# Mean log deviation

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{{Short description|Measure of income inequality}}
In [statistics](/source/statistics) and [econometrics](/source/econometrics), the '''mean log deviation (MLD)''' is a measure of [income inequality](/source/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&nbsp;ln(''x''). The last definition shows that MLD is nonnegative, since <math>\ln{\overline{x}} \geq \overline{\ln x}</math> by [Jensen's inequality](/source/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](/source/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](/source/generalized_entropy_index). Specifically, the MLD is the generalized entropy index with α=0.

== References ==

{{Reflist}}

== External links ==
* [US Census Bureau](/source/United_States_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

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