I have the mean (u) and the standard deviation (sd) of a continuous distribution (X). How do I solve for the mean (u_log) and standard deviation (sd_log) of the log of that continuous distribution (log(X))?

I am looking to draw random numbers from a log-normal distribution. The problem is, most statistical packages have a log-normal-random-number-generator function that takes the mean of log(X) and sd of log(X) as inputs.

I only have the mean of X and std of X.

How to I solve for mean(log(X)) and sd(log(X)) given those inputs.

  • 2
    $\begingroup$ see below for answer: en.wikipedia.org/wiki/Log-normal_distribution In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s.d., and v in this article. The two sets of parameters can be related as (see also #Arithmetic moments below):[3] $\endgroup$
    – patricio
    Commented Apr 28, 2014 at 13:21
  • 1
    $\begingroup$ This reads like routine bookwork. Is this for some subject? $\endgroup$
    – Glen_b
    Commented Jun 2, 2014 at 23:41

1 Answer 1


Let m and s be the mean and sd of $X$ on the original scale. The appropriate mean and sd on the log scale can be found after a little algebra to be

$E(\log(X)) = \log(m) - \frac{1}{2} \log [ (s/m)^2 +1]$

$sd(\log(X)) = \sqrt{\log [(s/m)^2 +1]}$


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