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How do I convert the $\text{MAD}$ (median absolute deviation from the median) of data that is drawn from a log-normal distribution to the standard deviation of a log-normal distribution? To clarify, if I calculate the $\text{MAD}$ of a sample that I assume follows a log-normal distribution ($\text{Lognormal}(\mu, \sigma^2)$), how do I calculate $\sigma$?

I know that such relationship exists for symmetrical distributions. E.g., for normal distribution that would be:

$\sigma=\Phi^{-1}(3/4)\cdot \text{MAD}\approx1.4826\cdot\text{MAD},$

where $\Phi()$ is the cumulative distribution function for the standard normal distribution.

Any help would be appreciated!

rp1
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  • Because medians are defined in terms of order only, they will be preserved under any order-preserving transformation of the variable. So: take logarithms and apply what you know about the Normal distribution, then back-transform. – whuber Oct 15 '19 at 13:38
  • Would you please be able to walk me through the steps? Once I take the logarithms I can calculate the standard deviation of the normally distributed data, but how do I convert it back to the sd of the lognormal distribution? – rp1 Oct 22 '19 at 16:10
  • See https://stats.stackexchange.com/questions/173715/calculate-variance-and-standard-deviation-for-log-normal-distribution. For more, search our site: https://stats.stackexchange.com/search?q=lognormal+standard+deviation+formula. – whuber Oct 22 '19 at 16:28

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