Mean log deviation

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

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

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. 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.

• US Census Bureau: Mean Log Deviation (MLD)