Hello Hyunjin,

It depends on the aims of the study. Standardizing scores on a set of three variables (regardless of whether they are the same measure/scale to begin with) guarantees that all three will be equal in average value and variance after. So, tests of equality of means across variables will be meaningless. Likewise, tests of variance equality across variables will also be meaningless.

However, correlations of scores on pairs of variables or correlations of scores on variables with other variables (beyond x, y, z) will be unaffected, because standardizing is merely the application of a linear transformation.

In some analyses, such as cluster analysis, standardizing variables is a good idea, since any distance-based measure of similarity/dissimilarity/proximity will be more heavily influenced by the variable(s) having larger variance than variables having smaller variance. As well, correlation coefficients (such as Pearson) represent standardized measures of linear association, whereas covariances represent unstandardized measures of linear association.

Sorry to be so general, but it depends on what you're trying to accomplish.

Good luck with your work.

Hyunjin Cho , complementing David Morse 's answer: standardization -even if the variables are in same unit- is extremely useful -even required- when one is estimating neural network models, as in this case standardization will greatly help/improve the convergence of the gradient algorithm. I personally try this with back-propagation.

Here's what Jim says: https://statisticsbyjim.com/regression/standardize-variables-regression

"Centering the variables and standardizing them will both reduce the multicollinearity. However, standardizing changes the interpretation of the coefficients. "

Btw, if you use numerical methods to fit your models, it is also useful to standardize. Many optimizers have this procedure implemented.