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.

Definition
The MLD of household income has been defined as

\mathrm{MLD}=\frac{1}{N}\sum_{i=1}^N \ln \frac{\overline{x}}{x_i} $$

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

\mathrm{MLD}=\frac{1}{N}\sum_{i=1}^N (\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



\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 $$X$$ 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.