mu + \sigma ^{2/2\}.}

Thus, asymptotically, MLD converges to:

ln \{\exp[\mu + \sigma ^{2/2]\}} - \mu = \sigma ^{2/2}

For the standard log-normal, SDL converges to 1 while MLD converges to 1/2.

Related statistics

The MLD is a special case of the generalized entropy index. Specifically, the MLD is the generalized entropy index with α=0.

External links

• US Census Bureau: Mean Log Deviation (MLD)

Notes and References

1. Jonathan Haughton and Shahidur R. Khandker. 2009. The Handbook on Poverty and Inequality. Washington, DC: The World Bank.