How to log transform Z-scores? The data for my variable is in the form of Z-scores only. I'd like to log transform the scores, but I don't know the mean or standard deviation in order to covert to raw scores. Can I assign an arbitrary mean and standard deviation, use that to convert to raw scores, then log transform the raw scores? Or is there some other way to log transform Z-scores? Thanks.
 A: A few quick points about logs
The following R code is a reminder that the log of a negative number is not a number and that the log of zero is negative infinity. Thus, if you are going to take a log of a z-score, you first need to make all values obtained greater than zero.
> values <- c(-2, -1, 0, .001, .1, 1, 10)
> data.frame(values=values, logvalues=log(values))
  values logvalues
1 -2.000       NaN
2 -1.000       NaN
3  0.000      -Inf
4  0.001 -6.907755
5  0.100 -2.302585
6  1.000  0.000000
7 10.000  2.302585
Warning message:
In log(values) : NaNs produced

A simple strategy of logs on z-scores
A simple strategy for log transforming a variable is to first add a constant to the variable such that the minimum value is one. i.e., 1 + x - min(x).
The following code shows a simple example of some standardised positively skewed data. The minimum of 1 + x - min(x) is 1. Thus, the variable can be log transformed.
The plot then shows the density before and after transformation.
> set.seed(4444)
> # some skewed raw data
> x <- scale((rnorm(1000) + 3)^2)
> 
> xnew <- 1 + x - min(x)
> min(xnew)
[1] 1
> min(x)
[1] -1.584252
> xnew <- log(xnew)
> 
> par(mfrow=c(2,1))
> plot(density(x))
> plot(density(xnew))


But exactly what transformation should you perform?


*

*There is a general issue of whether a log transformation is appropriate to your data, and if so, what constant you should add to your raw data.

*Presumably if you already have z-scores, then you don't care too much about the absolute metric.

*You'll find further discussion of this issue on this question
A: You cannot assign arbitrary Mean and SD to covert z-score data into Raw data (x). However, you can check a shape of the distribution of z-scores by calculating skewness or kurtosis. Log-transform only useful if you're data is positively skewed. Moreover, it would be good if you explain that what is your objective? as @Karl asked. It might be helpful to visit this URL.
A: I understood that you wanted to log-transform your data so that it looked more "normal" (that is, more symmetric). But if that is the goal, why don't you apply a transform to your data, that makes it exactly standard normal?
Suppose you have a variable $x$, and you estimated its CDF as $\hat{F}(x)$. Then you can apply the transformation $y=\Phi^{-1}(F(x))$, where $\Phi()$ is standard normal CDF. By definition, $y$ will be standard normal.
Different algorithms, like scikit-learn quantile transformation, will do this for you. 
