Lately, there have been numerous questions about normalization
What are some of the situations where you never ever ever should normalize your data, and what are the alternatives?
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Whether one can normalize a non-normal data set depends on the application. For example, data normalization is required for many statistical tests (i.e. calculating a z-score, t-score, etc.) Some tests are more prone to failure when normalizing non-normal data, while some are more resistant ("robust" tests).
One less-robust statistic is the mean, which is sensitive to outliers (i.e. non-normal data). Alternatively, the median is less sensitive to outliers (and therefore more robust).
A great example of non-normal data when many statistics fail is bi-modally distributed data. Because of this, it's always good practice to visualize your data as a frequency distribution (or even better, test for normality!)
Of course one should never try to blindly normalize data if the data does not follow a (single) normal distribution.
For example one might want to rescale observables $X$ to all be normal with $(X-\mu)/\sigma$, but this can only work if the data is normal and if both $\mu$ and $\sigma$ are the same for all data points (e.g. $\sigma$ doesn't depend on $\mu$ in a particular $X$ range).
I thought this was too obvious, until I saw this question!
When you normalise data, make sure you always have access to the raw data after normalisation. Of course, you could break this rule if you have a good reason, e.g. storage.