# Convert median absolute deviation (MAD) to SD for log-normal distribution

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!

• 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
Commented Oct 15, 2019 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
Commented Oct 22, 2019 at 16:10
• See stats.stackexchange.com/questions/173715/…. For more, search our site: stats.stackexchange.com/….
– whuber
Commented Oct 22, 2019 at 16:28