mean log deviation

The MLD of household income has been defined asJonathan Haughton and Shahidur R. Khandker. 2009. The Handbook on Poverty and Inequality. Washington, DC: The World Bank.

: $$

\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 $\backslash overline\{x\}$ 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 $\backslash overline\{\backslash ln\; x\}$ is the mean of ln(x). The last definition shows that MLD is nonnegative, since $\backslash ln\{\backslash overline\{x\}\}\; \backslash geq\; \backslash overline\{\backslash ln\; x\}$ 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 $\backslash log(X)$ being $\backslash mu$ and $\backslash sigma$, respectively, then

:$EX\; =\; \backslash exp\backslash \{\backslash mu\; +\; \backslash sigma^2/2\backslash \}.$

Thus, asymptotically, MLD converges to:

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

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.

{{Reflist}}

• 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