**Comment** on lognormal and normal distributions.

The variance of data tends to decrease when you take logs of the values.
Perhaps the most common case is that, if $X_1$ is lognormal, then $X_2 = \ln(X_1)$ is normal, which has smaller variance than $X_1.$ Also, $X_3 = \ln(X_2)$ has a still smaller variance. 

Below we begin with a random sample (using R) of $n = 10^4$ observations
from a lognormal distribution with parameters $\mu = 50, \sigma = 2.$ (It is customary to use lognormal parameters that match parameters of the related normal distribution. You can look at [Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution) on 'lognormal distributions' for details.) We show means and standard deviations for distributions of $X_1, X_2,$ and $X_3.$

    set.seed(720); n = 10^4
    x2 = rnorm(n, 50, 2);  x1 = exp(x2);  x3 = log(x2)
    mean(x1); mean(x2); mean(x3)
    [1] 3.686093e+22
    [1] 50.01289         # aprx E(X2) = 50
    [1] 3.911481
    sd(x1); sd(x2); sd(x3)
    [1] 2.308712e+23
    [1] 1.997261         # aprx SD(X2) = 2
    [1] 0.04004551

Then we show histograms of the three samples. At the left, notice that it
is difficult to make an informative histogram of $X_1$ because it is so
severely skewed to the right. In the center panel, we overlay the density
function of $\mathsf{Norm}(\mu =50, \sigma=2);$ which is symmetrical.
At the right, notice that taking (natural) logs once again results in a
slightly left-skewed distribution.

[![enter image description here][1]][1]

_Notes:_ (1) The the support of a lognormal distribution is $(0, \infty).$
A normal distribution may take negative values.

(2) R code for the figure:

    par(mfrow=c(1,3))
    hist(x1, prob=T, br=50, col="skyblue2")
    hist(x2, prob=T, col="skyblue2")
      curve(dnorm(x,50,2), add=T, col="red")
    hist(x3, prob=T, col="skyblue2")
    par(mfrow=c(1,1)) 


  [1]: https://i.sstatic.net/ICW5Y.png