Min-Max scaling on Z-score standardized data?

Question

For a specific task of score fusion I need to test my data on some different normalization techniques like typical Z-normalization or Sigmoid-normalization. This is my first step to do.

For a second step I need to have comparable data in the same data range like from 0 to 1.

I would ask myself right now, if the following is a possible (and probably recommended) process or is there some mistake. If so, is there any other solution?
My current process:

1. Do normalization technique on data -> the result is probably a different data range
2. To have comparable data -> do min-max-scaling on normalized data

Answer 1

TL;DR It is pointless to use both transformations.

Say that $X$ is your data. What you are trying to do is

$$ z = \frac{x-\mathrm{mean}(X)}{\mathrm{sd}(X)}, \qquad y = \frac{z-\min(Z)}{\max(Z)-\min(Z)} $$

Let us use thew $m,s,l,u$ symbols for the sample mean, standard deviation, minimum and maximum respectively. Notice that after $z$-transforming also the minimum and maximum get $z$-transformed, so $\min(Z) = \frac{l - m}{s}$ etc. Now, if we combine both equations, we have

$$ \require{cancel} \frac{\frac{x - m}{s} - \frac{l-m}{s}}{\frac{u-m}{s} - \frac{l-m}{s}} = \frac{\frac{x - \cancel{m} - (l - \cancel{m})}{s}}{\frac{u -\cancel{m}-(l-\cancel{m})}{s}} = \frac{\frac{x - l}{s}}{\frac{u-l}{s}} = \frac{x - l}{u-l} $$

So basically, using $z$-transformation and then min-max scaling, leads to the same result as min-max scaling alone. Same can be shown about using min-max scaling and then $z$-transformation, as it gives same result as $z$-transformation alone.

See also Transform data to have specific mean, minimum and maximum?.