# Does centering data destroy information if the data are not gaussian distributed?

I've alway wondered if you are given tabular data where things have different distributions (some features are normally distributed, some are log-normally distributed) if centering the data by...

$$\frac{x - \mu}{\sigma}$$

Do we destroy the information about the variables which do not have a normal distribution?

If your level of measurement is an interval scale, you can do positive linear transformations ($$x' = ax+b$$ with $$a>0$$) without losing information. This does not depend on the distribution of your data.
Note that for centering, you can set $$a = \frac{1}{\sigma}$$ and $$b = \frac{\mu}{\sigma}$$. Since $$\sigma>0$$, this will be a positive linear transformation.