Do we do data normalization just in case our initial data has gaussian distribution?

Question

We always do Data Normalization of our structered data when we have different ranges, and I found that normalization is just a translation followed by a multiplicative scaling, so it does not really change your distribution : a translated and scaled gaussian is still a gaussian ... But this is just in case our initial data is also gaussian, does it mean that if no gaussian data then we shouldn't do normalization? else we may lose our data variance because we change the whole distribution! Please if someone can clarify that foor me ...