Mean log deviation

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 as¹

${\displaystyle \mathrm {MLD} ={\frac {1}{N}}\sum _{i=1}^{N}\ln {\frac {\overline {x}}{x_{i}}}}$

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

Equivalent definitions are

${\displaystyle \mathrm {MLD} ={\frac {1}{N}}\sum _{i=1}^{N}(\ln {\overline {x}}-\ln x_{i})=\ln {\overline {x}}-{\overline {\ln x}}}$

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

MLD has been called "the standard deviation of ln(x)",¹ (SDL) but this is not correct. The SDL is

${\displaystyle \mathrm {SDL} ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(\ln x_{i}-{\overline {\ln x}})^{2}}}}$

and this is not equal to the MLD.

In particular, if a random variable ${\displaystyle X}$ follows a log-normal distribution with mean and standard deviation of ${\displaystyle \log(X)}$ being ${\displaystyle \mu }$ and ${\displaystyle \sigma }$, respectively, then

${\displaystyle EX=\exp\{\mu +\sigma ^{2}/2\}.}$

Thus, asymptotically, MLD converges to:

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

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

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