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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?

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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.

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  • $\begingroup$ Thanks for your answer, what qualifies as an "interval scale?" $\endgroup$
    – Joff
    Apr 27, 2020 at 12:04
  • $\begingroup$ When you can interpret the difference between values in a meaningful way. If you can compute mean and standard deviation, you should typically have an interval variable. You can find more information on levels of measurement for instance here $\endgroup$
    – mesolimbic
    Apr 27, 2020 at 12:56

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