Is there a word for making an assumption of something larger from a small external part (possibly erroneously)?
judging a person's entire moral character by one of his crimes.
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
When you extrapolate, you use specific details to make a general conclusion. For example, if you travel to Canada and encounter only friendly, kind natives, you might extrapolate that all Canadians are friendly.
As for the part about "possibly erroneously," there's the term hasty generalization. From Wikipedia:
Hasty generalization is an informal fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence—essentially making a hasty conclusion without considering all of the variables.
There's also the idiom jumping to conclusions.
This is called judging a book by its cover: judging the whole by some prominently visible part of the whole.
See Don't judge a book by its cover (Wikipedia).
It is some times called judging (yes, the same word as used in your example!).
5 : to form an estimate or evaluation of : trying to judge the amount of time required;
especially : to form a negative opinion about : shouldn't judge him because of his accent