if the variables do not have different scales and means, should standardization be done based on Z scores or by centering the scale? In a case where this might be appropriate, is it necessary to also transform the values to absolute values?
The standardization, normalization or linear transformations or transformation to absolute values etc., are data preprocessing steps as most of the researchers do. So in case comparing your results to other it is good to be of the same pattern to others.
This is a little bit ambiguous: what exactly do you mean by "do not have different scales and means". This would simply mean several things:
1. Similar mean: It becomes just a matter of scaling. In this case z-score works, but not necessarily only z-score, as other scalings would also work. But to be in accordance with what is done regularly, z-score is preferred.
2. Similar scale: It becomes a matter of translation. You can subtract mean, but like the case above not necessarily mean. Again, z-score is preferred.
3. Similar mean and scale. This is a bit problematic and may mean the distribution of different variables is the same (in extreme case the variables are the same and the correlated values (corr=1) themselves are source of problem).
Regarding the second part of question, necessity is not enforced. It all depends on your data. Sometimes it helps, sometimes it makes no difference, other times it deteriorates (yes, it does in some cases).
Dear Seyed Mehdi Ahmadi
These links might be useful, have a look:
1. https://www.listendata.com/2017/04/how-to-standardize-variable-in-regression.html
2. https://statisticsbyjim.com/regression/standardize-variables-regression/
3. https://stats.stackexchange.com/questions/29781/when-conducting-multiple-regression-when-should-you-center-your-predictor-varia
Kind Regards
Qamar Ul Islam