I'm trying to understand z scores and how to use them. As far as I understand, I can use the z transformation to be able to compare different variables with different value range, that were not comparable beforehand; i.e. to get comparable values across different variables, I can use the z transformation.
Now, the data is not normally distributed, so I understand I cannot use the normal z table to check for percentages.
However, as far as I understand, what I could do is to calculate the percentiles myself instead.
There is one thing more that I do not understand. To compare z-scores, someone transformed the data to a new scale (between 1 and 100), so that all variables are on the same scale. The reason being that z scores from variables with a high span result in a high span of the z scores as well, which makes them not-comparable. Is this even true? I don't understand why we would do that (as the values are already comparable?).
The questions I have are:
I can use the z transformation to standardize my variables, so that I can compare different variables. Is this correct?
I can calculate the percentiles myself and the values I get are still comparable across different variables. Is this correct?
Can I transform my z-scores to a new scale, so that all variables use the same scale and what would I expect the median and/or average to be? Important here is that my minimum has to be 1, maximum has to be 100.