In academic papers, is it always necessary to show the full mathematical description of well-accepted norms?
It depends entirely on the specific journal you publish in, as well as the topic of your paper. The paper is meant to be read by other experts in your field: hence if something is well known, it doesn't need to be defined. For the average reader of your paper, will they already know what z-normalization is? If so, you don't need to define it.