# What happens to the data distribution and results if we calculate z-score of a z-scored data?

The data that I am using is already z-scored and batch normalized. I accidentally calculated the z-score again and then performed further analysis and calculated results. Does it make sense to take z-score of a z-scored data distribution? Does it affect the distribution and the results? Thank you

Pretty much nothing happens. The mean of your z-score data is $$0$$, and the standard deviation is $$1$$. Thus, when you run your points through the formula, you subtract $$0$$ and then divide by $$1$$.

$$z=\dfrac{x-\bar x}{s}$$

Because you're likely doing this on a computer, it is possible that your $$\bar x = 0$$ and $$s=1$$ are stored as something like $$0.0000000710008$$ and $$1.00000004000502203$$. I would ignore that in every situation that comes to mind. Because you did something unusual, however, my recommendation is to redo your work with only the one z-score calculated, knowing that almost nothing is going to change. (You are setting yourself up for criticism that you can avoid.)

A simulation in R shows all $$1000$$ second z-scores to be within about $$10^{-16} = 0.0000000000000001$$ of the first z-scores. Since I simulated an exponential distribution, this fact does not depend on the data being normal.

set.seed(2021)
N <- 1000
z_score <- function(x){
return((x - mean(x))/sd(x))
}
x <- rexp(N)
z <- z_score(x) # Calculate the z-scores
z2 <- z_score(z) # Calculate the z-scores of the z-scores.
summary(z - z2)
Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
0.000e+00 9.368e-17 1.110e-16 9.113e-17 1.110e-16 1.110e-16