The MLD of household income has been defined as[1]
where N is the number of households, is the income of household i, and is the mean of . Naturally the same formula can be used for positive variables other than income and for units of observation other than households.
Equivalent definitions are
where is the mean of ln(x). The last definition shows that MLD is nonnegative, since by Jensen's inequality.
MLD has been called "the standard deviation of ln(x)",[1] (SDL) but this is not correct. The SDL is
and this is not equal to the MLD.
In particular, if a random variable follows a log-normal distribution with mean and standard deviation of being and , respectively, then
Thus, asymptotically, MLD converges to:
For the standard log-normal, SDL converges to 1 while MLD converges to 1/2.