{{Short description|Measure of income inequality}} 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<ref name="Haughter">Jonathan Haughton and Shahidur R. Khandker. 2009. ''The Handbook on Poverty and Inequality''. Washington, DC: The World Bank.</ref> : <math> \mathrm{MLD}=\frac{1}{N}\sum_{i=1}^N \ln \frac{\overline{x}}{x_i} </math>
where N is the number of households, <math>x_i</math> is the income of household ''i'', and <math>\overline{x}</math> is the mean of <math>x_i</math>. Naturally the same formula can be used for positive variables other than income and for units of observation other than households.
Equivalent definitions are : <math> \mathrm{MLD}=\frac{1}{N}\sum_{i=1}^N (\ln \overline{x} - \ln x_i) =\ln \overline{x} - \overline{\ln x} </math>
where <math>\overline{\ln x}</math> is the mean of ln(''x''). The last definition shows that MLD is nonnegative, since <math>\ln{\overline{x}} \geq \overline{\ln x}</math> by Jensen's inequality.
MLD has been called "the standard deviation of ln(''x'')",<ref name="Haughter" /> (SDL) but this is not correct. The SDL is
: <math> \mathrm{SDL} =\sqrt{\frac{1}{N}\sum_{i=1}^N (\ln x_i - \overline{\ln x})^2} </math>
and this is not equal to the MLD.
In particular, if a random variable <math>X</math> follows a log-normal distribution with mean and standard deviation of <math>\log(X)</math> being <math>\mu</math> and <math>\sigma</math>, respectively, then
:<math> EX = \exp\{\mu + \sigma^2/2\}.</math>
Thus, asymptotically, MLD converges to:
:<math> \ln\{\exp[\mu + \sigma^2/2]\} - \mu = \sigma^2/2</math>
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.
== References ==
{{Reflist}}
== External links == * US Census Bureau: ''[https://www.census.gov/topics/income-poverty/income-inequality/about/metrics/mld.html Mean Log Deviation (MLD)]''
Category:Descriptive statistics Category:Income inequality metrics