nonidentically

Nonidentically is an adverb formed with the prefix non- added to identically, and it means not in an identical fashion or not in a way that is identical. It is used to indicate that two or more things do not share the same characteristics, distribution, or form. In everyday language, speakers may prefer the simpler constructions “not identically” or “not identically distributed.”

In statistics and probability, nonidentically is most often encountered in phrases like nonidentically distributed or not identically distributed. This describes random variables that do not follow the same probability distribution. When variables are independent but not identically distributed, they are said to be independent and not identically distributed (often abbreviated as i.n.i.d.). Such distinctions matter in theoretical results and practice, as many theorems assume identical distributions (the i.i.d. assumption), while others extend to nonidentically distributed cases under additional conditions.

Outside mathematics, nonidentically is used less frequently and can sound unusual in plain prose. Editors and writers may opt for clearer alternatives such as not identically or not identical when describing similarities or forms that are not the same. The term remains a precise, if somewhat technical, modifier in formal contexts where exactness about distributional sameness is required.

See also identically distributed, nonidentical, independent and identically distributed, and the broader concept of distribution in probability theory.